%Paper: ewp-game/9403005
%From: <atlewis@students.wisc.edu>
%Date: Thu, 24 Mar 1994 13:52:00 -0600 (CST)
%Date (revised): Tue, 29 Mar 1994 10:14:25 -0600 (CST)

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%%%%%%%%%%%%%%larrybib.tex%%%%%%%%%%%%%%%%%%%%%%%%%%%

@TECHREPORT{Anderlini1990,
     AUTHOR = "L. Anderlini",
     TITLE = "Communication, Computability, and Common Interest
          Games",
     INSTITUTION = "St. John's College",
     ADDRESS = "Cambridge",
     YEAR = 1990,
     TYPE = "Working Paper",                       }

@BOOK{AndersondePalmaandThisse1992,
     AUTHOR = "S. Anderson and J. de Palma and J. Thisse",
     TITLE = "A Discrete Choice Theory of Product Differentiation",
     PUBLISHER = "MIT Press",
     YEAR = 1992,
     ADDRESS = "Cambridge, Massachusetts",           }

@ARTICLE{AndreoniandMiller1991, 
     AUTHOR = "John H. Miller and James Andreoni", 
     TITLE = "Can Evolutionary Dynamics Explain Free Riding in 
          Experiments?", 
     JOURNAL = "Economics Letters", 
     YEAR = 1991, 
     VOLUME = 36, 
     PAGES = "9-15",                    } 

@TECHREPORT{AndreoniandMiller1993,
     AUTHOR = "James Andreoni and John H. Miller",
     TITLE = "Auctions with Artificial Adaptive Agents",
     INSTITUTION = "University of Wisconsin and Carnegie Mellon University",
     ADDRESS = "Madison, Wisconsin and Pittsburg, Pennsylvania",
     YEAR = 1993,
     TYPE = "Mimeo",                       }


@INCOLLECTION{AraujoandScheinkman1979,
     AUTHOR = "A. Araujo and J. Scheinkman",
     TITLE = "Notes on Comparative Dynamics",
     BOOKTITLE = "General Equilibrium, Growth and Trade:  Essays in
          Honor of Lionel W. McKenzie",
     PUBLISHER = "Academic Press",
     YEAR = 1979,
     EDITOR = "J. Green and J. Scheinkman",
     ADDRESS = "New York",                    }

@ARTICLE{AumannandSorin1989,
     AUTHOR = "Robert Aumann and Sylvain Sorin",
     TITLE = "Cooperation and Bounded Recall",
     JOURNAL = "Games and Economic Behavior",
     YEAR = 1989,
     VOLUME = 1,
     PAGES = "5-39",                    }

@TECHREPORT{BandB89,
     AUTHOR = "Mark Bagnoli and Ted Bergstrom",
     TITLE = "Log-Concave Probability and Its Applications",
     INSTITUTION = "University of Michigan",
     YEAR = 1989,
     TYPE = "Mimeo",             }


@TECHREPORT{Balkenborg1992,
     AUTHOR = "Dieter Balkenborg",
     TITLE = "Repeated Games with Common Interests and the Notion
          of a Strict Outcome Path",
     INSTITUTION = "University of Bonn",
     YEAR = 1992,
     TYPE = "Mimeo",             }

@TECHREPORT{BandW93,
     AUTHOR = "Abhijit Banerjee and {J\"{o}rgen} W. Weibull",
     TITLE = "Evolutionary Selection with Discriminating Players",
     INSTITUTION = "University of Stockholm",
     YEAR = 1993,
     TYPE = "Research Paper in Economics",             }

@ARTICLE{BanksandSobel1987,
     AUTHOR = "J. Banks and J. Sobel",
     TITLE = "Equilibrium Selection in Signaling Games",
     JOURNAL = "Econometrica",
     YEAR = 1987,
     VOLUME = 55,
     PAGES = "647--662",            }

@TECHREPORT{Barany1988,
     AUTHOR = "I. Barany",
     TITLE = "Fair Distribution Protocols, or how the Players Replace
Fortune",
     INSTITUTION = "University College London",
     YEAR = 1988,
     TYPE = "Mimeo",             }

@ARTICLE{Basu1990,
     AUTHOR = "Kaushik Basu",
     TITLE = "On the Non-Existence of a Rationality Definition for 
           Extensive Games",
     JOURNAL = "International Journal of Game Theory",
     YEAR = 1990,
     VOLUME = 19,
     PAGES = "33--44",            }

@TECHREPORT{BendorMookherjeeandRay1991,
     AUTHOR = "Jonathan Bendor and Dilip Mookherjee and Debraj Ray",
     TITLE = "Aspiration--Based Adaptive Learning in Two Person Repeated
Games",
     INSTITUTION = "Indian Statistical Institute",
     YEAR = 1991,
     TYPE = "Mimeo",             }

@BOOK{BerlekampConwayandGuy1982, 
     AUTHOR="E. Berlekamp and J. Conway and R. Guy", 
      TITLE="Winning Ways for your Mathematical Plays", 
  PUBLISHER="Academic Press", 
    ADDRESS="New York", 
       YEAR=1982,                          } 
 
@ARTICLE{Bernheim1984,
     AUTHOR = "Douglas Bernheim",
     TITLE = "Rationalizable Stratetic Behavior",
     JOURNAL = "Econometrica",
     YEAR = 1984,
     VOLUME = 52,
     PAGES = "1007-1028",               }

@Book{Billingsley1968,
           AUTHOR = "Patrick Billingsley",
           TITLE = "Convergence of Probability Measures",
           PUBLISHER = "John Wiley and Sons",
           YEAR = 1968,
           ADDRESS = "New York",          }

@Book{Billingsley1986,
           AUTHOR = "Patrick Billingsley",
           TITLE = "Probability and Measure",
           PUBLISHER = "John Wiley and Sons",
           YEAR = 1986,
           ADDRESS = "New York",          }

@ARTICLE{Binmore198788,
     AUTHOR = "Ken Binmore",
     TITLE = "Modelling Rational Players: I and {I}{I}",
     JOURNAL = "Economics and Philosophy",
     YEAR = "1987-88",
     VOLUME = "3 and 4",
     PAGES = "179-214 and 9-55",          }

@INCOLLECTION{Binmore1991,
     AUTHOR = "Ken Binmore",
     TITLE = "Foundations of Game Theory",
     BOOKTITLE = "Advances in Economic Theory:  Sixth World
          Congress of the Econometric Society",
     PUBLISHER = "Cambridge University Press",
     YEAR = 1991,
     EDITOR = "J. J. Laffont",
     ADDRESS = "Cambridge",   }

@Book{Binmore1992,
           AUTHOR = "Ken Binmore",
           TITLE = "Fun and Games",
           PUBLISHER = "D. C. Heath",
           YEAR = 1992,
           ADDRESS = "Lexington, MA.",          }

@Book{Binmore1993, 
           AUTHOR = "Ken Binmore", 
           TITLE = "Playing Fair", 
           PUBLISHER = "MIT Press", 
           YEAR = 1993, 
           ADDRESS = "Cambridge, MA.",          } 
 
@Book{Binmore1990, 
           AUTHOR = "Ken Binmore", 
           TITLE = "Essays on the Foundations of Game Theory", 
           PUBLISHER = "Basil Blackwell", 
           YEAR = 1990, 
           ADDRESS = "Cambridge, Massachusetts",          } 

@ARTICLE{BinmoreMorganShakedandSutton1991,
           AUTHOR = "Ken Binmore and Peter Morgan and Avner Shaked and John
                      Sutton",
           TITLE = "Do People Exploit Their Bargaining Power?  {A}n
Experimental Study",
           JOURNAL = "Games and Economic Behavior",
           YEAR = 1991,
           VOLUME = 3,
           PAGES = "295--322",                 }

@ARTICLE{BinmoreandSamuelson1992,
     AUTHOR = "Ken Binmore and Larry Samuelson",
     TITLE = "Evolutionary Stability in Repeated Games Played by
          Finite Automata",
     JOURNAL = "Journal of Economic Theory",
     YEAR = 1992,
      VOLUME = 57,
           PAGES = "278--305",            }

@TECHREPORT{BinmoreandSamuelson1993,
           AUTHOR = "Ken Binmore and Larry Samuelson",
           TITLE = "Muddling Through:  {N}oisy Equilibrium Selection",
           Institution = "University College London and University of
Wisconsin",
           YEAR = 1993,
           TYPE = "Mimeo",                  }

@TECHREPORT{BinmoreandSamuelson1993Saar, 
     AUTHOR = "Ken Binmore and Larry Samuelson", 
     TITLE = "An Economist's Perspective on the Evolution of 
          Norms", 
     INSTITUTION = "University College London and University of Wisconsin",
           YEAR = 1993,
           TYPE = "Mimeo",    }

@ARTICLE{BinmoreandSamuelson1993D,
     AUTHOR = "Ken Binmore and Larry Samuelson",
     TITLE = "Drift",
     JOURNAL = "European Economic Review",
     YEAR = 1993,
      NOTE = "Forthcoming",            } 
     
@TECHREPORT{BinmoreSamuelsonandVaughan1993, 
           AUTHOR = "Ken Binmore and Larry Samuelson and Richard Vaughan", 
           TITLE = "Musical Chairs:  Modelling Noisy Evolution", 
           Institution = "University College London and University 
               of Wisconsin", 
           YEAR = 1993, 
           TYPE = "Mimeo",                  } 
 
@TECHREPORT{BinmoreGaleandSamuelson1993, 
           AUTHOR = "Ken Binmore and John Gale and Larry Samuelson", 
           TITLE = "Learning to be Imperfect:  The Ultimatum Game", 
           Institution = "University College London and University 
               of Wisconsin", 
           YEAR = 1993, 
           TYPE = "Mimeo",                  } 

@ARTICLE{BinmoreShakedandSutton1985,
           AUTHOR = "Ken Binmore and Avner Shaked and John Sutton",
           TITLE = "Testing Noncooperative Bargaining Theory:  {A} Preliminary
Study",
           JOURNAL = "American Economic Review",
           YEAR = 1985,
           VOLUME = 75,
           PAGES = "1178--1180",                 }

@UNPUBLISHED{BSHandP92, 
    AUTHOR="Binmore, K. and J. Swierzsbinski and S. Hsu and C. 
Proulx", 
     TITLE="Focal Points and Bargaining", 
      NOTE="University of Michigan Discussion Paper", 
      YEAR="1992"} 
 
@ARTICLE{BlumeandEasley1993,
           AUTHOR = "Lawrence E. Blume and David Easley",
           TITLE = "Economic Natural Selection",
           JOURNAL = "Economics Letters",
           YEAR = 1993,
           NOTE = "Forthcoming",      }                           

@ARTICLE{BlumeKimandSobel1992,
     AUTHOR = "Andreas Blume and Yong-Gwan Kim and Joel Sobel",
     TITLE = "Evolutionary Stability in Games of Communication",
     JOURNAL = "Games and Economic Behavior",
     YEAR = 1993,
       VOLUME = 5,
           PAGES = "547--575",     }

@ARTICLE{BlumeBrandenburgerandDekel1991a,
     AUTHOR = "Lawrence Blume and Adam Brandenburger and Eddie
          Dekel",
     TITLE = "Lexicographic Probabilities and Choice Under
          Uncertainty",
     JOURNAL = "Econometrica",
     VOLUME = 59,
     YEAR = 1991,
     PAGES = "61--80",                         }

@ARTICLE{BlumeBrandenburgerandDekel1991b,
     AUTHOR = "Lawrence Blume and Adam Brandenburger and Eddie
          Dekel",
     TITLE = "Lexicographic Probabilities and Equilibrium
          Refinements",
     JOURNAL = "Econometrica",
     VOLUME = 59,
     YEAR = 1991,
     PAGES = "81--98",                          }

@ARTICLE{Bolton1991,
           AUTHOR = "Gary E. Bolton",
           TITLE = "A Comparative Model of Bargaining: {T}heory and Evidence",
           JOURNAL = "American Economic Review",
           YEAR = 1991,
           VOLUME = 81,
           PAGES = "1096--1136",                 }

@TECHREPORT{BoltonandZwick1993,
           AUTHOR = "Gary E. Bolton and Rami Zwick",
           TITLE = "Anonymity versus Punishment in Ultimatum Bargaining",
           INSTITUTION = "Penn State University",
           YEAR = 1993,
           TYPE = "Mimeo",     }

@TECHREPORT{BandB92,
     AUTHOR = "Immanuel Bomze and Reinhard {B\"{u}rger}",
     TITLE = "On the Balance Between Mutation and Frequency-Depedent Selection
in
                      Evolutinary Games",
     INSTITUTION = "University of Vienna",
     YEAR = 1992,
     TYPE = "Institute of Statistics and Computer Science Technical Report",
      NUMBER = 121,                }

@TECHREPORT{BomzeandEichberger1992,
     AUTHOR = "Immanuel Bomze and Jurgen Eichberger",
     TITLE = "Evolutive versus Naive {B}ayesian Learning",
     INSTITUTION = "University of Vienna",
     YEAR = 1992,
     TYPE = "Institute of Statistics and Computer Science Technical Report",
      NUMBER = 118,                }

@TECHREPORT{BomzeandvanDamme1990,
     AUTHOR = "Immanuel M. Bomze and Eric van Damme",
     TITLE = "A Dynamical Characterization of Evolutionarily Stable
          State",
     INSTITUTION = "Tilburg University",
     YEAR = 1990,
     TYPE = "CentER for Economic Research Discussion Paper",
     NUMBER = 9045,                  }

@TECHREPORT{BorgersandSamuelson1992,
     AUTHOR = "Tilman {B\"{o}rgers} and Larry Samuelson",
     TITLE = "``{C}autious'' Utility Maximization and Iterated Weak
          Dominance",
     INSTITUTION = "University College London and University of
          Wisconsin",
     YEAR = 1992,
     TYPE = "Mimeo",     }

@ARTICLE{BoydandRicherson1990,
     AUTHOR = "Robert Boyd and Peter J. Richerson",
     TITLE = "Group Selection among Alternative Evolutionarily
          Stable Strategies",
     JOURNAL = "Journal of Theoretical Biology",
     YEAR = 1990,
     VOLUME = 145,
     PAGES = "331-342",               }

@TECHREPORT{Boylan1991,
           AUTHOR = "Richard T. Boylan",
           TITLE = "Continuous Approximation of Dynamical Systems with
Randomly
                      Matched Individuals",
           INSTITUTION = "Washington University",
           YEAR = 1991,
           TYPE = "Mimeo",          }
           
@ARTICLE{Boylan1992,
           AUTHOR = "Richard T. Boylan",
           TITLE = "Laws of Large Numbers for Dynamical Systems with Randomly
                      Matched Individuals",
           JOURNAL = "Journal of Economic Theory",
           YEAR = 1991,
           VOLUME = 57,
           PAGES = "473--504",       }
                                            
                 
                                                       
                      
@ARTICLE{BrandenburgerandDekel1987,
     AUTHOR = "Adam Brandenburger and Eddie Dekel",
     TITLE = "Rationalizability and Correlated Equilibria",
     JOURNAL = "Econometrica",
     YEAR = 1987,
     VOLUME = 55,
     PAGES = "1391-1402",            }

@TECHREPORT{Brock1989,
     AUTHOR = "William A. Brock",
     TITLE = "A Possible Unification of Some Results on Learning
          Dynamics in Games",
     INSTITUTION = "University of Wisconsin",
     YEAR = 1989,
     TYPE = "Mimeo",         }

@TECHREPORT{BrockandSamuelson1992,
     AUTHOR = "William A. Brock and Larry Samuelson",
     TITLE = "Convergence of Learning Schemes on Supermodular
           Games",
     INSTITUTION = "University of Wisconsin",
     YEAR = 1992,
     TYPE = "SSRI Working Paper",
      NUMBER = 9219,         }

@INCOLLECTION{Brown1951,
     AUTHOR = "G. W. Brown",
     TITLE = "Iterative Solution of Games by Fictitious Play",
     BOOKTITLE = "Activity Analysis of Production and Allocation",
     PUBLISHER = "Wiley",
     YEAR = 1951,           }

@ARTICLE{BulowGeanakoplosandKlemperer1985,
     AUTHOR = "Jeremy I. Bulow and John D. Geanakoplos and Paul D.
          Klemperer",
     TITLE = "MultiMarket Oligopoly:  Strategic Substitutes and
          Complements",
     JOURNAL = "Journal of Political Economy",
     YEAR = 1985,
     VOLUME = 93,
     PAGES = "488--511",                 }

@TECHREPORT{Cabrales1993,
           AUTHOR = "Antonio Cabrales",
           TITLE = " Stochastic Replicator Dynamics",
           INSTITUTION = "University of California, San Diego",
           YEAR = 1993,
           TYPE = "Mimeo",       }

@ARTICLE{CabralesandSobel1992,
           AUTHOR = "Antonio Cabrales and Joel Sobel",
           TITLE = "On the Limit Points of Discrete Selection Dynamics",
           JOURNAL = "Journal of Economic Theory",
           VOLUME = 57,
           YEAR = 1992,
           PAGES = "407-419",                 }

@TECHREPORT{CFandM91,
           AUTHOR = "Charles Bram Cadsby and Murray Frank and Vojislav
Maksimovic",
           TITLE = "Equilibrium Dominance in Experimental Financial Markets",
           INSTITUTION = "University of Guelph and University of British
Columbia",
           YEAR = 1991,
           TYPE = "Mimeo",       }

@TECHREPORT{CanalsandRedondo1992,
           AUTHOR = "{Jos\`{e}} Canals and Fernando {Vega-Redondo}",
           TITLE = " Multi--Level Evolution in Games",
           INSTITUTION = "Universidad de Alicante",
           YEAR = 1992,
           TYPE = "Mimeo",       }

@TECHREPORT{Canning1989,
     AUTHOR = "David Canning",
     TITLE = "Convergence to Equilibrium in a Sequence of Games with
Learning",
     INSTITUTION = "London School of Economics",
     YEAR = 1989,
     TYPE = " STICERD Discussion Paper",
     NUMBER = "89/190",                          }

@ARTICLE{Canning1991,
     AUTHOR = "David Canning",
     TITLE = "Average Behavior in Learning Models",
     JOURNAL = "Journal of Economic Theory",
     YEAR = 1991,
      VOLUME = 57,
     PAGES = "442--472",                          }

@TECHREPORT{Canning1992,
     AUTHOR = "David Canning",
     TITLE = "Learning Language Conventions in Common Interest
          Signaling Games",
     INSTITUTION = "Columbia University",
     YEAR = 1992,
     TYPE = "Department of Economics Discussion Paper Series",
     NUMBER = "607",                          }

@TECHREPORT{Canning1992b,
           AUTHOR = "David Canning",
           TITLE = "Learning the Subgame Perfect Equilibrium",
           INSTITUTION = "Columbia University",
           YEAR = 1992,
           TYPE = "Discussion Paper",
           NUMBER = 608,                      }

@TECHREPORT{Canning1992c,
           AUTHOR = "David Canning",
           TITLE = "Learning and Efficiency in Common Interest Signaling
Games",
           INSTITUTION = "Columbia University",
           YEAR = 1992,
           TYPE = "Mimeo",                      }



@ARTICLE{CarlssonandvanDamme1992,
     AUTHOR = "Hans Carlsson and Eric van Damme",
     TITLE = "Global Games and Equilibrium Selection",
    JOURNAL = "Econometrica", 
     YEAR = 1993,                      
      VOLUME = 61,
       PAGES = "989--1018",  }

@ARTICLE{ChoandKreps1987,
     AUTHOR = "In-Koo Cho and David M. Kreps",
     TITLE = "Signaling Games and Stable Equilibria",
     JOURNAL = "Quarterly Journal of Economics",
     YEAR = 1987,
     VOLUME = 102,
     PAGES = "179--221",                   }

@ARTICLE{ChoandSobel1990,
     AUTHOR ="In-Koo Cho and Joel Sobel",
     TITLE = "Strategic Stabililty and Uniqueness in Signalling
          Games",
     JOURNAL = "Journal of Economic Theory",
     YEAR = 1990,
     VOLUME = 50,
     PAGES = "381-413",                  }

@TECHREPORT{Cooper1993,
           AUTHOR = "David J. Cooper",
           TITLE = "Supergames Played by Finite Automata with Finite Costs of
Complexity
                      in an Evolutionary Setting",
           INSTITUTION = "University of Pittsburgh",
           YEAR = 1993,
           TYPE = "Mimeo",                      }

@ARTICLE{CooperandJohn1988,
     AUTHOR = "Russell Cooper and Andrew John",
     TITLE = "Coordinating Coordination Failures in {K}eynesian
          Models",
     JOURNAL = "Quarterly Journal of Economics",
     YEAR = 1988,
     VOLUME = 103,
     PAGES = "441--463",                      }

@ARTICLE{Crawford1991,
           AUTHOR = "Vincent P. Crawford",
           TITLE = "An ``Evolutionary'' Interpretation of {V}an {H}uyck,
{B}attalio, and {B}eil'zs
                      Experimental Results on Coordination",
           JOURNAL = "Games and Economic Behavior",
           YEAR = 1991,
           VOLUME = 3,
           PAGES = "25--59",            }

@TECHREPORT{Crawford1992,
     AUTHOR = "Vincent P. Crawford",
     TITLE = "Adaptive Dynamics in Coordination Games",
     INSTITUTION = "University of California, San Diego",
     YEAR = "1992",
     TYPE = "Department of Economics Working Paper",  
      NUMBER = "92--02R",         }

@ARTICLE{CrawfordandHaller1990,
     AUTHOR = "Vincent P. Crawford and Hans Haller",
     TITLE = "Learning How to Cooperate: Optimal Play in Repeated
          Coordination Games",
     JOURNAL = "Econometrica",
     YEAR = 1990,
     VOLUME = 58,
     PAGES = 571,                          }

@Book{Dawkins1976, 
           AUTHOR = "R. Dawkins", 
           TITLE = "The Selfish Gene", 
           PUBLISHER = "Oxford University Press", 
           YEAR = 1976, 
           ADDRESS = "Oxford",          } 

@ARTICLE{DekelandScotchmer1992,
           AUTHOR = "Eddie Dekel and Suzanne Scotchmer",
           TITLE =  "On the Evolution of Optimizing Behavior",
           JOURNAL = "Journal of Economic Theory",
           YEAR = 1992,
           VOLUME = 57,
           PAGES = "392--406",          }

@TECHREPORT{EichbergerHallerandMilne1990,
     AUTHOR = "Jurgen Eichberger and Hans Haller and Frank Milne",
     TITLE = "Naive {Bayesian} Learning in 2 x 2 Matrix Games",
     INSTITUTION = "University of Melbourne",
     YEAR = 1990,
     TYPE = "Mimeo",                }

@TECHREPORT{Ellison1991,
     AUTHOR = "Glenn Ellison",
     TITLE = "Learning, Local Interaction, and Coordination",
     INSTITUTION = "MIT",
     YEAR = 1992,
     TYPE = "Mimeo",                  }

@TECHREPORT{Ellison1991pd,
     AUTHOR = "Glenn Ellsion",
     TITLE = "Cooperation in the Prisoner's Dilemma with Anonymous
          Random Matching",
     INSTITUTION = "MIT",
     YEAR = 1991,
     TYPE = "Mimeo",              }

@TECHREPORT{EllisonandFudenberg1992,
     AUTHOR = "Glenn Ellsion and Drew Fudenberg",
     TITLE = "Rules of Thumb for Social Learning",
     INSTITUTION = "University of Toulouse",
     YEAR = 1992,
     TYPE = "{IDEI} Discussion paper 17",              }

@ARTICLE{Eshel1972,
     AUTHOR = "Ilan Eshel",
     TITLE = "On the Neighbor Effect and the Evolution of
          Altruistic Traits",
     JOURNAL = "Theoretical Population Biology",
     YEAR = 1972,
     VOLUME = 3,
     PAGES = "258--277",             }

@ARTICLE{Farrell1992,
     AUTHOR = "Joseph Farrell",
     TITLE = "Meaning and Credibility in Cheap Talk Games",
     JOURNAL = "Games and Economic Behavior",  
     YEAR = 1993,
      NOTE = "Forthcoming",    }   

@TECHREPORT{FarrellandGibbons1988,
     AUTHOR = "Joseph Farrell and Robert Gibbons",
     TITLE = "Cheap Talk, Neologisms, and Bargaining",
     INSTITUTION = "Massachusetts Institute of Technology",
     YEAR = 1988,
     TYPE = "Department of Economics Working Paper",
     NUMBER = "500",                    }

@ARTICLE{FarrellandGibbons1989,
     AUTHOR = "Joseph Farrell and Robert Gibbons",
     TITLE = "Cheap Talk Can Matter in Bargaining",
     JOURNAL = "Journal of Economic Theory",
     YEAR = 1989,
     VOLUME = 48,
     PAGES = "221-237",             }

@ARTICLE{ForsytheHorowitzSavinandSefton1993,
     AUTHOR = "R. Forsythe and J. L. Horowitz and N. E. Savin and M. Sefton",
     TITLE = "The Statistical Analysis of Experiments with Simle Bargaining
Games",
           JOURNAL = "Games and Economic Behavior",
     YEAR = 1993,
           NOTE = "Forthcoming",                   }

@ARTICLE{FosterandYoung1990,
     AUTHOR = "Dean Foster and Peyton Young",
     TITLE = "Stochastic Evolutionary Game Dynamics",
     JOURNAL = "Journal of Theoretical Biology",
     YEAR = 1990,
     VOLUME = 38,
     PAGES = "219--232", }          

@Book{FreidlinandWentzell1984,
           AUTHOR = "M. I. Freidlin and A. D. Wentzell",
           TITLE = "Random Perturbations of Dynamical Systems",
           PUBLISHER = "Springer--Verlag",
           YEAR = 1984,
           ADDRESS = "New York",          }
           
@ARTICLE{Friedman1991,
     AUTHOR = "D. Friedman",
     TITLE = "Evolutionary Games in Economics",
     JOURNAL = "Econometrica",
     YEAR = 1991,
     VOLUME = 59,
     PAGES = "637--666",                    }

@ARTICLE{FudenbergandHarris1992,
     AUTHOR = "Drew Fudenberg and Chris Harris",
     TITLE = "Evolutionary Dynamics with Aggregate Shocks",
     JOURNAL = "Journal of Economic Theory",
     YEAR = 1992,
     VOLUME = 57,
     PAGES = "420-441",      }

@TECHREPORT{FudenbergandKreps1988,
     AUTHOR = "Drew Fudenberg and David M. Kreps",
     TITLE = "A Theory of Learning, Experimentation, and
          Equilibrium in Games",
     INSTITUTION = "Stanford University and Massachusetts Institute
          of Technology",
     YEAR = 1988,
     TYPE = "Mimeo",                          }

@ARTICLE{FudenbergandLevine1986,
     AUTHOR = "Drew Fudenberg and David K. Levine",
     TITLE = "Limit Games and Limit Equilibria",
     JOURNAL = "Journal of Economic Theory",
     YEAR = 1986,
     VOLUME = 38,
     PAGES = "261--279",                }

@TECHREPORT{FudenbergandLevine1990,
     AUTHOR = "Drew Fudenberg and David K. Levine",
     TITLE = "Steady State Learning and Self-Confirming
               Equilibium",
     INSTITUTION = "Massuchusetts Institute of Technology",
     YEAR = 1990,
     TYPE = "Mimeo",                 }

@ARTICLE{FudenbergandLevine1993,
     AUTHOR = "Drew Fudenberg and David K. Levine",
     TITLE = "Self-Confirming Equilibrium",
     YEAR = 1993,
     JOURNAL = "Econometrica",
           VOLUME = 61,
           PAGES = "523--446",    }

@ARTICLE{FudenbergandLevine1993b,
     AUTHOR = "Drew Fudenberg and David K. Levine",
     TITLE = "Steady State Learning and {N}ash Equilibrium",
       YEAR = 1993,
     JOURNAL = "Econometrica",
           VOLUME = 61,
           PAGES = "547-574",    }


@ARTICLE{FudenbergandMaskin1986,
     AUTHOR = "Drew Fudenberg and Eric Maskin",
     TITLE = "The Folk Theorem in Repeated Games with Discounting
          and Incomplete Information",
     JOURNAL = "Econometrica",
     YEAR = 1986,
     VOLUME = 54,
     PAGES = "533-554",                   }

@ARTICLE{FudenbergandMaskin1990,
     AUTHOR = "Drew Fudenberg and Eric Maskin",
     TITLE = "Evolution and Cooperation in Noisy Repeated Games",
     JOURNAL = "American Economic Review",
     YEAR = 1990,
     VOLUME = 80,
     PAGES = "274-279",
     MONTH = "May",                           }

@Book{FandT91,
     AUTHOR = "Drew Fudenberg and Jean Tirole",
     TITLE = "Game Theory",
     PUBLISHER = "MIT Press",
     YEAR = 1991,
     ADDRESS = "Cambridge, Massachusetts",           }

@Book{Futuyma1986,
     AUTHOR = "Douglas J. Futuyma",
     TITLE = "Evolutionary Biology",
     PUBLISHER = "Sinauer Associates, Inc.",
     YEAR = 1986,
     ADDRESS = "Sunderland, Massachusetts",           }

@ARTICLE{Gale1992,
     AUTHOR = "Douglas Gale",
     TITLE = " A {W}alrasian Theory of Markets with Adverse Selection",
     JOURNAL = "Review of Economics Studies",
     YEAR = 1992,
     VOLUME = 59,
     PAGES = "229--256",                    }

@ARTICLE{GilboaandMatsui1991,
           AUTHOR = "Itzhak Gilboa and Akihiko Matsui",
           TITLE = "Social Stability and Equilibrium",
           JOURNAL = "Econometrica",
           YEAR = 1991,
           VOLUME = 59,
           PAGES = "859--868",             }

@TECHREPORT{GilboaandSchmeidler1992, 
           AUTHOR = "Itzhak Gilboa and David Schmeidler", 
           TITLE = "Case--Based Decision Theory", 
            INSTITUTION = "Northwestern University", 
             YEAR = 1992, 
             TYPE = "Mimeo",         } 
 
@TECHREPORT{GilboaandSchmeidler1993a, 
           AUTHOR = "Itzhak Gilboa and David Schmeidler", 
           TITLE = "Case--Based Optimization", 
            INSTITUTION = "Northwestern University", 
             YEAR = 1993, 
             TYPE = "Mimeo",         } 
 
@TECHREPORT{GilboaandSchmeidler1993b, 
           AUTHOR = "Itzhak Gilboa and David Schmeidler", 
           TITLE = "Case--Based consumer Theory", 
            INSTITUTION = "Northwestern University", 
             YEAR = 1993, 
             TYPE = "Mimeo",         } 

           
@ARTICLE{GrossmanandPerry1986,
     AUTHOR = "Sanford J. Grossman and Motty Perry",
     TITLE = "Perfect Sequential Equilibrium",
     JOURNAL = "Journal of Economic Theory",
     YEAR = 1986,
     VOLUME = 39,
     PAGES = "97-119",                    }

@ARTICLE{GuthandTietz1990,
     AUTHOR = "Werner {G\"{u}th} and Reinhard Tietz",
     TITLE = "Ultimatum Bargaining Behavior: {A} Survey and Comparison of
Experimental
                      Results",
     JOURNAL = "Journal of Economic Psychology",
     YEAR = 1990,
     VOLUME = 11,
     PAGES = "417--49",                    }

@ARTICLE{Harley1981, 
     AUTHOR = "C. B. Harley", 
     TITLE = "Learning the Evolutionarily Stable Strategy", 
     JOURNAL = "Journal of Theoretical Biology", 
     YEAR = 1981, 
     VOLUME = 89, 
     PAGES = "611--633",                } 

@TECHREPORT{HarrisonandMcCabe1992,
           AUTHOR = "G. W. Harrison and K. A. McCabe",
           TITLE = "Expectations and Fairness in a Simple Bargaining
Experiment",
            INSTITUTION = "University of South Carolina",
             YEAR = 1992,
             TYPE = "College of Business Administration Working Paper",
              NUMBER = "B--92--10",             }

@ARTICLE{Harsanyi73, 
     AUTHOR = "John C. Harsanyi", 
     TITLE = "Oddness of the Number of Equilibrium Points:  A New Proof",
     JOURNAL = "International Journal of Game Theory", 
     YEAR = 1973, 
     VOLUME = 2, 
     PAGES = "235--250",                } 

@BOOK{HarsanyiandSelten1988,
     AUTHOR = "John C. Harsanyi and Reinhard Selten",
     TITLE = "A General Theory of Equilibrium Selection in Games",
     PUBLISHER = "MIT Press",
     YEAR = 1988,
     ADDRESS = "Cambridge, Massachusetts",      } 

@ARTICLE{Hillas1990,
           AUTHOR = "John Hillas",
           TITLE = "On the Definition of the Strategic Stability of
Equilibria",
           JOURNAL = "Econometrica",
           VOLUME = 58,
           YEAR = 1990,
           PAGES = "1365--1390",          }

@ARTICLE{Hirsch1988,
     AUTHOR = "Morris W. Hirsch",
     TITLE = "Stability and Convergence in Strongly Monotone
          Dynamical Systems",
     JOURNAL = "Journal {f\"{ur}} die Reine und Angewandte
          Mathematik",
     YEAR = 1988,
     VOLUME = 383,
     PAGES = "1--53",     }

@BOOK{HofbauerandSigmund1988,
     AUTHOR = "J. Hofbauer and K. Sigmund",
     TITLE = "The Theory of Evolution and Dynamical Systems",
     PUBLISHER = "Cambridge University Press",
     YEAR = 1988,
     ADDRESS = "Cambridge",          }

@TECHREPORT{HoffmanMcCabeShachatandSmith1992,
           AUTHOR = "Elizabeth Hoffman and Kevin McCabe and Keith Shachat and
Vernon
                      Smith",
           TITLE = "Preferences, Property Rights and Anonymity in Bargaining
Games",
           INSTITUTION = "University of Arizona",
           YEAR = 1992,
           TYPE = "Mimeo",       }

@TECHREPORT{Howard1988,
     AUTHOR = "J. V. Howard",
     TITLE = "Cooperation in the Prisoner's Dilemma",
     INSTITUTION = "London School fo Economics",
     YEAR = 1988,
     TYPE = "STICERD Discussion Paper",
     NUMBER = "88/171",         }

@ARTICLE{Jordan1991,
     AUTHOR = "James S. Jordan",
     TITLE = "Bayesian Learning in Normal Form Games",
     JOURNAL = "Games and Economic Behavior",
     YEAR = 1991,
     VOLUME = 3,
     PAGES = "60--81",           }

@ARTICLE{Jordan1992,
     AUTHOR = "James S. Jordan",
     TITLE = "The Exponential Convergence of {B}ayesian Learning in
          Normal Form Games",
     JOURNAL = "Games and Economic Behavior",
     YEAR = 1992,
     VOLUME = 4,
     PAGES = "202--217",       }

@TECHREPORT{Jordan1992a,
     AUTHOR = "James S. Jordan",
     TITLE = "Three Problems in Learning Mixed-Strategy {Nash}
          Equilibria",
     INSTITUTION = "University of Minnesota",
     YEAR = 1992,
     TYPE = "Mimeo",             }

@INCOLLECTION{KandT87, 
     AUTHOR="Kahneman, D. and A. Tversky", 
   BOOKTITLE="Decision Making", 
      TITLE="Rational Choice and the Framing of Decisions", 
  PUBLISHER={Cambridge University Press}, 
    ADDRESS={Cambridge}, 
       YEAR=1988} 
 

@TECHREPORT{KalaiandLehrer1991,
     AUTHOR = "Ehud Kalai and Ehud Lehrer",
     TITLE = "Private-Beliefs Equilibrium",
     INSTITUTION = "Northwestern University",
     YEAR = 1991,
     TYPE = "Center for Mathematical Studies in Economics and
          Management Science Discussion Paper",
     NUMBER = 926,                     }

@TECHREPORT{KalaiandLehrer1991a,
     AUTHOR = "Ehud Kalai and Ehud Lehrer",
     TITLE = "Rational Learning Leads to {N}ash Equilibria",
     INSTITUTION = "Northwestern Universty",
     YEAR = 1991,
     TYPE = "Mimeo",             }

@ARTICLE{Kandori1992,
     AUTHOR = "Michihiro Kandori",
     TITLE ="Social Norms and Community Enforcement",
     JOURNAL = "Review of Economic Studies",
     YEAR = 1991,
     VOLUME = 59,
     PAGES = "61--80",        }

@ARTICLE{KandoriMailathandRob1992,
     AUTHOR = "Michihiro Kandori and George J. Mailath and Rafael
          Rob",
     TITLE = "Learning, Mutation, and Long Run Equilibria in
          Games",
           JOURNAL = "Econometrica",
     YEAR = 1993,
         VOLUME = 61,
           PAGES = "29--56",                 }

@Book{KemenyandSnell1960,
           AUTHOR = "John G. Kemeny and J. Laurie Snell",
           TITLE = "Finite Markov Chains",
           PUBLISHER = "D. Van Nostrand Company, Inc.",
           YEAR = 1960,
           ADDRESS = "Princeton",         }
           
@TECHREPORT{KimandSobel1992,
     AUTHOR = "Yong-Gwan Kim and Joel Sobel",
     TITLE = "An Evolutionary Approach to Pre-Play Communication",
     INSTITUTION = "University of Iowa and University of
          California, San Diego",
     YEAR = 1992,
     TYPE = "Mimeo",                      }

@BOOK{Kimura1983,
     AUTHOR = "Motoo Kimura",
     TITLE = "The Neutral Theory of Molecular Evolution",
     PUBLISHER = "Cambridge University Press",
     YEAR = 1983,
     ADDRESS ="Cambridge",      }

@TECHREPORT{Kohlberg1989,
     AUTHOR = "Elon Kohlberg",
     TITLE = "Refinement of {N}ash Equilibrium:  The Main Ideas",
     INSTITUTION = "Harvard Business School",
     YEAR = 1989,
     TYPE = "Working Paper",
     NUMBER = "89-073",               }

@ARTICLE{KohlbergandMertens1986,
     AUTHOR = "Elon Kohlberg and Jean-Francois Mertens",
     TITLE = "On the Strategic Stability of Equilibria",
     JOURNAL = "Econometrica",
     YEAR = 1986,
     VOLUME = 54,
     PAGES = "1003--1038",                }

@ARTICLE{KrepsandWilson1982,
     AUTHOR = "David M. Kreps and Robert J. Wilson",
     TITLE = "Sequential Equilibrium",
     JOURNAL = "Econometrica",
     YEAR = 1982,
     VOLUME = 50,
     PAGES = "863--894",             }

@TECHREPORT{Krishna1992,
     AUTHOR = "Vijay Krishna",
     TITLE = "Learning in Games with Strategic Complementarities",
     INSTITUTION = "Harvard Business School",
     YEAR = 1992,
     TYPE = "Working Paper",
     NUMBER = "92--073",           }

@UNPUBLISHED{Ledyard92, 
      AUTHOR="Ledyard, J.", 
       TITLE="Public Goods{:} {A} Survey of Experimental 
Research", 
        YEAR="1992", 
        NOTE="Working Paper{,} California Institute of 
     Technology"} 

@TECHREPORT{Lipman93,
     AUTHOR = "Bart Lipman",
     TITLE = "Long-Run Equilibrium with Varying Mutation Rates",
     INSTITUTION = "Queen's University",
     YEAR = 1993,
     TYPE = "Mimeo"           }

@TECHREPORT{Lomborg,
     AUTHOR = "{Bj\o"rn} Lomborg",
     TITLE = "The Structure of Solutions in the Iterated Prisoners's Dilemma",
     INSTITUTION = "University of Copenhagen",
     YEAR = 1993,
     TYPE = "Mimeo",     }

@ARTICLE{Ljung1977,
     AUTHOR = "L. Ljung",
     TITLE ="Analysis of Recursive Stochastic Algorithms",
     JOURNAL = "IEEE Trans. Automat. Control",
     YEAR = 1977,
     VOLUME = "AC-33",
     PAGES = "551-557",             }

@BOOK{LjungandSoderstrom1983,
     AUTHOR = "L. Ljung and T. Soderstrom",
     TITLE = "Theory and Practice of Recursive Identification",
     PUBLISHER ="MIT Press",
     YEAR = 1983,
     ADDRESS = "Cambridge",        }


@ARTICLE{Mailath1993,
           AUTHOR = "George J. Mailath",
           TITLE = "Perpetual Randomness in Evolutionary Economics",
           JOURNAL = "Economics Letters",
           YEAR = 1993,
           NOTE = "Forthcoming",       }

@ARTICLE{MOandP93,
           AUTHOR = "George J. Mailath and Masahiro Okuno-Fujiwara and Andrew
                      Postlewaite",
           TITLE =  "Belief-Based Refinements in Signalling Games",
           JOURNAL = "Journal of Economic Theory",
           YEAR = 1993,
           VOLUME = 60,
           PAGES = "241--276",         }

@ARTICLE{1992,
           AUTHOR = "George J. Mailath",
           TITLE =  "Introduction: {S}ymposium on Evolutionary Game Theory",
           JOURNAL = "Journal of Economic Theory",
           YEAR = 1992,
           VOLUME = 57,
           PAGES = "259--277",         }


@TECHREPORT{MailathSamuelsonandSwinkels1992,
     AUTHOR = "George Mailath and Larry Samuelson and Jeroen
          Swinkels",
     TITLE = "Extensive Form Reasoning in Normal Games",
     INSTITUTION = "University of Pennsylvania",
     YEAR = 1992,
     TYPE = "Department of Economics Working Paper",
     NUMBER = "92-01",       }

@ARTICLE{MarcetandSargent1989,
     AUTHOR = "Albert Marcet and Thomas J. Sargent",
     TITLE = "Convergence of Least Squares Learning Mechanisms in
          Self-Referential Linear Stochastic Models",
     JOURNAL = "Journal of Economic Theory",
     YEAR = 1989,
     VOLUME =48,
     PAGES = "337--368",          }

@INCOLLECTION{MarimonandMcGratten1992,
     AUTHOR = "Ramon Marimon and Ellen McGratten",
     TITLE = "On Adaptive Learning in Strategic Games",
     BOOKTITLE = "Learning and Rationality in Economics",
     PUBLISHER = "Basil Blackwell",
     YEAR = 1992,
     EDITOR = "Alan Kirman and Mark Salmon",
     NOTE = "Forthcoming",                    }

@ARTICLE{Matsui1991,
     AUTHOR = "Akihiko Matsui",
     TITLE = "Cheap-Talk and Cooperation in Society",
     JOURNAL = "Journal of Economic Theory",
     YEAR = 1991,
     VOLUME = 54,
     PAGES = "245--258",            }     

@ARTICLE{Matsui1992,
           AUTHOR = "Akihiko Matsui",
           TITLE = "Best Response Dynamics and Socially Stable Strategies",
           JOURNAL = "Journal of Economic Theory",
           YEAR = 1992,
           VOLUME = 57,
           PAGES = "343--362",        }
           
@TECHREPORT{MatsuiandMatsuyama1991,
     AUTHOR = "Akihiko Matsui and Kiminori Matsuyama",
     TITLE = "An Approach to Equilibrium Selection",
     INSTITUTION = "Northwestern University",
     YEAR = "1992",
     TYPE = "Center for Mathematical Studies in Economics and
          Management Science Discussion Paper",
     NUMBER = "970",               }

@TECHREPORT{MatsuiandRob1991,
     AUTHOR = "Akihiko Matsui and Rafael Rob",
     TITLE = "The Roles of Public Information and Preplay
          Communication in Evolutionary Games",
     INSTITUTION = "University of Pennsylvania",
     YEAR = 1991,
     TYPE = "CARESS Working Paper",
     NUMBER = "91-10",                       }

@ARTICLE{MatthewsOkunoFujiwaraandPostlewaite1991,
     AUTHOR = "Steven A. Matthews and Masahiro Okuno-Fujiwara and
          Andrew Postlewaite",
     TITLE = "Refining Cheap-Talk Equilibria",
     JOURNAL = "Journal of Economic Theory",
     YEAR = 1991,
     VOLUME = 55,
     PAGES = "247--273",                }

@BOOK{MaynardSmith1982,
     AUTHOR = "{Maynard Smith}, John",
     TITLE = "Evolution and the Theory of Games",
     PUBLISHER = "Cambridge University Press",
     YEAR = 1982,
     ADDRESS = "Cambridge",                 }

@ARTICLE{MaynardSmithandPrice1973,
     AUTHOR = "{Maynard Smith}, John and G. R. Price",
     TITLE = "The Logic of Animal Conflict",
     JOURNAL = "Nature",
     YEAR = 1973,
     VOLUME = 273,
     PAGES = "15-18",                   }

@INCOLLECTION{McKenzie1959,
     AUTHOR = "Lionel McKenzie",
     TITLE = "Matrices with Dominant Diagonals and Economic
          Theory",
     BOOKTITLE = "Mathematical Methods in the Social Sciences",
     PUBLISHER = "Stanford University Press",
     YEAR = 1959,
     EDITOR = "K. Arrow, S. Karlin and P. Suppes",
     ADDRESS = "Stanford",             }

@ARTICLE{Mertens1989,
           AUTHOR = "Jean--Francois Mertens",
           TITLE = "Stable Equilibria--a Reformulation, Part I",
           JOURNAL = "Mathematics of Operations Research",
           YEAR = 1989,
           VOLUME = 14,
           PAGES = "575--624",             }

@TECHREPORT{Mertens1990,
           AUTHOR = "Jean--Francois Mertens",
           TITLE = "The Small Worlds Axiom for Stable Equilibria",
           INSTITUTION = "CORE",
           YEAR = "1989",
           TYPE = "Discussion Paper 9007",   }

@ARTICLE{MilgromandRoberts1990,
     AUTHOR = "Paul Milgrom and John Roberts",
     TITLE = "Rationalizability, Learning, and Equilibrium in Games
          with Strategic Complementarities",
     JOURNAL = "Econometrica",
     YEAR = 1990,
     VOLUME = 58,
     PAGES = "1255-1278",         }

@ARTICLE{MilgromandRoberts1991,
     AUTHOR = "Paul Milgrom and John Roberts",
     TITLE = "Adaptive and Sophisticated Learning in Normal From
          Games",
     JOURNAL = "Games and Economic Behavior",
     YEAR = 1991,
     VOLUME = 3,
     PAGES = "82--100",           }



@TECHREPORT{Miyasawa1961,
     AUTHOR = "K. Miyasawa",
     TITLE = "On the Convergence of the Learning Process in 2x2
          Non-zero Sum Games",
     INSTITUTION = "Princeton University",
     YEAR = 1961,
     TYPE = "Econometric Research Program Research Memorandum",
     NUMBER = 33,                }

@TECHREPORT{MondererandShapley1992,
     AUTHOR = "Dov Monderer and Lloyd Shapley",
     TITLE = "Potential Games",
     INSTITUTION = "Technion",
     YEAR = 1992,
     TYPE = "Mimeo",          }

@BOOK{Murray1989,
     AUTHOR = "J. D. Murray",
     TITLE = "Mathematical Biology",
     PUBLISHER = "Springer--Verlag",
     YEAR = 1989,
     ADDRESS ="New York",      }

@BOOK{Myerson1991, 
     AUTHOR = "Roger B. Murray", 
     TITLE = "Game Theory: Analysis of Conflict", 
     PUBLISHER = "Harvard University Press", 
     YEAR = 1991, 
     ADDRESS ="Cambridge, Mass"} 

@ARTICLE{MyersonPollockandSwinkels1991,
     AUTHOR = "Roger B. Myerson and Gregory B. Pollock and Jeroen
          M. Swinkels",
     TITLE = "Viscous Population Equilibria",
     JOURNAL = "Games and Economic Behavior",
     YEAR = 1991,
     VOLUME = 3,
     PAGES = "101-109",           }

@ARTICLE{Nachbar1990,
     AUTHOR = "John H. Nachbar",
     TITLE = "`{E}volutionary' Selection Dynamics in Games: 
          Convergence and Limit Properties",
     JOURNAL = "International Journal of Game Theory",
     YEAR = 1990,
     VOLUME = 19,
     PAGES = "59-89",             }

@ARTICLE{Nachbar1992,
     AUTHOR = "John H. Nachbar",
     TITLE = "Evolution in the Finitely Repeated Prisoner's Dilemma",
     JOURNAL = "Journal of Economic Behavior and Organization",
     YEAR = 1992,
     VOLUME = 19,
     PAGES = "307--326",             }

@TECHREPORT{Nachbar1993,
     AUTHOR = "John H. Nachbar",
     TITLE = "Perturbed Best--Response Dynamics",
     INSTITUTION = "Washington University, St. Louis",
     YEAR = 1993,
     TYPE = "Mimeo",             }


@ARTICLE{Nash1950,
     AUTHOR = "John F. Nash",
     TITLE = "Equilibrium Points in n-Person Games",
     JOURNAL = "Proceedings of lthe National Academy of Sciences",
     YEAR = 1950,
     VOLUME = 36,
     PAGES = "48-49",              }

@BOOK{NelsonandWinter1982,
     AUTHOR = "R. Nelson and S. Winter",
     TITLE = "An Evolutionary Theory of Economic Change",
     PUBLISHER = "Harvard University Press",
     YEAR = 1982,
     ADDRESS ="Cambridge",      }

@TECHREPORT{NoldekeandSamuelson1992wp,
           AUTHOR = "Georg {N\"{o}ldeke} and Larry Samuelson",
           TITLE = "The Evolutionary Foundations of Backward and Forward
Induction",
           INSTITUTION = "University of Bonn",
           YEAR = 1992,
           TYPE = "{SFB} Discussion Paper {B}-216",        }

@ARTICLE{NoldekeandSamuelson1993,
           AUTHOR = "Georg {N\"oldeke} and Larry Samuelson",
           TITLE = "An Evolutionary Analysis of Backward and Forward
                      Induction",
           JOURNAL = "Games and Economic Behavior",
           YEAR = 1993,
            VOLUME = 5,
           PAGES = "425--454",    }
           
@ARTICLE{NoldekeandvanDamme1990,
           AUTHOR = "Georg {N\"{o}ldeke} and Eric van Damme",
           TITLE = "Signalling in a Dynamic Labour Market",
           JOURNAL = "Review of Economic Studies",
           VOLUME = 57,
           YEAR = 1990,
           PAGES = "1--24",       }

@Book{Norman1972, 
     AUTHOR = "M. Frank Norman", 
     TITLE = "Markov Processes and Learning Models", 
     YEAR = 1972, 
     PUBLISHER = "Academic Press", 
     ADDRESS = "New York",           } 

@ARTICLE{Nowack90,
     AUTHOR = "Martin Nowak",
     TITLE = "Stochastic Stategies in the Prisoner's Dilemma",
     JOURNAL = "Theoretical Population Biology",
           VOLUME = 38,
     YEAR = 1990,
     PAGES = "93--112",        }

@ARTICLE{NS90,
     AUTHOR = "Martin Nowak and Karl Sigmund",
     TITLE = "The Evolution of Stochastic Strategies in the Prisoner's
Dilemma",
     JOURNAL = "Acta Applicandae Mathematica",
           VOLUME = 20,
     YEAR = 1990,
     PAGES = "247--265",        }

@ARTICLE{NS92,
     AUTHOR = "Martin Nowak and Karl Sigmund",
     TITLE = "Tit for Tat in Heterogeneous Populations", 
     JOURNAL = "Nature",
           VOLUME = 355,
     YEAR = 1992,
     PAGES = "250-252",        }

@ARTICLE{NS93a,
     AUTHOR = "Martin Nowak and Karl Sigmund",
     TITLE = "Win-Stay, Lose-Shift Outperforms Tit for Tat", 
     JOURNAL = "Nature",
           VOLUME = 364,
     YEAR = 1992,
     PAGES = "56-58",        }

@ARTICLE{NS93b,
     AUTHOR = "Martin Nowak and Karl Sigmund",
     TITLE = "Chaos and the Evolution of Cooperation",
     JOURNAL = "Proceedings of the National Academy of Sciences USA",
     YEAR = 1993,
     NOTE = "Forthcoming",        }

@TECHREPORT{NS93,
     AUTHOR = "Martin Nowak and Karl Sigmund",
     TITLE = "The Alternating Prisoner's Dilemma",
     INSTITUTION = "University of Oxford",
     YEAR = 1993,
     TYPE = "Mimeo"           }

@TECHREPORT{NSE93,
     AUTHOR = "Martin Nowak, Karl Sigmund and Esam El-Sedy",
     TITLE = "Automata, Repeated Games and Noise",
     INSTITUTION = "University of Oxford",
     YEAR = 1993,
     TYPE = "Mimeo"           }

@TECHREPORT{OkunoFujiwaraandPostlewaite1989,
     AUTHOR = "Masahiro Okuno-Fujiwara and Andrew Postlewaite",
     TITLE = "Social Norms in Random Matching Games",
     INSTITUTION = "University of Tokyo and University of
          Pennsyvlania",
     YEAR = 1989,
     TYPE = "Mimeo",             }

@ARTICLE{Pearce1984,
     AUTHOR ="David Pearce",
     TITLE = "Rationalizable Strategic Behavior and the Problem of
          Perfection",
     JOURNAL = "Econometrica",
     YEAR = 1984,
     VOLUME = 52,
     PAGES = "1029--1050",           }

@ARTICLE{Rabin1990,
     AUTHOR ="Matthew Rabin",
     TITLE = "Communication between Rational Agents",
     JOURNAL = "Journal of Economic Theory",
     YEAR = 1990,
     VOLUME = 51,
     PAGES = "144--170",           }

@ARTICLE{Reny1985,
     AUTHOR = "Philip J. Reny",
     TITLE = "Rationality, Common Knowledge and the Theory of
          Games",
     JOURNAL = "Journal of Economic Theory",
     YEAR = 1993,
     NOTE = "Forthcoming",          }

@ARTICLE{Riley1979,
     AUTHOR ="John Riley",
     TITLE = "Informational Equilibrium",
     JOURNAL = "Econometrica",
     YEAR = 1979,
     VOLUME = 47,
     PAGES = "331--359",           }

@TECHREPORT{Ritz93,
           AUTHOR = "Klaus Ritzberger",
           TITLE = "The Theory of Normal Form Games from the Differentiable
Viewpoint",
           INSTITUTION = "Institute for Advanced Studies",
           YEAR = 1993,
           TYPE = "Mimeo",        }

@TECHREPORT{RandV92,
           AUTHOR = "Klaus Ritzberger and Karl Vogelsberger",
           TITLE = "The Nash Field",
           INSTITUTION = "Institute for Advanced Studies",
           YEAR = 1992,
           TYPE = "Mimeo",        }

@TECHREPORT{RitzbergerandWeibull1993,
           AUTHOR = "Klaus Ritzberger and {J\"{o}rgen} Weibull",
           TITLE = "Evolutionary Selection in Normal Form Games",
           INSTITUTION = "Institute for Advanced Studies, Vienna and Stockholm
University,
                      Sweden",
           YEAR = 1993,
           TYPE = "Mimeo",        }

@ARTICLE{Robinson1951,
     AUTHOR = "Julia Robinson",
     TITLE = "An Iterative Method of Solving a Game",
     JOURNAL = "Annals of Mathematics",
     YEAR = 1951,
     VOLUME = 54,
     PAGES = "296--301",          }

@ARTICLE{Robson1990,
     AUTHOR = "Arthur J. Robson",
     TITLE = "Efficiency in Evolutionary Games: {D}arwin, {N}ash,
          and the Secret Handshake",
     JOURNAL = "Journal of Theoretical Biology",
     YEAR = 1990,
     VOLUME = 144,
     PAGES = "379--396",                    }

@TECHREPORT{Robson1992,
           AUTHOR = "Arthur J. Robson",
           TITLE = "The Biological Basis of Expected Utility, Knightian
Uncertainty and the
                      Ellsberg Paradox",
           INSTITUTION = "University of Western Ontario",
           YEAR = 1992,
           TYPE = "Mimeo",        }

@ARTICLE{Rosenmuller1971,
     AUTHOR = "Joachim {Rosenm\"{u}ller}",
     TITLE = "{U}ber {Perioizit\"{a}tseigenschaften}
          Spieltheoretischer Lernprozesse",
     JOURNAL = "Zeitschrift {f\"{u}r} Wahrscheinlichkeitstheorie
          und Verwandte Gebeite",
     YEAR = 1971,
     VOLUME = 17,
     PAGES = "259--308",              }

@ARTICLE{Rosenthal1973,
     AUTHOR = "Robert W. Rosenthal",
     TITLE = "A Class of Games Possessing Pure-Strategy {N}ash
          Equilibria",
     JOURNAL = "International Journal of Game Theory",
     YEAR = 1973,
     VOLUME = 2,
     PAGES = "65--67",           }

@ARTICLE{Rosenthal1992,
           AUTHOR = "Edward C. Rosenthal",
           TITLE = "Coalition Formation under Limited Communication",
           JOURNAL = "Games and Economic Behavior",
           YEAR = 1992,
           VOLUME = 4,
           PAGES = "402--421",       }

@TECHREPORT{RothandErev1993,
     AUTHOR = "Alvin E. Roth and Ido Erev",
     TITLE = "Learning in Extensive--Form Games:  {E}xperimental Data and
Simple
                      Dynamic Models in the Intermediate Term",
     INSTITUTION = "University of Pittsburgh and The Technion",
     YEAR = 1993,
     TYPE = "Mimeo",           }

@ARTICLE{RothMaloufandMurnighan1981, 
           AUTHOR = "Alvin E. Roth and Michael Malouf and J. Keith 
Murnighan", 
           TITLE = "Sociological versus Strategic Facators in 
Bargaining", 
           JOURNAL = "Journal of Economic Behavior and 
Orgainization", 
           YEAR = 1981, 
           VOLUME = 2, 
           PAGES = "153--177",       } 

@INCOLLECTION{Roth1991,
     AUTHOR = "Alvin E. roth",
     TITLE = "Bargaining Experiments",
     BOOKTITLE = "Handbook of Experimental Economics",
     PUBLISHER = "Princeton University Press",
     YEAR = 1991,
     EDITOR = "John Kagel and Alvin E. Roth",
     NOTE = "Forthcoming",                    }

@ARTICLE{RothschildandStiglitz1976,
     AUTHOR ="M. Rothschild and J. Stiglitz",
     TITLE = "Equilibrium in Competitive Insurance Markets: An Essay on the
Economics of
                      Imperfect Information",
     JOURNAL = "Quarterly Journal of Economics",
     YEAR = 1976,
     VOLUME = 80,
     PAGES = "629--649",           }

@INCOLLECTION{Samuelson1988, 
     AUTHOR="Larry Samuelson", 
   BOOKTITLE="Theoretical Aspects of Reasoning About Knowledge", 
      TITLE="Evolutinary Foundations of Solution Concepts for Finite,
Two-Player, 
                      Normal-Form Games", 
  PUBLISHER={Morgan Kaufmann Publishers, Inc.}, 
       YEAR=1988,
           EDITOR="Moshe Y. Vardi",    } 

@ARTICLE{Samuelson1991,
     AUTHOR = "Larry Samuelson",
     TITLE = "Stochastic Stability in Games with Alternative Best Replies",
     JOURNAL = "Journal of Economic Theory",
     YEAR = 1993,
     NOTE = "Forthcoming",                        }

@TECHREPORT{Samuelson1991dom,
     AUTHOR = "Larry Samuelson",
     TITLE = "Does Evolution Eliminate Dominated Strategies",
     INSTITUTION = "University of Wisconsin",
     YEAR = 1991,
     TYPE = "Mimeo",           }

@ARTICLE{Samuelson1991l,
     AUTHOR = "Larry Samuelson",
     TITLE = "Limit Evolutionarily Stable Strategies in Two-Player,
          Normal Form Games",
     JOURNAL = "Games and Economic Behavior",
     YEAR = 1991,
     VOLUME = 3,
     PAGES = "110--119",           }

@ARTICLE{Samuelson1992,
     AUTHOR = "Larry Samuelson",
     TITLE = "Dominated Strategies and Common Knowledge",
     JOURNAL = "Games and Economic Behavior",
     YEAR = 1992,
     VOLUME = 4,
     PAGES = "284--313",           }

@ARTICLE{SamuelsonandZhang1992,
     AUTHOR = "Larry Samuelson and Jianbo Zhang",
     TITLE = "Evolutionary Stability in Asymmetric Games",
     JOURNAL = "Journal of Economic Theory",
     YEAR = 1992,
           VOLUME = 57,
           PAGES = "363--391",  }

@TECHREPORT{Schlag1993,
     AUTHOR = "Karl H. Schlag",
     TITLE = "Dynamic Stability in the Repeated Prisoners' Dilemma Played by
Finite
                      Automata",
     INSTITUTION = "University of Bonn",
     YEAR = 1993,
     TYPE = "Mimeo",           }

@ARTICLE{Selten1965,
     AUTHOR = "Reinhard Selten",
     TITLE = "Spieltheoretische Behandlung eines Oligopolmodells mit 
                      Nachfragetragheit",
     JOURNAL = "Zeitschrift fur die gesamte Staatswissenschaft",
     YEAR = 1965,
     VOLUME = 12,
     PAGES = "301--324",       }


@ARTICLE{Selten1975,
     AUTHOR = "Reinhard Selten",
     TITLE = "Reexamination of the Perfectness Concept for
          Equilibrium Points in Extensive-Form Games",
     JOURNAL = "International Journal of Game Theory",
     YEAR = 1975,
     VOLUME = 4,
     PAGES = "25-55",       }

@ARTICLE{Selten1978, 
      AUTHOR="Selten, R.", 
       TITLE="The Chain-Store Paradox", 
     JOURNAL="Theory and Decision", 
        YEAR="1978", 
      VOLUME="9", 
       PAGES="127-159"} 

@ARTICLE{Selten1980,
           AUTHOR = "Reinhard Selten",
           TITLE = "A Note on Evolutionarily Stable Strategies in Asymmetric 
                      Animal Contests",
           JOURNAL = "Journal of Theoretical Biology",
           YEAR = 1980,
           VOLUME = 84,
           PAGES = "93--101",          }
           
@ARTICLE{Selten1983,
     AUTHOR = "Reinhard Selten",
     TITLE = "Evolutionary Stability in Extensive Two-Person
          Games",
     JOURNAL = "Mathematical Social Sciences",
     YEAR = 1983,
     VOLUME = 5,
     PAGES = "269-363",       }

@ARTICLE{Selten1988,
     AUTHOR = "Reinhard Selten",
     TITLE = "Evolutionary Stability in Extensive Two-Person
          Games--Correction and Further Development",
     JOURNAL = "Mathematical Social Sciences",
     YEAR = 1988,
     VOLUME = 16,
     PAGES = "223--266",       }
 
@ARTICLE{Shapley1964,
     AUTHOR = "Lloyd Shapley",
     TITLE = "Some Topics in Two-Person Games",
     JOURNAL = "Advances in Game Theory, Annals of Mathematical
          Studies",
     YEAR = 1964,
     VOLUME = 5,
     PAGES = "1--28",        }

@TECHREPORT{Skyrms1991,
     AUTHOR = "Brian Skyrms",
     TITLE = "Chaos in Game Dynamics",
     INSTITUTION = "University of California, Irvine",
     YEAR = 1991,
     TYPE = "Mathematical Behavior Sciences Working Paper",
     NUMBER = "91-18",              }

@ARTICLE{VernonSmith76, 
    AUTHOR="Smith, V.", 
     TITLE="Experimental Economics: {I}nduced Value Theory", 
   JOURNAL="American Economic Review", 
      YEAR=1976, 
    VOLUME=66, 
     PAGES="274-279"} 
 

@ARTICLE{Sobel1993,
           AUTHOR = "Joel Sobel",
           TITLE = "Evolutionary Stabiliy in Communication Games",
           JOURNAL = "Economics Letters",
           YEAR = 1993,
           NOTE = "Forthcoming",          }

@ARTICLE{Spence1973,
           AUTHOR = "A. M. Spence",
           TITLE = "Job Market Signaling",
           JOURNAL = "Quarterly Journal of Economics",
           YEAR = 1973,
           VOLUME = 90,
           PAGES = "225--243",           }

@BOOK{Spence1974,
           AUTHOR = "A. M. Spence", 
           TITLE = "Market Signaling, Information Transfer in Hiring and
Related                  
                     Processes",
           PUBLISHER = "Harvard University Press",
           ADDRESS = "Cambridge",
           YEAR = 1974,                    }      

@TECHREPORT{StanleyAshlockandTesfatsion1992,
     AUTHOR = "E. Ann Stanley and Dan Ashlock and Leigh Tesfatsion",
     TITLE = "Iterated Prisoner's Dilemma with Choice and Refusal",
     INSTITUTION = "Iowa State University",
     YEAR = 1992,
     TYPE  = "Economic Report Series",
     NUMBER = 30,                  }

@INCOLLECTION{SandW1993,
     AUTHOR = "J. Stiglitz and A. Weiss",
     TITLE = "Sorting out the Differences Between Screening and Signaling
Models",
     BOOKTITLE = "Oxford Essakys in Mathematical Economics",
     PUBLISHER = "Oxford University Press",
     YEAR = 1993,
     EDITOR = "M. Bachrach",
     ADDRESS = "Oxford",                    }


@TECHREPORT{Straub93,
     AUTHOR = "Paul G. Straub",
     TITLE = "Risk Dominance and Coordination Failures in Static Games",
     INSTITUTION = "Northwestern University",
     YEAR = 1993,
      NUMBER = 106,
     TYPE = "Dispute Resolution Research Center Working Paper",
      }


                
@ARTICLE{Swinkels1992a,
     AUTHOR = "Jeroen Swinkels",
     TITLE = "Evolutionary Stability with Equilibrium Entrants",
     JOURNAL = "Journal of Economic Theory",
     YEAR = 1992,
     VOLUME = 57,
     PAGES = "306--332",        }

@ARTICLE{Swinkels1992b,
     AUTHOR = "Jeroen Swinkels",
     TITLE = "Evolution and Strategic Stability:  From {Maynard
          Smith} to {Kohlberg--Mertens}",
     JOURNAL = "Journal of Economic Theory",
     YEAR = 1992,
     VOLUME = 57,
     PAGES = "333-342",           }

@ARTICLE{Swinkels1992d,
     AUTHOR = "Jeroen Swinkels",
     TITLE = "Adjustment Dynamics and Rational Play in Games",
      JOURNAL = "Games and Economic Behavior",
     YEAR = 1993,
       VOLUME = 5,
           PAGES = "455--484",             }

@ARTICLE{TanandWerlang1988,
     AUTHOR = "Tommy Chin-Chiu Tan and Sergio Ribeiro da Costa
          Werlang",
     TITLE = "The {Bayesian} Foundations of Solution Concepts of
          Games",
     JOURNAL = "Journal of Economic Theory",
     YEAR = 1988,
     VOLUME = 45,
     PAGES = "370--391",            }

@ARTICLE{TaylorandJonker1978,
           AUTHOR = "Peter D. Taylor and Leo B. Jonker",
           TITLE = "Evolutionarily Stable Strategies and Game Dynamics",
           JOURNAL = "Mathematical Biosciences",
           YEAR = 1978,
           VOLUME = 40,
           PAGES = "145-156",          }


@ARTICLE{Thaler1988,
           AUTHOR = "Richard H. Thaler",
           TITLE = "Anomalies:  {T}he Ultimatum Game",
           JOURNAL = "Journal of Economic Perspectives",
           YEAR = 1988,
           VOLUME = 2,
           PAGES = "195--206",          }


@ARTICLE{Thomas1985a,
     AUTHOR = "B. Thomas",
     TITLE = "On Evolutionarily Stable Sets",
     JOURNAL = "Journal of Mathematical Biology",
     YEAR = 1985,
     VOLUME = 22,
     PAGES = "105--115",                    }

@ARTICLE{Thomas1985b,
     AUTHOR = "B. Thomas",
     TITLE = "Evolutionarily Stable Sets in Mixed--Strategist
          Models",
     JOURNAL = "Theoretical Population Biology",
     YEAR = 1985,
     VOLUME = 28,
     PAGES = "332--41",                    }

@ARTICLE{ThorlundPetersen1990,
     AUTHOR = "Lars Thorlund-Petersen",
     TITLE = "Iterative Computation of {C}ournot Equilibrium",
     JOURNAL = "Games and Economic Behavior",
     YEAR = 1990,
     VOLUME = 2,
     PAGES = "61--75",     }

@BOOK{Tirole1988,
     AUTHOR = "Jean Tirole",
     TITLE = "The Theory of Industrial Organization",
     PUBLISHER = "MIT PRESS",
     ADDRESS = "Cambridge, Massachusetts",
     YEAR = 1988,                }

@ARTICLE{Topkis1978,
     AUTHOR = "Donald M. Topkis",
     TITLE = "Minimizing a Submodular Function on a Lattice",
     JOURNAL = "Operations Research",
     YEAR = 1978,
     VOLUME = 26,
     PAGES = "305--321",          }

@ARTICLE{Topkis1979,
     AUTHOR = "Donald M. Topkis",
     TITLE = "Equilibrium Points in Nonzero-Sum n-Person Submodular
          Games",
     JOURNAL = "SIAM Journal of Control and Optimization",
     YEAR = 1979,
     VOLUME = 17,
     PAGES = "773--787",              }

@BOOK{vanDamme1987,
     AUTHOR = "Eric van Damme",
     TITLE = "Stability and Perfection of {N}ash Equilibria",
     PUBLISHER = "Springer-Verlag",
     ADDRESS = "Berlin",
     YEAR = 1987,                             }

@ARTICLE{vanDamme1984,
     AUTHOR = "Eric van Damme",
     TITLE = "A Relation between Perfect Equilibria in Extensive
          Form Games and Proper Equilibria in Normal Form Games",
     JOURNAL = "International Journal of Game Theory",
     YEAR = 1984,
     VOLUME = 13,
     PAGES = "1--13",              }

@TECHREPORT{vanDamme1987fi,
     AUTHOR = "Eric van Damme",
     TITLE = "Stable Equilibria and Forward Induction",
     INSTITUTION = "University of Bonn",
     YEAR = 1987,
     TYPE  = "{SFB} Discussion Paper {A}-128",
     NUMBER = 128,                  }

@ARTICLE{vanDamme1989,
     AUTHOR = "Eric van Damme",
     TITLE = "Stable Equilibria and Forward Induction",
     JOURNAL = "Journal of Economic Theory",
     YEAR = 1989,
     VOLUME = 48,
     PAGES = "476-509",               }

@TECHREPORT{vanDamme1991,
           AUTHOR = "Eric van Damme",
           TITLE = "Refinements of {N}ash Equilibrium",
           INSTITUTION = "Cent{ER} for Economics Research",
           YEAR = 1991,
           TYPE = "Discussion Paper",
           NUMBER = 9107,                 }

@TECHREPORT{Redondo1991,
     AUTHOR = "Fernando Vega-Redondo",
     TITLE = "Competitioin and Culture in the Evolution of Economic
               Behavior",
     INSTITUTION = "University of Alicante",
     YEAR = 1991,
     TYPE = "Mimeo",     }

@ARTICLE{vanHuyckBattalioandBeil1990,
     AUTHOR = "J. {van Huyck} and R. Battalio and R. Beil",
     TITLE = "Tacit Coordination Games, Strategic Uncertainty, and Coordination
Failure",
     JOURNAL = "American Economic Review",
     YEAR = 1990,
     VOLUME = 80,
     PAGES = "234--248",          }

@ARTICLE{Vives1990,
     AUTHOR = "Xavier Vives",
     TITLE = "Nash Equilibrium with Strategic Complementarities",
     JOURNAL = "Journal of Mathematical Economics",
     YEAR = 1990,
     VOLUME = 19,
     PAGES = "305--321",          }

@TECHREPORT{Warneryd1990,
     AUTHOR = "{W\"{a}rneryd}, Karl",
     TITLE = "Cheap Talk, Coordination, and Evolutionary
          Stability",
     INSTITUTION = "Stockholm School of Economics",
     YEAR = 1990,
     TYPE = "Mimeo",                     }

@ARTICLE{Wilson1977,
     AUTHOR ="Charles Wilson",
     TITLE = "A Model of Insurance Markets with Incomplete Information",
     JOURNAL = "Journal of Economic Theory",
     YEAR = 1977,
     VOLUME = 16,
     PAGES = "167--207",           }

@ARTICLE{Young1992,
     AUTHOR = "Peyton Young",
     TITLE = "The Evolution of Conventions", 
      JOURNAL = "Econometrica", 
     YEAR = "1993",
    VOLUME = 61,
           PAGES = "57--84",                }

@ARTICLE{YoungandFoster1990,
     AUTHOR = "Peyton Young and Dean Foster",
     TITLE = "Cooperation in the Short and in the Long Run",
     JOURNAL = "Games and Economic Behavior",
     YEAR = 1990,
     VOLUME = 3,
     PAGES = "145--156",                }
%%%%%%%%%%%%%%%%%%kenbib.tex%%%%%%%%%%%%%%%%%%%%%%%
%A 
 
%Abreu and Rubinstein 
 
@ARTICLE{AandR88, 
      AUTHOR="D. Abreu and A. Rubinstein", 
       TITLE="The Structure of {N}ash Equilibrium in Repeated Games 
with Finite Automata", 
     JOURNAL="Econometrica", 
        YEAR="1988", 
      VOLUME="56", 
       PAGES="1259-1282"} 
 
 
%Albin 
 
@ARTICLE{Albin82, 
    AUTHOR="Albin, P. S.", 
     TITLE="The Metalogic of Economic Predictions{,} Calculations 
and Propositions", 
   JOURNAL="Mathematical Social Sciences", 
      YEAR=1982, 
    VOLUME=4, 
     PAGES="329-358"} 
 
%Allais 
 
@ARTICLE{Allais53, 
    AUTHOR="Allais, M.", 
     TITLE="Le comportement de l'homme rationnel devant le risque: 
critiqye des postulants et axiomes de l'ecole {A}mericaine", 
   JOURNAL="Econometrica", 
      YEAR=1953, 
    VOLUME=21, 
     PAGES="503-546"} 
 
%Anderlini 
 
@BOOKLET{Anderlini88, 
    AUTHOR="Anderlini, L.", 
     TITLE="Some Notes on {C}hurch's Thesis and the Theory of 
Games", 
HOWPUBLISHED="Cambridge Economic Theory Discussion Paper", 
      YEAR=1988, 
    NUMBER="126"} 
 
@UNPUBLISHED{Anderlini90, 
      AUTHOR="Anderlini, L.", 
       TITLE="Communication{,} Computability and Common Interest 
Games", 
        YEAR="1990", 
        NOTE="Working Paper{,} St. John's College{,} Cambridge"} 
 
%Anscombe and Aumann 
 
@ARTICLE{AandA, 
    AUTHOR="Anscombe, F. and Aumann, R.", 
     TITLE="A Definition of Subjective Probability", 
   JOURNAL="Annals of Mathematical Statistics", 
      YEAR=1963, 
    VOLUME=34, 
     PAGES="199-205"} 
 
%Arbib 
 
@BOOK{Arbib87, 
      AUTHOR="Arbib, M.", 
       TITLE="Brains{,} Machines and Mathematics", 
   PUBLISHER="Springer-Verlag", 
     ADDRESS="Berlin", 
        YEAR="1987"} 
 
@BOOK{Arbib85, 
      AUTHOR="Arbib, M.", 
       TITLE="In Search of the Person", 
   PUBLISHER="University of Massachusetts Press", 
     ADDRESS="Amherst", 
        YEAR="1985"} 
 
@BOOK{Arbib69, 
      AUTHOR="Arbib, M.", 
       TITLE="Theories of Abstract Automata", 
   PUBLISHER="Prentice-Hall", 
     ADDRESS="Englewood Cliffs, NJ", 
        YEAR="1969"} 
 
 
 
%Aristotle 
 
@BOOK{Aristotle63, 
      AUTHOR="Aristotle", 
       TITLE="The Philosophy of Aristotle", 
   PUBLISHER="Mentor Books", 
     ADDRESS="New York", 
        YEAR="1963", 
        NOTE={(Translated by  R. Brambrough.)}} 
 
%Arrow 
 
@BOOK{Arrow63, 
    AUTHOR="Arrow, K.", 
     TITLE="Social Choice and Individual Values", 
 PUBLISHER="Yale University Press", 
   ADDRESS="New Haven", 
       YEAR=1963} 
 
@ARTICLE{Arrow73, 
    AUTHOR="Arrow, K.", 
     TITLE="Rawls' Principle of Just Savings", 
   JOURNAL="Swedish Journal of Economics", 
      YEAR=1973, 
    VOLUME=75, 
     PAGES="323-335"} 
 
 
@ARTICLE{Arrow78, 
    AUTHOR="Arrow, K.", 
     TITLE="Extended Sympathy and the Problem of Social Choice", 
   JOURNAL="Philosophia", 
      YEAR=1978, 
    VOLUME=7, 
     PAGES="233-237"} 
 
%Artemov 
 
@INCOLLECTION{Artemov90, 
      AUTHOR="Artemov, S.", 
       TITLE="Kolmogorov's logic of problems and a provability 
interpretation of intuitionistic logic", 
   BOOKTITLE="Theoretical Aspects of Reasoning About Knowledge", 
      EDITOR="R. Parikh", 
   PUBLISHER="Morgan Kaufmann", 
     ADDRESS="San Mateo, CA", 
        YEAR="1990"} 
 
%Aubrey 
 
@BOOK{Aubrey, 
    AUTHOR="Aubrey, J.", 
     TITLE="Brief Lives", 
 PUBLISHER="Penguin Books", 
   ADDRESS="Harmondsworth, Middlesex, UK", 
       YEAR=1962, 
       NOTE="(Edited by O. Lawson.)"} 
 
%Aumann 
 
@ARTICLE{Aumann76, 
     AUTHOR="Aumann, R.", 
      TITLE="Agreeing to Disagree", 
    JOURNAL="The Annals of Statistics", 
       YEAR=1976, 
     VOLUME=4, 
      PAGES="1236-1239"} 
 
@INCOLLECTION{Aumann81, 
     AUTHOR="Aumann, R.", 
      TITLE="Survey of Repeated Games", 
  BOOKTITLE="Essays in Game Theory and Mathematical Economics in 
Honor of Oskar Morgenstern", 
  PUBLISHER={Wissenschaftsverlag, Bibliographisches Institut}, 
    ADDRESS={Mannheim/Wien/Zurich}, 
       YEAR=1981} 
 
@INCOLLECTION{Aumann85, 
      AUTHOR="Aumann, R.", 
       TITLE="What is Game Theory Trying to Accomplish?", 
   BOOKTITLE="Frontiers of Economics", 
     EDITOR="K. Arrow and S. Honkapohja", 
      PAGES="28-88", 
   PUBLISHER="Basil Blackwell", 
     ADDRESS="Oxford", 
        YEAR="1985"} 
 
 
 
@ARTICLE{Aumann87b, 
    AUTHOR="Aumann, R.", 
     TITLE="Correlated Equilibrium as an Expression of {B}ayesian 
Rationality", 
   JOURNAL="Econometrica", 
      YEAR=1987, 
    VOLUME=55, 
     PAGES="1-18"} 
 
@UNPUBLISHED{Aumann88, 
    AUTHOR="Aumann, R.", 
     TITLE="{I}rrationality in game theory (preliminary notes)", 
      NOTE={Hebrew University of Jerusalem"}, 
      YEAR=1988} 
 
@UNPUBLISHED{Aumann89a, 
      AUTHOR="Aumann, R.", 
       TITLE="Irrationality in Game Theory", 
        YEAR="1989", 
        NOTE="Working Paper{,} Hebrew University"} 
 
@UNPUBLISHED{Aumann89b, 
      AUTHOR="Aumann, R.", 
       TITLE="Interactive Epistemology", 
        YEAR="1989", 
        NOTE="Working Paper{,} Cowles Foundation{,} Yale 
University"} 
 
@INCOLLECTION{Aumann90a, 
     AUTHOR="Aumann, R.", 
  BOOKTITLE="Economic Decision-Making: Econometrics and 
Optimization", 
      TITLE="Nash Equilibria are not Self-enforcing", 
     EDITOR="J.-J. Gabsewicz, J.-F. Richard and L. Wolsley", 
  PUBLISHER={Elsevier Science Publishers}, 
    ADDRESS={Amsterdam}, 
       YEAR=1990, 
       NOTE="(Essays in honor of Jacques Dreze.)"} 
 
 
@UNPUBLISHED{Aumann93, 
      AUTHOR="Aumann, R.", 
       TITLE="Backwards Induction and Common Knowledge of Rationality", 
        YEAR="1993", 
        NOTE="Paper presented at the Nobel Symposium on Game 
Theory{,} to appear in {\em Games and Economic Behavior}"} 
 
 
%Aumann and Brandenburger 
 
 
@UNPUBLISHED{AandB90, 
      AUTHOR="Aumann, R. and A. Brandenburger", 
       TITLE="Epistemic Conditions for {N}ash Equilibrium", 
        YEAR="1990", 
        NOTE="Working Paper{,} Harvard Business School"} 
 
%Aumann and Sorin 
 
@ARTICLE{AandS89, 
      AUTHOR="Aumann, R. and S. Sorin", 
       TITLE="Cooperation and Bounded Recall", 
     JOURNAL="Games and Economic Behavior", 
        YEAR="1989", 
      VOLUME="1", 
       PAGES="5-39"} 
 
 
 
%Axelrod 
 
@BOOK{Axelrod84, 
    AUTHOR="Axelrod, R.", 
    TITLE="The Evolution of Cooperation", 
   PUBLISHER={Basic Books}, 
     ADDRESS={New York}, 
        YEAR=1984} 
 
@INCOLLECTION{Axelrod87, 
      AUTHOR="Axelrod, R.", 
       TITLE="The Evolution of Strategies in the Iterated 
Prisoners' 
Dilemma", 
   BOOKTITLE="Genetic Algorithms and Simulated Annealing", 
      EDITOR="L. Davis", 
   PUBLISHER="Morgan Kaufmann", 
     ADDRESS="Los Altos, Calif.", 
        YEAR="1987"} 
 
%Axelrod and Hamilton 
 
@ARTICLE{AandH81, 
      AUTHOR="Axelrod, R. and Hamilton, W.", 
       TITLE="The Evolution of Cooperation", 
     JOURNAL="Science", 
        YEAR="1981", 
      VOLUME="211", 
       PAGES="1390-1396"} 
 
%B 
 
%Bacharach 
 
@ARTICLE{Bacharach85, 
    AUTHOR="Bacharach, M.", 
     TITLE="{S}ome Extensions to a Claim of {A}umann in an 
Axiomatic Model of Knowledge", 
   JOURNAL="Journal of Economic Theory", 
      YEAR=1985, 
    VOLUME=37, 
     PAGES="167-190"} 
 
@UNPUBLISHED{Bacharach87a, 
     AUTHOR="Bacharach, M.", 
      TITLE="{W}hen Do We Have Information Partitions?", 
       NOTE="Working Paper{,} Oxford University", 
       YEAR="1987"} 
 
@ARTICLE{Bacharach87b, 
    AUTHOR="Bacharach, M.", 
     TITLE="A Theory of Rational Decision in Games", 
   JOURNAL="Erkerntnis", 
      YEAR=1987, 
    VOLUME=27, 
     PAGES="17-55"} 
 
%Bailey, Olson and Wonnacott 
 
@ARTICLE{BOandW, 
    AUTHOR="Bailey, M. and M. Olson and P. Wonnacott", 
     TITLE="The Marginal Utility of Income does not increase: 
Borrowing, 
Lending and Friedman-Savage Gambles", 
   JOURNAL="American Economic Review", 
      YEAR=1980, 
    VOLUME=70, 
     PAGES="372-379"} 
 
%Banks and Sundaram 
 
@ARTICLE{BandS90, 
      AUTHOR="Banks, J. and  R. Sundaram", 
       TITLE="Repeated Games{,} Finite Automata and Complexity", 
        YEAR="1990", 
      VOLUME=2, 
     JOURNAL="Games and Economic Behavior", 
       PAGES="97-117"} 
 
%Barfield 
 
@BOOK{Barfield, 
    AUTHOR="Barfield, O.", 
    TITLE="This Ever-Diverse Pair", 
   PUBLISHER={Floris Classics}, 
     ADDRESS={Edinburgh}, 
        YEAR=1985} 
 
 
%Barry 
 
@BOOK{Barry89, 
    AUTHOR="Barry, B.", 
    TITLE="Theories of Justice", 
   PUBLISHER={University of California Press}, 
     ADDRESS={Berkeley and Los Angeles}, 
        YEAR=1989} 
 
%Basu 
 
@ARTICLE{Basu90, 
    AUTHOR="Basu, K.", 
     TITLE="On the Non-Existence of a Rationality Definition for 
Extensive 
Games", 
   JOURNAL="International Journal of Game Theory", 
      YEAR=1990, 
    VOLUME="19-1", 
     PAGES="33-44"} 
 
@UNPUBLISHED{Basu85, 
      AUTHOR="Basu, K.", 
       TITLE="Strategic Irrationality in Games", 
        YEAR="1985", 
        NOTE="Working Paper{,} Institute of Advanced Study{,} 
Princeton University"} 
 
 
%Bentham 
 
@INCOLLECTION{Bentham, 
     AUTHOR="Bentham, J.", 
  BOOKTITLE="Utilitarianism and Other Essays", 
      TITLE="An Introduction to the Principles of Morals and 
Legislation", 
  PUBLISHER={Penguin Books}, 
    ADDRESS={Harmondsworth, Middlesex, U.K.}, 
       YEAR=1987, 
       NOTE="(Introduction by A. Ryan{,} essay first published 
1789)"} 
 
 
 
%Ben-Porath 
 
@UNPUBLISHED{Ben-Porath86, 
      AUTHOR="Ben-Porath, E.", 
       TITLE="Repeated Games with Finite Automata", 
        YEAR="1986", 
        NOTE="Working Paper{,} Hebrew University"} 
 
%Bergstrom 
 
@ARTICLE{Bergstrom89, 
    AUTHOR="Bergstrom, T.", 
     TITLE="Love and Spaghetti, The Opportunity Cost of Virtue", 
   JOURNAL="Journal of Economic Perspectives", 
      YEAR=1989, 
    VOLUME=3, 
     PAGES="165-173"} 
 
%Berkeley 
 
@BOOK{Berkeley13, 
     AUTHOR="Berkeley, G.", 
      TITLE="Three Dialogues Between Hylas and Philonus", 
  PUBLISHER={Bobbs-Merrill}, 
    ADDRESS={Indianapolis}, 
       YEAR=1954, 
       NOTE="(Edited by C. Turbayne; first published 1713)"} 
 
 
%Berlekamp, Conway and Guy 
 
@BOOK{BCandG, 
     AUTHOR="Berlekamp, E., J. Conway and R. Guy", 
      TITLE="Winning Ways for your Mathematical Plays", 
  PUBLISHER={Academic Press}, 
    ADDRESS={New York}, 
       YEAR=1982} 
 
%Bernheim 
 
@ARTICLE{Bernheim84, 
    AUTHOR="Bernheim, D.", 
     TITLE="Rationalizable Strategic Behavior", 
   JOURNAL="Econometrica", 
      YEAR=1984, 
    VOLUME=52, 
     PAGES="1007-1028"} 
 
@ARTICLE{Bernheim86, 
    AUTHOR="Bernheim, B. D.", 
     TITLE="Axiomatic Characterizations of Rational Choice in 
Strategic 
Environments", 
   JOURNAL="Scandinavian Journal of Economics", 
      YEAR=1986, 
    VOLUME=88, 
     PAGES="473-488"} 
 
%Bianchi 
 
@UNPUBLISHED{Bianchi90, 
    AUTHOR="Bianchi, M.", 
     TITLE="How to Learn Sociality: {T}rue and False Solutions to 
{M}andeville's 
Problem", 
      NOTE="University of Rome Paper", 
      YEAR=1990} 
 
%Bicchieri 
 
@INCOLLECTION{Bicchieri88, 
      AUTHOR="Bicchieri, C.", 
       TITLE="Common Knowledge and Backwards Induction:  A Solution 
to the Paradox", 
   BOOKTITLE="Theoretical Aspects of Reasoning About Knowledge", 
     EDITOR="M. Vardi", 
   PUBLISHER="Morgan Kaufmann", 
     ADDRESS="Los Altos", 
        YEAR="1988"} 
 
 
@ARTICLE{Bicchieri89, 
    AUTHOR="Bicchieri, C.", 
     TITLE="Strategic Behavior and Counterfactuals", 
   JOURNAL="Erkerntnis", 
      YEAR=1989, 
    VOLUME=30, 
     PAGES="69-85"} 
 
%Bicchieri and Gilboa 
 
@UNPUBLISHED{BandG92, 
      AUTHOR="Bicchieri, C. and I. Gilboa", 
       TITLE="Etude for: {C}an Free Choice be Known", 
        YEAR="1992", 
        NOTE="Working Paper{,} Philosophy Department, 
Carnegie-Mellon University."} 
 
%Binmore 
 
@ARTICLE{Binmore75, 
    AUTHOR="Binmore, K.", 
     TITLE="An Example in Group Preference", 
   JOURNAL="Journal of Economic Theory", 
      YEAR=1975, 
    VOLUME=10, 
     PAGES="377-385"} 
 
 
@ARTICLE{Binmore87c, 
    AUTHOR="Binmore, K.", 
     TITLE="Experimental Economics", 
   JOURNAL="European Economic Review", 
      YEAR=1987, 
    VOLUME=31, 
     PAGES="257-264"} 
 
@TECHREPORT{Binmore84, 
    AUTHOR="Binmore, K.", 
     TITLE="Game Theory and the Social Contract {I}", 
INSTITUTION="London School of Economics", 
      YEAR=1984, 
    NUMBER="ST/ICERD 84/108"} 
 
@ARTICLE{Binmore84a, 
    AUTHOR="Binmore, K.", 
     TITLE="Equilibria in Extensive Games", 
   JOURNAL="Economic Journal", 
      YEAR=1984, 
    VOLUME= 95, 
     PAGES="51-59"} 
 
 
@ARTICLE{Binmore87, 
    AUTHOR="Binmore, K.", 
     TITLE="Modeling Rational Players{,} {I} and {II}", 
   JOURNAL="Economics and Philosophy", 
      YEAR=1987, 
    VOLUME="3 and 4", 
     PAGES="179-214 and 9-55"} 
 
@ARTICLE{Binmore87a, 
      AUTHOR="Binmore, K.", 
       TITLE="Modeling Rational Players {I}", 
     JOURNAL="Economics and Philosophy", 
        YEAR="1987", 
      VOLUME="3", 
       PAGES="9-55"} 
 
@ARTICLE{Binmore87b, 
      AUTHOR="Binmore, K.", 
       TITLE="Modeling Rational Players {II}", 
     JOURNAL="Economics and Philosophy", 
        YEAR="1987", 
      VOLUME="4", 
       PAGES="179-214"} 
 
 
@TECHREPORT{Binmore88a, 
    AUTHOR="Binmore, K.", 
     TITLE="Game Theory and the Social Contract {II}", 
INSTITUTION="London School of Economics", 
      YEAR="1988", 
    NUMBER="ST/ICERD 88/170", 
    NOTE="(supersedes part I)"} 
 
@TECHREPORT{Binmore88b, 
    AUTHOR="Binmore, K.", 
     TITLE="Social Contract {II}: {G}authier and {N}ash", 
INSTITUTION="University of Michigan", 
      YEAR=1988, 
    NUMBER="CREST 89-04"} 
 
@TECHREPORT{Binmore88c, 
    AUTHOR="Binmore, K.", 
     TITLE="Social Contract {III}: {E}volution and Utilitarianism", 
INSTITUTION="University of Michigan", 
      YEAR=1988, 
    NUMBER="CREST 89-05", 
    NOTE="(forthcoming in {\it Constitutional Political 
Economy\/})"} 
 
 
@ARTICLE{Binmore89a, 
    AUTHOR="Binmore, K.", 
     TITLE="Social Contract {I}: {H}arsanyi and {R}awls", 
   JOURNAL="Economic Journal", 
      YEAR=1989, 
    VOLUME=99, 
     PAGES="84-102"} 
 
@BOOK{Binmore90, 
      AUTHOR="Binmore, K.",  
       TITLE="Essays on Foundations of Game Theory", 
   PUBLISHER="Basil Blackwell", 
     ADDRESS="Oxford", 
        YEAR="1990"} 
 
@ARTICLE{Binmore90a, 
    AUTHOR="Binmore, K.", 
     TITLE="Social Contract {III}: {E}volution and Utilitarianism", 
   JOURNAL="Constitutional Political Economy", 
      YEAR=1990, 
    VOLUME=1, 
     PAGES="1-26"} 
 
 
@INPROCEEDINGS{Binmore92a, 
     AUTHOR="Binmore, K.", 
      TITLE="Foundations of Game Theory", 
   BOOKTITLE="Advances in Economic Theory", 
     EDITOR="J.-J. Laffont", 
   PUBLISHER="Cambridge University Press", 
     ADDRESS="Cambridge", 
        YEAR="1992", 
ORGANIZATION="Sixth World Congress of the Econometric Society"} 
 
 
@INCOLLECTION{Binmore91d, 
      AUTHOR="Binmore, K.", 
     TITLE="Social Contract {IV}: {E}volution and Convention", 
   BOOKTITLE="Rational Interaction: {E}ssays in Honor of John 
Harsanyi", 
     EDITOR="R. Selten", 
   PUBLISHER="Springer", 
     ADDRESS="Berlin", 
        YEAR="1992"} 
 
 
@BOOK{Binmore91e, 
      AUTHOR="Binmore, K.",  
       TITLE="Fun and Games", 
   PUBLISHER="D. C. Heath", 
     ADDRESS="Lexington, Mass.", 
        YEAR="1991"} 
 
 
@INCOLLECTION{Binmore92g, 
     AUTHOR="Binmore, K.", 
      TITLE="Social Contract {IV}: {C}onvention and Evolution", 
  BOOKTITLE="Rational Interaction: Essays in Honor of {J}ohn {C}.\ 
{H}arsanyi", 
  PUBLISHER="Springer-Verlag", 
    ADDRESS="Berlin", 
       YEAR=1992, 
     EDITOR="R. Selten"} 
 
@INCOLLECTION{Binmore92b, 
     AUTHOR="Binmore, K.", 
      TITLE="Bargaining and Morality: {S}ocial Contract {II}", 
  BOOKTITLE="Rationality, Justice and the Social Contract: Themes 
from 
`Morals by Agreement'", 
  PUBLISHER="Simon and Schuster", 
    ADDRESS="Hemel Hempstead, U.K.", 
       YEAR=1992, 
     EDITOR="R. Sugden"} 
 
@INCOLLECTION{Binmore91f, 
     AUTHOR="Binmore, K.", 
      TITLE="Game Theory and the Social Contract", 
  BOOKTITLE="Game Equilibrium Models {II}: Methods, Morals and 
Markets", 
  PUBLISHER="Springer-Verlag", 
    ADDRESS="Berlin", 
       YEAR=1991, 
     EDITOR="R. Selten"} 
 
@INCOLLECTION{Binmore92c, 
     AUTHOR="Binmore, K.", 
      TITLE="DeBayesing Game Theory", 
  BOOKTITLE="Studies in Logic and the Foundations of Game Theory: 
Proceedings of the Ninth International Congress of 
Logic, Methodology and the Philosophy of Science", 
  PUBLISHER="Kluwer", 
    ADDRESS="Dordrecht", 
       YEAR=1992, 
     EDITOR="B. Skyrms"} 
 
 
@BOOK{Binmore93, 
      AUTHOR="Binmore, K.",  
       TITLE="Playing Fair: Game Theory and the Social Contract I", 
   PUBLISHER="MIT Press", 
     ADDRESS="Cambridge, Mass.", 
        YEAR="1993", 
        NOTE="forthcoming"} 
 
%Binmore and Brandenburger 
 
@INCOLLECTION{BandB, 
     AUTHOR="Binmore, K. and A. Brandenburger", 
      TITLE="Common Knowledge and Game Theory", 
  BOOKTITLE="Essays on the Foundations of Game Theory", 
  PUBLISHER="Blackwell", 
    ADDRESS="Oxford", 
       YEAR=1990, 
     EDITOR="K. Binmore"} 
 
 
@INCOLLECTION{BandB90, 
      AUTHOR="Binmore, K. and A. Brandenburger", 
       TITLE="Common Knowledge and Game Theory", 
   BOOKTITLE="Essays on Foundations of Game Theory", 
   PUBLISHER="Basil Blackwell", 
     ADDRESS="Oxford", 
      EDITOR="{K}. {G}. {B}inmore", 
        YEAR="1990"} 
 
 
%Binmore and Dasgupta 
 
@BOOK{BandDb, 
    AUTHOR="Binmore, K. and Dasgupta, P.", 
     TITLE="The Economics of Bargaining", 
 PUBLISHER="Basil Blackwell", 
   ADDRESS="Oxford", 
       YEAR=1987} 
 
 
 
%Binmore, Morgan, Shaked and Sutton 
 
@ARTICLE{BMSandS, 
    AUTHOR="Binmore, K. and Morgan, P. and Shaked, A. and Sutton, 
J.", 
     TITLE="Do People Exploit their Bargaining Power? {A}n 
Experimental Study", 
   JOURNAL="Games and Economic Behavior", 
      YEAR=1991, 
    VOLUME=3, 
     PAGES="295-322"} 
 
%Binmore and Samuelson 
 
@ARTICLE{BandS90a, 
    AUTHOR="Binmore, K. and L. Samuelson", 
     TITLE="Evolutionary Stability in Repeated Games Played by 
Finite Automata", 
   JOURNAL="Journal of Economic Theory", 
      YEAR=1992, 
    VOLUME=57, 
     PAGES="278-305"} 
 
@UNPUBLISHED{BandS93, 
    AUTHOR="Binmore, K. and L. Samuelson", 
     TITLE="Muddling Through: Noisy Equilibrium Selection", 
      NOTE="University of Wisconsin Discussion Paper", 
      YEAR="1993"} 
 
@ARTICLE{BandS93a, 
    AUTHOR="Binmore, K. and L. Samuelson", 
     TITLE="An Economist's Perspective on the Evolution of Norms", 
      JOURNAL = "Journal of Institutional 
                      and Theoretical Economics", 
      YEAR=1993,
           NOTE = "Forthcoming", } 
 
%Binmore, Gale and Samuelson 
 
@UNPUBLISHED{BGandS93, 
    AUTHOR="Binmore, K. and J. Gale and L. Samuelson.", 
     TITLE="Learning and Subgame-Perfection", 
      NOTE="University of Wisconsin Discussion Paper", 
      YEAR="1993"} 
 
 
%Binmore, Shaked and Sutton 
 
@ARTICLE{BSandSa, 
    AUTHOR="Binmore, K. and Shaked, A. and Sutton, J.", 
     TITLE="Testing Noncooperative Game Theory: {A} Preliminary 
Study",  
   JOURNAL="American Economic Review", 
      YEAR=1985, 
    VOLUME=75, 
     PAGES="1178-1180"} 
 
@ARTICLE{BSandSb, 
    AUTHOR="Binmore, K. and Shaked, A. and Sutton, J.", 
     TITLE="Noncooperative Bargaining Theory: a Further Test: 
Reply",  
   JOURNAL="American Economic Review", 
      YEAR=1988, 
    VOLUME=78, 
     PAGES="837-839"} 
 
@ARTICLE{BSandS, 
    AUTHOR="Binmore, K. and Shaked, A. and Sutton, J.", 
     TITLE="An Outside Option Experiment",  
   JOURNAL="Quarterly Journal of Economics", 
      YEAR=1989, 
    VOLUME=104, 
     PAGES="753-770"} 
 
%Binmore and Shin 
 
@INCOLLECTION{BandShin90, 
    AUTHOR="Binmore, K. and H. Shin", 
     TITLE="Algorithmic Knowledge and Game Theory", 
   BOOKTITLE="Knowledge, Belief and Strategic Interaction", 
 PUBLISHER="Cambridge University Press", 
    ADDRESS="Cambridge", 
       YEAR=1992, 
     EDITOR="C. Bicchieri and M. Chiara"} 
 
%Binmore, Swierzsbinski, Hsu and Proulx 
 
@UNPUBLISHED{BSHandP92, 
    AUTHOR="Binmore, K. and J. Swierzsbinski and S. Hsu and C. 
Proulx", 
     TITLE="Focal Points and Bargaining", 
      NOTE="University of Michigan Discussion Paper", 
      YEAR="1992"} 
 
%Bishop 
 
@ARTICLE{Bishop63, 
    AUTHOR="Bishop, R.", 
     TITLE="Game-theoretic Analyses of Bargaining", 
   JOURNAL="Quarterly Journal of Economics", 
      YEAR=1963, 
    VOLUME=77, 
     PAGES="559-602"} 
 
%Blackwell and Girshick 
 
@BOOK{BandG, 
    AUTHOR="Blackwell, D. and M. Girshick", 
     TITLE="Theory of Games and Statistical Decisions", 
 PUBLISHER="Dover", 
   ADDRESS="New York", 
      YEAR=1979, 
      NOTE="({F}irst published by {W}iley, 1954)"} 
 
%Blume, Brandenburger and Dekel 
 
@UNPUBLISHED{BBandD89a, 
      AUTHOR="Blume, L. and A. Brandenburger and E. Dekel", 
       TITLE="Lexicographic Probabilities and Choice Under 
Uncertainty", 
        YEAR="1989", 
        NOTE="Working Paper{,} Harvard Business School"} 
 
@UNPUBLISHED{BBandD89b, 
      AUTHOR="Blume, L. and A. Brandenburger and E. Dekel", 
       TITLE="Equilibrium Refinements and Lexicographic 
Probabilities", 
        YEAR="1989", 
        NOTE="Working Paper{,} Harvard Business School"} 
 
 
%Blume and Easley 
 
@UNPUBLISHED{BandE90, 
      AUTHOR="Blume, L. and D. Easley", 
       TITLE="Bayesian Learning in Repeated Games: An Example 
of Non-Convergence to {N}ash Equilibrium", 
        YEAR="1990", 
        NOTE="Working Paper{,} Cornell University"} 
 
 
%Bolton 
 
@ARTICLE{Bolton91, 
    AUTHOR="Bolton, G.", 
     TITLE="A Comparative Model of Bargaining: Theory and 
Evidence", 
   JOURNAL="American Economic Review", 
      YEAR=1991, 
    VOLUME=81, 
     PAGES="1096-1136"} 
 
 
@TECHREPORT{BandZ93, 
           AUTHOR = "Bolton, G. and R. Zwick", 
           TITLE = "Anonymity Versus Punishment in Ultimatum Bargaining", 
           INSTITUTION = "Pennsylvania State University", 
           YEAR = 1993, 
           TYPE = "Mimeo"} 
 
 
%Bonnano 
 
@UNPUBLISHED{Bonnano, 
      AUTHOR="Bonnano, G.", 
       TITLE="A Set-Theoretic Approach to Noncooperative Games {I} 
and {II}", 
        YEAR="1988", 
        NOTE="Working Paper No. 330{,} {UC} Davis"} 
 
 
%Boolos 
 
@BOOK{Boolos, 
     AUTHOR="Boolos, G.", 
      TITLE="The Unprobability of Consistency", 
  PUBLISHER={Cambridge University Press}, 
    ADDRESS={Cambridge}, 
       YEAR=1979} 
 
 
%Boolos and Jeffrey 
 
@BOOK{BandJ, 
     AUTHOR="Boolos, G. and R. Jeffrey", 
      TITLE="Computability and Logic", 
  PUBLISHER={Cambridge University Press}, 
    ADDRESS={Cambridge}, 
       YEAR=1974} 
 
%Borgers and Sarin 
 
@TECHREPORT{BorgersandS93, 
    AUTHOR="{B\"{o}rgers}, T. and R. Sarin", 
     TITLE="Learning Through Reinforcement and Replicator Dynamics", 
INSTITUTION="University College London", 
      YEAR=1993} 
 
%Boyd and Loberbaum 
 
@ARTICLE{BandL87, 
    AUTHOR="Boyd, R. and J. Loberbaum", 
     TITLE="No Pure Strategy is Evolutionarily Stable in the 
Repeated {P}risoners{'} {D}ilemma Game", 
   JOURNAL="Nature", 
      YEAR=1987, 
    VOLUME=327, 
     PAGES="58-59"} 
 
%Boyd and Richerson 
 
@BOOK{BandR85, 
      AUTHOR="Boyd, R. and P. Richerson", 
       TITLE="Culture and the Evolutionary Process", 
   PUBLISHER="University of Chicago Press", 
     ADDRESS="Chicago", 
        YEAR="1985"} 
 
 
 
%Braithwaite 
 
@BOOK{Braithwaite, 
     AUTHOR="Braithwaite, R.", 
      TITLE="The Theory of Games as a Tool for the Moral 
Philosopher", 
  PUBLISHER={Cambridge University Press}, 
    ADDRESS={Cambridge}, 
       YEAR=1955} 
 
%Brams 
 
@ARTICLE{Brams75, 
    AUTHOR="Brams, S.", 
     TITLE="Newcomb's Problem and {P}risoners' {D}ilemma", 
   JOURNAL="Journal of Conflict Resolution", 
      YEAR=1975, 
    VOLUME=19, 
     PAGES="592-612"} 
 
@BOOK{Brams83, 
     AUTHOR="Brams, S.", 
      TITLE="Superior Beings", 
  PUBLISHER={Springer-Verlag}, 
    ADDRESS={New York}, 
       YEAR=1983} 
 
%Brandenburger 
 
@INCOLLECTION{Brandenburger, 
     AUTHOR="Brandenburger, A.", 
      TITLE="The Economics of Missing Markets{,} Information and Games", 
    CHAPTER="The Role of Common Knowledge Assumption in Game Theory", 
  PUBLISHER="Oxford University Press", 
    ADDRESS="London", 
       YEAR=1989, 
     EDITOR="F. Hahn"} 
 
%Brandenburger and Dekel 
 
@TECHREPORT{BandD85, 
    AUTHOR="Brandenburger, A. and E. Dekel", 
     TITLE="Hierarchies of Beliefs and Common Knowledge", 
INSTITUTION="Graduate School of Business, Stanford University", 
      YEAR=1985, 
    NUMBER="Research Paper No. 841"} 
 
@ARTICLE{BandD87a, 
    AUTHOR="Brandenburger, A. and E. Dekel", 
     TITLE="Common Knowledge with Probability 1", 
   JOURNAL="Journal of Mathematical Economics", 
      YEAR=1987, 
    VOLUME=16, 
     PAGES="237-245"} 
 
 
 
@ARTICLE{BandD87b, 
    AUTHOR="Brandenburger, A. and E. Dekel", 
     TITLE="Rationalizability and Correlated Equilibria", 
   JOURNAL="Econometrica", 
      YEAR=1987, 
    VOLUME=55, 
     PAGES="1391-1402"} 
 
@UNPUBLISHED{BandD90, 
      AUTHOR="Brandenburger, A. and E. Dekel", 
       TITLE="Heirarchies of Beliefs and Common Knowledge", 
        YEAR="1990", 
        NOTE="Working Paper{,} Harvard Business School"} 
 
 
 
%Brandenburger, Dekel and Geanakoplos 
 
@UNPUBLISHED{BDandG90, 
      AUTHOR="Brandenburger, A. and E. Dekel and J. Geanakoplos", 
       TITLE="Correlated Equilibrium with Generalized Information 
Structures", 
        YEAR="1989", 
        NOTE="Harvard Business School Discussion Paper"} 
 
 
%Brandenburger and Geanakoplos 
 
@UNPUBLISHED{BandG, 
     AUTHOR="Brandenburger, A. and J. Geanakoplos", 
      TITLE="Common Knowledge of Summary Statistics", 
       YEAR="1988", 
     NUMBER="Cowles Foundation Discussion Paper No. 864{,} Yale 
University"} 
 
%Brock, Marimon, Rust and Sargent 
 
@UNPUBLISHED{BMRandS, 
     AUTHOR="Brock, W. and R. Marimon and J. Rust and T. Sargent", 
      TITLE="Informationally Decentralized Learning Algorithms for 
Finite-Player, Finite-Action Games of Incomplete Information", 
       YEAR="1988", 
       NOTE="Working paper{,} University of Wisconsin"} 
 
%Broome 
 
@BOOK{Broome91, 
     AUTHOR="Broome, J.", 
      TITLE="Weighing Goods", 
  PUBLISHER={Basil Blackwell}, 
    ADDRESS={Oxford}, 
       YEAR=1991} 
 
%Buchanan 
 
@BOOK{Buchanan75, 
     AUTHOR="Buchanan, J.", 
      TITLE="The Limits of Liberty", 
  PUBLISHER={University of Chicago Press}, 
    ADDRESS={Chicago}, 
       YEAR=1975} 
 
@ARTICLE{Buchanan76, 
    AUTHOR="Buchanan, J.", 
     TITLE="A {H}obbsian Interpretation of the {R}awlsian 
Difference Principle", 
   JOURNAL="Kyklos", 
      YEAR=1976, 
    VOLUME=29, 
     PAGES="5-25"} 
 
@ARTICLE{Buchanan87, 
    AUTHOR="Buchanan, J.", 
     TITLE="Towards the Simple Economics of Natural Liberty", 
   JOURNAL="Kyklos", 
      YEAR=1987, 
    VOLUME=40} 
 
@BOOK{Burton51, 
     AUTHOR="Burton, R.", 
      TITLE="The Anatomy of Melancholy", 
  PUBLISHER={Tudor}, 
    ADDRESS={New York}, 
       YEAR=1927, 
      NOTES="(Edited by F. Dill and P. Jordan Smith; 6th edition, 
first published in 1651.)"} 
 
%C 
 
 
%Cabrales 
 
@UNPUBLISHED{Cabrales93, 
      AUTHOR="Cabrales, A.", 
       TITLE="Stochastic Replicator Dynamics", 
        YEAR="1993", 
        NOTE="Working Paper{,} University of California at San Diego"} 
 
%Calvo 
 
@ARTICLE{Calvo, 
    AUTHOR="Calvo, G.", 
     TITLE="Some Notes on Time Inconsistency and {R}awls' Maximin 
Principle", 
   JOURNAL="Review of Economic Studies", 
      YEAR=1978, 
    VOLUME=45, 
     PAGES="97-102"} 
 
%Canning 
 
@UNPUBLISHED{Canning87, 
    AUTHOR="Canning, D.", 
     TITLE="Convergence to Equilibrium in a Sequence of Gmaes with Learning", 
      NOTE={European University Institute working paper}, 
      YEAR="1987"} 
 
 
@UNPUBLISHED{Canning88, 
    AUTHOR="Canning, D.", 
     TITLE="Rationality and Game Theory when Players are Turing Machines", 
      NOTE={London School of Economics {ST/ICERD} Discussion Paper 
88/183}, 
      YEAR="1988"} 
 
%Cass 
 
@ARTICLE{CandS, 
    AUTHOR="Cass, D. and K. Shell", 
     TITLE="Do Sunspots Matter?", 
   JOURNAL="Journal of Political Economy", 
      YEAR=1983, 
    VOLUME=91, 
     PAGES="193-227"} 
 
%Carlsson and Van Damme 
 
 
@UNPUBLISHED{CandVD90, 
    AUTHOR="Carlsson, H. and E. Van Damme", 
     TITLE="Global Games and Equilibrium Selection", 
      NOTE={CenTER discussion paper 9052, Tilburg University}, 
      YEAR="1990"} 
 
@UNPUBLISHED{CandVD91, 
    AUTHOR="Carlsson, H. and E. Van Damme", 
     TITLE="Equilibrium Selection in {S}tag {H}unt Games", 
      NOTE={CenTER discussion paper, Tilburg University}, 
      YEAR="1991"} 
 
%Cave 
 
@ARTICLE{Cave, 
    AUTHOR="Cave, J.", 
     TITLE="Learning to Agree", 
   JOURNAL="Economic Letters", 
      YEAR="1983", 
    VOLUME="12", 
     PAGES="147-152"} 
 
%Chatterjee and Samuelson 
 
@ARTICLE{CandS83, 
    AUTHOR="Chatterjee, K. and Samuelson, W.", 
     TITLE="Bargaining under Incomplete Information", 
   JOURNAL="Operations Research", 
      YEAR=1983, 
    VOLUME=31, 
     PAGES="835-851"} 
 
%Coase 
 
@BOOK{Coase66, 
     AUTHOR="Coase, R.", 
      TITLE="The Problem of Social Costs", 
  PUBLISHER={W. A. Benjamin}, 
    ADDRESS={Amsterdam}, 
       YEAR=1966} 
 
@ARTICLE{Coase60, 
    AUTHOR="Coase, R.", 
     TITLE="The Problem of Social Costs", 
   JOURNAL="Journal of Law and Economics", 
      YEAR=1960, 
    VOLUME=3, 
     PAGES="1-44"} 
 
%Cohen 
 
@BOOK{Cohen66, 
     AUTHOR="Cohen, P.", 
      TITLE="Set Theory and the Continuum Hypothesis", 
  PUBLISHER={W. A. Benjamin}, 
    ADDRESS={Amsterdam}, 
       YEAR=1966} 
 
%Cooper 
 
@UNPUBLISHED{Cooper93, 
    AUTHOR="Cooper, D.", 
     TITLE="Supergames Played by Finite Automata with Finite 
Costs of Complexity in an Evolutionary Setting", 
      NOTE={Working paper{,} {U}niversity of {P}ittsburgh}, 
      YEAR="1993"} 
 
%Crawford 
 
@UNPUBLISHED{Crawford88, 
    AUTHOR="Crawford, V.", 
     TITLE="Learning and Mixed Strategy Equilibria in Evolutionary Games", 
      NOTE={Working paper{,} {U}niversity of {C}alifornia at {S}an 
{D}iego}, 
      YEAR="1988"} 
 
@ARTICLE{Crawford91, 
      AUTHOR="Crawford, V.", 
       TITLE="An Evolutionary Interpretation of {V}an {H}uyck, {B}attalio 
and {B}eil's Experimental Results in Coordination Games", 
     JOURNAL="Games and Economic Behavior", 
        YEAR="1991", 
      VOLUME="3", 
       PAGES="25-59"} 
 
%Cross 
 
@BOOK{Cross69, 
     AUTHOR="Cross, J.", 
      TITLE="The Economics of Bargaining", 
  PUBLISHER={Basic Books}, 
    ADDRESS={New York}, 
       YEAR=1969} 
 
%D 
 
%Dalkey 
 
@TECHREPORT{Dalkey, 
     AUTHOR="Dalkey, N.", 
      TITLE="The {D}elphi Method: An Experimental Study of Group Opinion", 
INSTITUTION={RAND Corporation}, 
       YEAR=1969} 
 
%Dasgupta 
 
@ARTICLE{Dasgupta74, 
    AUTHOR="Dasgupta, P.", 
     TITLE="Of some Alternative Criteria for Justice between 
Generations", 
   JOURNAL="Journal of Public Economics", 
      YEAR=1974, 
    VOLUME=3, 
     PAGES="405-423"} 
 
@UNPUBLISHED{Dasgupta93, 
    AUTHOR="Dasgupta, P.", 
     TITLE="Population and Savings: {E}thical Issues", 
      NOTE={Working paper{,} {U}niversity of {C}ambridge}, 
      YEAR="1993"} 
 
%Dawes, van de Kragt and Orbell 
 
@ARTICLE{DKO88, 
    AUTHOR="Dawes, R. and A. van der Kragt and J. Orbell", 
     TITLE="Not Thee or Me but We", 
   JOURNAL="Acta Psychologica", 
      YEAR=1988, 
    VOLUME=68, 
     PAGES="83-97"} 
 
 
%Dawkins 
 
@BOOK{Dawkins76, 
     AUTHOR="Dawkins, R.", 
      TITLE="The Selfish Gene", 
  PUBLISHER={Oxford University Press}, 
    ADDRESS={Oxford}, 
       YEAR=1976} 
 
@BOOK{Dawkins86, 
     AUTHOR="Dawkins, R.", 
      TITLE="The Blind Watchmaker", 
  PUBLISHER={Penguin Books}, 
    ADDRESS={London}, 
       YEAR=1986} 
 
%Defoe 
 
@BOOK{Defoe27, 
     AUTHOR="Defoe, D.", 
      TITLE="The Complete English Tradesman", 
  PUBLISHER={Augustus Kelley}, 
    ADDRESS={New York}, 
       YEAR=1969, 
       NOTE="(First published 1727.)"} 
 
%de Jasay 
 
@BOOK{Jasay89, 
     AUTHOR="Jasay, A. de", 
      TITLE="Social Contract Free Ride: A Study of the Publci Goods Problem", 
  PUBLISHER={Clarendon Press}, 
    ADDRESS={Oxford}, 
       YEAR=1989} 
 
%de Morgan 
 
@BOOK{deMorgan, 
     AUTHOR="Morgan, A. de", 
      TITLE="The Encyclopaedia of Eccentrics", 
  PUBLISHER={Open Court}, 
    ADDRESS={La Salle, Ill.}, 
       YEAR=1974} 
 
%Dekel and Fudenberg 
 
@UNPUBLISHED{DandF, 
    AUTHOR="Dekel, E. and D. Fudenberg", 
     TITLE="Rational Behavior with Payoff Uncertainty", 
      NOTE={University of California, Berkeley}, 
      YEAR="1987"} 
 
%Dennett 
 
@BOOK{Dennett78, 
     AUTHOR="Dennett, C.", 
      TITLE="Brainstorms: Philosphical Essays on Mind and Psychology", 
  PUBLISHER={Harvester Press}, 
    ADDRESS={Hassocks, Sussex, UK}, 
       YEAR=1978} 
 
@BOOK{Dennett91, 
     AUTHOR="Dennett, C.", 
      TITLE="Consciousness Explained", 
  PUBLISHER={Allen Lane: The Penguin Press}, 
    ADDRESS={London}, 
       YEAR=1991} 
 
%Diamond 
 
@ARTICLE{Diamond67, 
    AUTHOR="Diamond, P.", 
     TITLE="Cardinal Welfare, Individualistic Ethics and 
Interpersonal Comparison of Utility: {C}omment", 
   JOURNAL="Journal of Political Economy", 
      YEAR=1967, 
    VOLUME=75, 
     PAGES="765-766"} 
 
%Dowling 
 
@ARTICLE{Dowling89, 
    AUTHOR="Dowling, W.", 
     TITLE="There are no safe virus tests", 
   JOURNAL="American Mathematical Monthly", 
      YEAR=1989, 
    VOLUME=96, 
     PAGES="835-836"} 
 
%E 
 
%Edgeworth 
 
@BOOK{Edgeworth, 
     AUTHOR="Edgeworth, F.", 
      TITLE="Mathematical Psychics", 
  PUBLISHER={Kegan Paul}, 
    ADDRESS={London}, 
       YEAR=1881} 
 
%Ellsberg 
 
@BOOK{Ellsberg, 
     AUTHOR="Ellsberg, D.", 
      TITLE="The Theory and Practice of Blackmail: Formal Theories of 
Negotiation", 
  PUBLISHER={University of Illinois Press}, 
    ADDRESS={Urbana}, 
       YEAR=1975} 
 
%Elster 
 
@BOOK{Elster79, 
    AUTHOR="Elster, J.", 
    TITLE="Ulysses and the Syrens: Studies in Rationality and 
Irrationality", 
   PUBLISHER={Cambridge University Press}, 
     ADDRESS={Cambridge}, 
        YEAR=1979} 
 
@INCOLLECTION{Elster82, 
     AUTHOR="Elster, J.", 
      TITLE="Sour Grapes--{U}tilitarianism and the Genesis of Wants", 
  BOOKTITLE="Utilitarianism and Beyond", 
  PUBLISHER={Cambridge University Press}, 
    ADDRESS={Cambridge}, 
       YEAR=1982, 
     EDITOR="A. Sen and B. Williams"} 
 
@BOOK{Elster89, 
    AUTHOR="Elster, J.", 
    TITLE="The Cement of Society: A Study of Social Order", 
   PUBLISHER={Cambridge University Press}, 
     ADDRESS={Cambridge}, 
        YEAR=1989} 
 
@BOOK{Elster79, 
    AUTHOR="Elster, J.", 
    TITLE="Ulysses and the Syrens: Studies in Rationality and 
Irrationality", 
   PUBLISHER={Cambridge University Press}, 
     ADDRESS={Cambridge}, 
        YEAR=1979} 
 
@BOOK{Elster85, 
    AUTHOR="Elster, J.", 
    TITLE="Making Sense of Marx", 
   PUBLISHER={Cambridge University Press}, 
     ADDRESS={Cambridge}, 
        YEAR=1985} 
 
%F 
 
%Farrell 
 
@INCOLLECTION{Farrell90, 
      AUTHOR="Farrell, J.", 
       TITLE="Meaning and Credibility in Cheap-Talk Games", 
   BOOKTITLE="Mathematical Models in Economics", 
     EDITOR="M. Dempster", 
   PUBLISHER="Oxford University Press", 
     ADDRESS="Oxford", 
        YEAR="(Forthcoming.)"} 
 
%Farrell and Ware 
 
@ARTICLE{FandW88, 
      AUTHOR="Farrell, J. and R. Ware", 
       TITLE="Evolutionary Stability in the Repeated {P}risoners' 
{D}ilemma Game", 
     JOURNAL="Theoretical Population Biology", 
        YEAR="1988", 
       PAGES="161-166", 
      VOLUME="36"} 
 
%Field 
 
@BOOK{Field21, 
     AUTHOR="Field, G.", 
      TITLE="Moral Theory", 
  PUBLISHER={Methuen}, 
    ADDRESS={London}, 
       YEAR=1988, 
       NOTE="(First published 1921.)"} 
 
%Foster and Young 
 
@ARTICLE{FandY90, 
      AUTHOR="Foster, D. and P. Young", 
       TITLE="Stochastic Evolutionary Game Dynamics", 
     JOURNAL="Journal of Theoretical Biology", 
        YEAR="1990", 
       PAGES="219-232", 
      VOLUME="38"} 
 
%Forsythe 
 
@ARTICLE{CDFandR91, 
      AUTHOR="Cooper, J. and D. DeJong and R. Forsythe and T. 
Ross", 
       TITLE="Selection Criteria in Coordination Games: Some 
Experimental 
Results", 
     JOURNAL="American Economic Review", 
        YEAR="1991", 
       PAGES="218-233", 
      VOLUME="80"} 
 
%Frank 
 
@BOOK{Frank88, 
     AUTHOR="Frank, R.", 
      TITLE="Passions within Reason", 
  PUBLISHER={Norton}, 
    ADDRESS={New York}, 
       YEAR=1988} 
 
%Freidlin and Wentzill 
 
@BOOK{FandW84, 
     AUTHOR="Freidlin, M. and A. Wentzell", 
      TITLE="Random Perturbations of Dynamical Systems", 
  PUBLISHER={Springer-Verlag}, 
    ADDRESS={New York}, 
       YEAR=1984} 
 
%Friedman 
 
@BOOK{Friedman77, 
     AUTHOR="Friedman, J.", 
      TITLE="Oligopoly and the Theory of Games", 
  PUBLISHER={North Holland}, 
    ADDRESS={Amsterdam}, 
       YEAR=1977} 
 
@BOOK{Friedman86, 
     AUTHOR="Friedman, J.", 
      TITLE="Game Theory with Applications to Economics", 
  PUBLISHER={Oxford University Press}, 
    ADDRESS={Oxford}, 
       YEAR=1986} 
 
@ARTICLE{Friedman47, 
      AUTHOR="Friedman, M.", 
       TITLE="Lerner on the Economics of Control", 
     JOURNAL="Journal of Political Economy", 
        YEAR="1947", 
       PAGES="405-416", 
      VOLUME="55"} 
 
 
%Friedman and Savage 
 
@ARTICLE{FandS48, 
      AUTHOR="Friedman, M. and L. Savage", 
       TITLE="Utility Analysis of Choices Involving Risk", 
     JOURNAL="Journal of Political Economy", 
        YEAR="1948", 
       PAGES="179-304", 
      VOLUME="56"} 
 
 
%Foldes 
 
@ARTICLE{Foldes64, 
      AUTHOR="Foldes, L.", 
       TITLE="A Determinate Model of Bilateral Monopoly", 
     JOURNAL="Economica", 
        YEAR="1964", 
       PAGES="117-131", 
      VOLUME="31"} 
 
%Fudenberg 
 
@INPROCEEDINGS{Fudenberg92a, 
     AUTHOR="Fudenberg, D.", 
      TITLE="Repeated Game Explanations of Commitment and 
Cooperation", 
  BOOKTITLE="Advances in Economic Theory", 
  PUBLISHER="Cambridge University Press", 
    ADDRESS="Cambridge", 
     EDITOR="J.-J. Laffont", 
       YEAR="1992", 
ORGANIZATION="Sixth World Congress of the Econometric Society"} 
 
 
@UNPUBLISHED{Fudenberg91, 
      AUTHOR="Fudenberg, D.", 
       TITLE="Repeated Learning{,} Experimentation and Equilibrium in Games", 
        YEAR="1988", 
        NOTE="MIT Working Paper"} 
 
 
 
%Fudenberg and Kreps 
 
@UNPUBLISHED{FandK88, 
      AUTHOR="Fudenberg, D. and D. Kreps", 
       TITLE="Learning{,} Experimentation and Equilibrium in 
Games", 
        YEAR="1988", 
        NOTE="MIT Working Paper"} 
 
 
%Fudenberg, Kreps and Levine 
 
@ARTICLE{FKandL, 
    AUTHOR="Fudenberg, D. and D. Kreps and D. Levine", 
     TITLE="On the Robustness of Equilibrium Refinements", 
   JOURNAL="Journal of Economic Theory", 
      YEAR="1988", 
    VOLUME="44", 
     PAGES="354-380"} 
 
%Fudenberg and Maskin 
 
@ARTICLE{FandM90, 
      AUTHOR="Fudenberg, D. and E. Maskin", 
       TITLE="Evolution and Cooperation in Noisy Repeated Games", 
     JOURNAL="American Economic Review", 
        YEAR="1990", 
      VOLUME="80", 
       PAGES="274-279"} 
 
%Fung 
 
@BOOK{Fung52, 
      AUTHOR="Fung, Y.-L.", 
       TITLE="A History of {C}hinese Philosophy{,} Volume 1", 
   PUBLISHER="Princeton University Press", 
     ADDRESS="Princeton", 
        YEAR="1952"} 
 
%G 
 
%Gaertner, Pattanaik and Suzumura 
 
@ARTICLE{GPandS92, 
      AUTHOR="Gaertner, P. and P. Pattanaik and K. Suzumura", 
       TITLE="Evolution and Cooperation in Noisy Repeated Games", 
     JOURNAL="Individual Rights Revisited", 
        YEAR="1992", 
      VOLUME="59", 
       PAGES="161-178"} 
 
%Gale 
 
@ARTICLE{Gale86a, 
    AUTHOR="Gale, D.", 
     TITLE="Bargaining and Competition Part I: Characterization", 
   JOURNAL="Econometrica", 
      YEAR=1986, 
    VOLUME=54, 
     PAGES="785-806"} 
 
@ARTICLE{Gale86b, 
    AUTHOR="Gale, D.", 
     TITLE="Bargaining and Competition Part II: Existence", 
   JOURNAL="Econometrica", 
      YEAR=1986, 
    VOLUME=54, 
     PAGES="807-818"} 
 
@ARTICLE{Gale87, 
    AUTHOR="Gale, D.", 
     TITLE="Limit Theorems for Markets with Sequential Bargaining", 
   JOURNAL="Journal of Economic Theory", 
      YEAR=1987, 
    VOLUME=43, 
     PAGES="20-54"} 
 
 
%Gardenfors 
 
@ARTICLE{Gardenfors, 
    AUTHOR="{Gardenfors}, P.", 
     TITLE="Qualitative Probability as an Intentional Logic", 
   JOURNAL="Journal of Philosophical Logic", 
      YEAR=1975, 
    VOLUME=4, 
     PAGES="171-185"} 
 
 
@BOOK{Gardenfors88, 
     AUTHOR="Gardenfors, P.", 
      TITLE="Knowledge in Flux", 
  PUBLISHER={MIT Press}, 
    ADDRESS={Cambridge, Mass.}, 
       YEAR=1988} 
 
%Gardner 
 
@ARTICLE{Gardner73, 
    AUTHOR="Gardner, M.", 
     TITLE="Mathematical Games", 
   JOURNAL="Scientific American", 
      YEAR=1973, 
    VOLUME=July, 
     PAGES="104-108"} 
 
@ARTICLE{Gardner74, 
    AUTHOR="Gardner, M.", 
     TITLE="Mathematical Games", 
   JOURNAL="Scientific American", 
      YEAR=1974, 
    VOLUME=March, 
     PAGES="102-108"} 
 
%Gauthier 
 
@BOOK{Gauthier69, 
     AUTHOR="Gauthier, D.", 
      TITLE="The Logic of Leviathan", 
  PUBLISHER={Oxford University Press}, 
    ADDRESS={Oxford}, 
       YEAR=1969} 
 
@ARTICLE{Gauthier79, 
    AUTHOR="Gauthier, D.", 
     TITLE="David {H}ume: {C}ontractarian", 
   JOURNAL="Philosophical Review", 
      YEAR=1979, 
    VOLUME=88, 
     PAGES="3-38"} 
 
 
@BOOK{Gauthier86, 
     AUTHOR="Gauthier, D.", 
      TITLE="Morals by Agreement", 
  PUBLISHER={Clarendon Press}, 
    ADDRESS={Oxford}, 
       YEAR=1986} 
 
@INCOLLECTION{Gauthier93, 
      AUTHOR="Gauthier, D.", 
       TITLE="Uniting Separate Persons", 
   BOOKTITLE="Rationality, Justice and the Social Contract", 
      EDITOR="D. Gauthier and R. Sugden", 
   PUBLISHER="Harvester Wheatsheaf", 
     ADDRESS="Hemel Hempstead, UK", 
        YEAR="1993"} 
 
 
 
%Gauthier and Sugden 
 
@BOOK{GandS93, 
     AUTHOR="Gauthier, D. and R. Sugden", 
      TITLE="Rationality, Justice and the Social Contract", 
  PUBLISHER={Harvester Wheatsheaf}, 
    ADDRESS={Hemel Hempstead, U. K.}, 
       YEAR=1993} 
 
%Geanakoplos 
 
@BOOK{Geanakoplos88a, 
     AUTHOR="Geanakoplos, J.", 
      TITLE="Common Knowledge{,} {B}ayesian Learning and Market 
Speculation 
with Bounded Rationality", 
  PUBLISHER={unpublished{,} Yale University}, 
    ADDRESS="{N}ew {H}aven", 
       YEAR=1988} 
 
@UNPUBLISHED{Geanakoplos88b, 
    AUTHOR="Geanakoplos, J.", 
     TITLE="Common Knowledge Without Partitions{,} with an 
Application to Speculation", 
      YEAR=1988, 
      NOTE="Cowles Foundation Discussion Paper No. 914{,} Yale 
University"} 
 
 
%Geanakoplos and Polemarchakis 
 
@ARTICLE{GandP, 
    AUTHOR="Geanakoplos, J.  and H. Polemarchakis", 
     TITLE="We Can't Disagree Forever", 
   JOURNAL="Journal of Economic Theory", 
      YEAR=1982, 
    VOLUME=28, 
     PAGES="192-200"} 
 
%Gellner 
 
@BOOK{Gellner88, 
    AUTHOR="Gellner, E.", 
    TITLE="Plough, Sword and Book", 
   PUBLISHER={Paladin, Grafton Books}, 
     ADDRESS={London}, 
        YEAR=1988} 
 
%Gibbard 
 
@ARTICLE{Gibbard74, 
    AUTHOR="Gibbard, A.", 
     TITLE="A {P}areto-Consistent Libertarian Claim", 
   JOURNAL="Journal of Economic Theory", 
      YEAR=1974, 
    VOLUME=7, 
     PAGES="388-410"} 
 
@BOOK{Gibbard90, 
    AUTHOR="Gibbard, A.", 
    TITLE="Wise Choices and Apt Feelings: A Theory of Normative Judgment", 
   PUBLISHER={Clarendon Press}, 
     ADDRESS={Oxford}, 
        YEAR=1990} 
 
%Gibbard and Harper 
 
@ARTICLE{GandH78, 
    AUTHOR="Gibbard, A. and W. Harper", 
     TITLE="Counterfactuals and Two Kinds of Expected Utility", 
   JOURNAL="Foundations and Applications of Decision Theory", 
      YEAR=1978, 
    VOLUME=1, 
     PAGES="125-162"} 
 
 
%Gilboa 
 
@TECHREPORT{Gilboa86, 
    AUTHOR="Gilboa, I.", 
     TITLE="Information and Meta-information", 
INSTITUTION="Tel Aviv University", 
      YEAR=1986, 
    NUMBER="Working Paper 3086"} 
 
@INPROCEEDINGS{Gilboa88a, 
      AUTHOR="Gilboa, I.", 
       TITLE="Information and Meta-information", 
   BOOKTITLE="Proceedings of the Second Conference of TARK", 
        YEAR=1988, 
      EDITOR="Moshe Yandi", 
   PUBLISHER={Morgan Kaufman}, 
ORGANIZATION="The Theoretical Aspects of Reasoning About 
Knowledge 
Conference"} 
 
@ARTICLE{Gilboa88b, 
      AUTHOR="Gilboa, I.", 
       TITLE="The Complexity of Computing Best-Response Automata in 
Repeated Games", 
     JOURNAL="Journal of Economic Theory", 
        YEAR="1988", 
      VOLUME="45", 
       PAGES="343-353"} 
 
 
 
@INCOLLECTION{Gilboa90, 
      AUTHOR="Gilboa, I.", 
       TITLE="A Note on the Consistency of Game Theory", 
   BOOKTITLE="Reasoning About Knowledge", 
     EDITOR="R. Parikh", 
   PUBLISHER="Morgan Kaufmann", 
     ADDRESS="San Mateo, CA", 
        YEAR="1990"} 
 
%Goldberg 
 
@BOOK{Goldberg89, 
      AUTHOR="Goldberg, A.", 
       TITLE="Genetic Algorithms in Search, Optimization and 
Machine Learning", 
   PUBLISHER="Addison-Wesley", 
        YEAR="1989"} 
 
 
%Good 
 
@BOOK{Good50, 
      AUTHOR="Good, I. J.", 
       TITLE="Probability and the Weighing of Evidence", 
   PUBLISHER="Hafner", 
     ADDRESS="New York", 
        YEAR="1950"} 
 
@BOOK{Good83, 
      AUTHOR="Good, I. J.", 
       TITLE="Good Thinking: The Foundations 
of Probability and its Applications", 
   PUBLISHER="University of Minnesota Press", 
     ADDRESS="Minneapolis", 
        YEAR="1983"} 
 
%Goodin 
 
@INCOLLECTION{Goodin93, 
      AUTHOR="Goodin, R.", 
       TITLE="Equal Rationality and Initial Endowments", 
   BOOKTITLE="Rationality, Justice and the Social Contract", 
      EDITOR="D. Gauthier and R. Sugden", 
   PUBLISHER="Harvester Wheatsheaf", 
     ADDRESS="Hemel Hempstead, U. K.", 
        YEAR="1993"} 
 
%Gough 
 
@BOOK{Gough, 
     AUTHOR="Gough, J. W.", 
      TITLE="The Social Contract", 
  PUBLISHER={Clarendon Press}, 
    ADDRESS={Oxford}, 
       YEAR=1938} 
 
%Green and Osband 
 
@ARTICLE{GandO91, 
    AUTHOR="Green, E. and K. Osband", 
     TITLE="A Revealed Preference Theory for Expected Utility", 
   JOURNAL="Review of Economic Studies", 
      YEAR=1991, 
    VOLUME=58, 
     PAGES="677-697"} 
 
%Guth, Schmittberger and Schwarze 
 
@ARTICLE{GSandS82, 
    AUTHOR="Guth, W. and R. Schmittberger and B. Schwarze", 
     TITLE="An Experimental Analysis of Ultimatum Bargaining", 
   JOURNAL="Journal of Behavior and Organization", 
      YEAR=1982, 
    VOLUME=3, 
     PAGES="367-388"} 
 
%Guth and Tietz 
 
@ARTICLE{GandT90, 
     AUTHOR = "Guth, W. and R. Tietz", 
     TITLE = "Ultimatum Bargaining Behavior: {A} Survey and 
Comparison of Experimental Results", 
     JOURNAL = "Journal of Economic Psychology", 
     YEAR = 1990, 
     VOLUME = 11, 
     PAGES = "417--49"} 
 
%H 
 
%Halpern 
 
@BOOK{Halpern86, 
     AUTHOR="{Halpern, (ed.)}, J.", 
      TITLE="Theoretical Aspects of Reasoning about Knowledge", 
  PUBLISHER="Morgan Kaufmann Publishers, Inc.", 
    ADDRESS="Los Altos", 
       YEAR=1986} 
 
@ARTICLE{Halpern87, 
    AUTHOR="Halpern, J. Y.", 
     TITLE="Using Reasoning about Knowledge to Analyse Distributed Systems", 
   JOURNAL="Annual Review of Computer Science", 
      YEAR=1987, 
    VOLUME=2, 
     PAGES="37-68"} 
 
 
%Hammond 
 
@ARTICLE{Hammond76, 
    AUTHOR="Hammond, P.", 
     TITLE="Why Ethical Measures of Inequality Need Interpersonal 
Comparisons", 
   JOURNAL="Theory and Decision", 
      YEAR=1976, 
    VOLUME=7, 
     PAGES="263-274"} 
 
@ARTICLE{Hammond88b, 
    AUTHOR="Hammond, P.", 
     TITLE="Consequentialist Foundations for Expected Utility", 
   JOURNAL="Theory and Decision", 
      YEAR=1988, 
    VOLUME=25, 
     PAGES="25-78"} 
 
@INCOLLECTION{Hammond92, 
      AUTHOR="Hammond, P.", 
       TITLE="{H}arsanyi's Utilitarian Theorem: {A} Simpler Proof 
and some Ethical Connotations", 
   BOOKTITLE="Rational Interaction: Essays in Honor of {J}ohn 
{H}arsanyi", 
      EDITOR="R. Selten", 
   PUBLISHER="Springer-Verlag", 
     ADDRESS="Berlin", 
        YEAR="1992"} 
 
%Hampshire 
 
@INCOLLECTION{Hampshire82, 
     AUTHOR="Hampshire, S.", 
      TITLE="Morality and Convention", 
  BOOKTITLE="Utilitarianism and Beyond", 
  PUBLISHER={Cambridge University Press}, 
    ADDRESS={Cambridge}, 
       YEAR=1982, 
     EDITOR="A. Sen and B. Williams" } 
 
%Hampton 
 
@BOOK{Hampton86, 
    AUTHOR="Hampton, J.", 
    TITLE="Hobbes and the Social Contract Tradition", 
   PUBLISHER={Cambridge University Press}, 
     ADDRESS={Cambridge}, 
        YEAR=1986} 
 
%Hardin, Garrett 
 
@ARTICLE{HardinG, 
    AUTHOR="Hardin, G.", 
     TITLE="The Tragedy of the Commons", 
   JOURNAL="Science", 
      YEAR=1968, 
    VOLUME=162, 
     PAGES="1243-1248"} 
 
%Hardin, Russell 
 
@BOOK{HardinR, 
    AUTHOR="Hardin, R.", 
    TITLE="Collective Action", 
   PUBLISHER={Johns Hopkins Press}, 
     ADDRESS={Baltimore}, 
        YEAR=1982} 
 
@BOOK{HardinR88, 
    AUTHOR="Hardin, R.", 
    TITLE="Morality within the Limits of Reason", 
   PUBLISHER={University of Chicago Press}, 
     ADDRESS={Chicago}, 
        YEAR=1988} 
 
%Harper 
 
@INCOLLECTION{Harper88, 
      AUTHOR="Harper, W.", 
       TITLE="Causal Decision Theory and Game Theory", 
   BOOKTITLE="Causation in Decision{,} Belief Change{,} and 
Statistics", 
      EDITOR="{W. Harper and B. Skyrms}", 
   PUBLISHER="Kluwer", 
     ADDRESS="Dordrecht", 
        YEAR="1988"} 
 
 
%Hare 
 
@ARTICLE{Hare64, 
    AUTHOR="Hare, R.", 
     TITLE="The Promising Game", 
   JOURNAL="Revue Internationale de Philosophie", 
      YEAR=1964, 
    VOLUME=70, 
     PAGES="398-412"} 
 
@INCOLLECTION{Hare75, 
     AUTHOR="Hare, R.", 
      TITLE="{R}awls' Theory of Justice", 
  BOOKTITLE="Reading {R}awls", 
  PUBLISHER={Basil Blackwell}, 
    ADDRESS={Oxford}, 
       YEAR=1975, 
     EDITOR="B. Daniels" } 
 
 
@BOOK{Hare81, 
    AUTHOR="Hare, R.", 
    TITLE="Moral Thinking: Its Levels{,} Method and Point", 
   PUBLISHER={Clarendon Press}, 
     ADDRESS={Cambridge}, 
        YEAR=1981} 
 
@INCOLLECTION{Hare82, 
     AUTHOR="Hare, R.", 
      TITLE="Ethical Theory and Utilitarianism", 
  BOOKTITLE="Utilitarianism and Beyond", 
  PUBLISHER={Cambridge University Press}, 
    ADDRESS={Cambridge}, 
       YEAR=1982, 
     EDITOR="A. Sen and B. Williams" } 
 
@INCOLLECTION{Hare84, 
     AUTHOR="Hare, R.", 
      TITLE="Rights{,} Utility and Universalization", 
  BOOKTITLE="Utility and Rights", 
  PUBLISHER={University of Minnesota Press}, 
    ADDRESS={Minneapolis}, 
       YEAR=1984, 
     EDITOR="R. G. Frey"} 
 
%Harrison and McCabe 
 
@UNPUBLISHED{HandM92, 
    AUTHOR="Harrison, G. and K. McCabe", 
     TITLE="Expectations and Fairness in a Simple Bargaining Experiment", 
      NOTE={University of Southern Carolina, College of Business 
Administration, 
Working Paper B-92-10}, 
      YEAR="1992"} 
 
 
 
%Harsanyi 
 
@ARTICLE{Harsanyi53, 
    AUTHOR="Harsanyi, J.", 
     TITLE="Cardinal Utility in Welfare Economics and in the Theory 
of Risk-Taking", 
   JOURNAL="Journal of Political Economy", 
      YEAR=1953, 
    VOLUME=61, 
     PAGES="434-435"} 
 
@ARTICLE{Harsanyi55, 
    AUTHOR="Harsanyi, J.", 
     TITLE="Cardinal Welfare, Individualistic Ethics, and the 
Interpersonal Comparison of Utility", 
   JOURNAL="Journal of Political Economy", 
      YEAR=1955, 
    VOLUME=63, 
     PAGES="309-321"} 
 
@ARTICLE{Harsanyi58, 
    AUTHOR="Harsanyi, J.", 
     TITLE="Ethics in Terms of Hypothetical Imperatives", 
   JOURNAL="Mind", 
      YEAR=1958, 
    VOLUME=47, 
     PAGES="305-316"} 
 
@ARTICLE{Harsanyi67, 
    AUTHOR="Harsanyi, J.", 
     TITLE="Games with Incomplete Information Played by 
`{B}ayesian' Players, {P}arts {I} - {III}", 
   JOURNAL="Management Science", 
      YEAR=1967, 
    VOLUME="14", 
     PAGES="159-182"} 
 
@ARTICLE{Harsanyi75, 
    AUTHOR="Harsanyi, J.", 
     TITLE="Can the Maximin Principle Serve as a Basis for 
Morality? {A} Critique of {J}ohn {R}awls' Theory ", 
   JOURNAL="American Political Science Review", 
      YEAR=1975, 
    VOLUME=69, 
     PAGES="594-606"} 
 
 
@BOOK{Harsanyi77, 
    AUTHOR="Harsanyi, J.", 
    TITLE="Rational Behavior and Bargaining Equilibrium in Games 
and Social Situations", 
   PUBLISHER={Cambridge University Press}, 
     ADDRESS={Cambridge}, 
        YEAR=1977} 
 
@INCOLLECTION{Harsanyi82, 
     AUTHOR="Harsanyi, J.", 
      TITLE="Morality and the Theory of Rational Behavior", 
  BOOKTITLE="Utilitarianism and Beyond", 
  PUBLISHER={Cambridge University Press}, 
    ADDRESS={Cambridge}, 
       YEAR=1982, 
     EDITOR="A. Sen and B. Williams" } 
 
@ARTICLE{Harsanyi87, 
    AUTHOR="Harsanyi, J.", 
     TITLE="Review of {G}authier's ``{M}orals by {A}greement''", 
   JOURNAL="Economics and Philosophy", 
      YEAR=1987, 
    VOLUME=3, 
     PAGES="339-343"} 
 
 
@INCOLLECTION{Harsanyi92a, 
    AUTHOR="Harsanyi, J.", 
     TITLE="Normative Validity and Meaning of {V}on {N}eumann and 
{M}orgenstern Utilities", 
  BOOKTITLE="Studies in Logic and the Foundations of Game Theory: 
Proceedings of the Ninth International Congress of 
Logic, Methodology and the Philosophy of Science", 
  PUBLISHER="Kluwer", 
    ADDRESS="Dordrecht", 
       YEAR=1992, 
     EDITOR="B. Skyrms"} 
 
%Harsanyi and Selten 
 
@BOOK{HandS88, 
      AUTHOR="Harsanyi, J. and R. Selten", 
       TITLE="A General Theory of Equilibrium Selection in Games", 
   PUBLISHER="MIT Press", 
     ADDRESS="Cambridge", 
        YEAR="1988"} 
 
%Hayek 
 
@BOOK{Hayek52, 
    AUTHOR="Hayek, F.", 
    TITLE="The Sensory Order", 
   PUBLISHER={University of Chicago Press}, 
     ADDRESS={Chicago}, 
        YEAR=1952} 
 
@BOOK{Hayek60, 
    AUTHOR="Hayek, F.", 
    TITLE="The Constitution of Liberty", 
   PUBLISHER={University of Chicago Press}, 
     ADDRESS={Chicago}, 
        YEAR=1960} 
 
@BOOK{Hayek67, 
    AUTHOR="Hayek, F.", 
    TITLE="Studies in Philosophy, Politics and Economics", 
   PUBLISHER={University of Chicago Press}, 
     ADDRESS={Chicago}, 
        YEAR=1967} 
 
%Hicks 
 
@BOOK{Hicks34, 
      AUTHOR="Hicks, J.", 
       TITLE="The Theory of Wages", 
   PUBLISHER="Macmillan", 
     ADDRESS="London", 
        YEAR="1934"} 
 
%Hobbes 
 
@BOOK{Hobbes, 
     AUTHOR="Hobbes, T.", 
      TITLE="Leviathan", 
  PUBLISHER={Penguin Classics}, 
    ADDRESS={London}, 
       YEAR=1986, 
       NOTE={(Edited by C. B. Macpherson, first published 1651.)}} 
 
@BOOK{Hobbes82, 
     AUTHOR="Hobbes, T.", 
      TITLE="Behemoth", 
  PUBLISHER={University of Chicago Press}, 
    ADDRESS={Chicago}, 
       YEAR=1990, 
       NOTE={(Edited by F. T\"{o}nnies, first published 1682.)}} 
 
 
@BOOK{Hobbes69, 
     AUTHOR="Hobbes, T.", 
      TITLE="Elements of Law", 
  PUBLISHER={Frank Cass}, 
    ADDRESS={London}, 
       YEAR=1969, 
       NOTE={(Edited by F. T\"{o}nnies)}} 
 
%Holland 
 
@BOOK{Holland75, 
      AUTHOR="Holland, J.", 
       TITLE="Adaption in Natural and Artificial Systems", 
   PUBLISHER="University of Michigan Press", 
     ADDRESS="Ann Arbor, Mich.", 
        YEAR="1975"} 
 
%Hofbauer and Sigmund 
 
@BOOK{HofbauerandS88, 
      AUTHOR="Hofbauer, S. and Sigmund, K.", 
       TITLE="The Theory of Evolution and Dynamical Systems", 
   PUBLISHER="Cambridge University Press", 
     ADDRESS="Cambridge", 
        YEAR="1988"} 
 
%Hofstadter 
 
@INCOLLECTION{Hofstadter81, 
      AUTHOR="Hofstadter, D.", 
       TITLE="A Conversation with {E}instein's Brain", 
   BOOKTITLE="The Mind's I", 
      EDITOR="D. Hofstadter and D. Dunnett", 
   PUBLISHER="Basic Books", 
     ADDRESS="Harmondsworth, UK", 
        YEAR="1981"} 
 
@ARTICLE{Hofstadter83a, 
    AUTHOR="Hofstadter, D.", 
     TITLE="Metamagical Themes: Questing for the Essence of Mind 
and Pattern", 
   JOURNAL="Scientific American", 
      YEAR=1983, 
    VOLUME=248, 
    NUMBER=5,  
    PAGES="16-28"} 
 
@ARTICLE{Hofstadter83b, 
    AUTHOR="Hofstadter, D.", 
     TITLE="Metamagical Themes", 
   JOURNAL="Scientific American", 
      YEAR=1983, 
    VOLUME=248, 
    NUMBER=6, 
     PAGES="14-20"} 
 
%Holt 
 
@UNPUBLISHED{Holt, 
     AUTHOR="Holt, D.", 
      TITLE="Models of Knowledge and Information", 
        YEAR="1988", 
        NOTE="Working Paper{,}  University of Michigan"} 
 
%Hopcraft and Ullman 
 
@BOOK{HandU79, 
     AUTHOR="Hopcraft, J. and Ullman, J.", 
      TITLE="Introduction to Automata Theory{,} Languages and 
Computation", 
  PUBLISHER={Addison Wesley}, 
    ADDRESS={Reading, Mass.}, 
       YEAR=1979} 
 
%Hopkins 
 
@BOOK{Hopkins78, 
     AUTHOR="Hopkins, K.", 
      TITLE="Conquerors and Slaves", 
  PUBLISHER={Cambridge University Press}, 
    ADDRESS={Cambridge}, 
       YEAR=1978} 
 
%Howard, Nigel 
 
@BOOK{Howard71, 
     AUTHOR="Howard, N.", 
      TITLE="Paradoxes of Rationality: Theory of Metagames and 
Political Behavior", 
  PUBLISHER={MIT Press}, 
    ADDRESS={Cambridge, Mass.}, 
       YEAR=1971} 
 
%Howard, John 
 
@ARTICLE{Howard88, 
    AUTHOR="Howard, J.", 
     TITLE="Cooperation in the {P}risoners' {D}ilemma", 
   JOURNAL="Theory and Decision", 
      YEAR=1988, 
    VOLUME=24, 
     PAGES="203-213"} 
 
%Hintikka 
 
@BOOK{Hintikka, 
      AUTHOR="Hintikka, J.", 
       TITLE="Knowledge and Belief", 
   PUBLISHER="Cornell University Press", 
     ADDRESS="Ithaca", 
        YEAR="1962"} 
 
%Howe and Roemer 
 
@ARTICLE{HandR, 
    AUTHOR="Howe, R. and Roemer, J.", 
     TITLE="{R}awlsian Justice as the Core of a Game", 
   JOURNAL="American Economic Review", 
      YEAR=1981, 
    VOLUME=71, 
     PAGES="880-895"} 
 
%Hughes, G and Cresswell, M. 
 
@BOOK{HandC, 
     AUTHOR="Hughes, G. and Cresswell, M.", 
      TITLE="An Introduction to Modal Logic", 
  PUBLISHER="Methuen", 
       YEAR=1985} 
 
 
%Hume 
 
@BOOK{Hume39, 
     AUTHOR="Hume, D.", 
      TITLE="A Treatise of Human Nature. {\rm 2nd edition}", 
  PUBLISHER={Clarendon Press}, 
    ADDRESS={Oxford}, 
       YEAR=1978, 
       NOTE="(Edited by L. A. Selby-Bigge{,} revised by 
              P. Nidditch{,} first published 1739.)"} 
 
@INCOLLECTION{Hume42, 
     AUTHOR="Hume, D.", 
      TITLE="Essays Moral, Political and Literary{,} Part I", 
    CHAPTER="Of the Independency of {P}arliament", 
  PUBLISHER={Liberty Classics}, 
    ADDRESS={Indianapolis}, 
       YEAR=1985, 
       NOTE="(Edited by E. Miller{,} essay first published 1742.)"} 
 
@INCOLLECTION{Hume58, 
     AUTHOR="Hume, D.", 
   BOOKTITLE="Essays Moral, Political and Literary{,} Part I", 
      TITLE="Of the First Principles of Government", 
  PUBLISHER={Liberty Classics}, 
    ADDRESS={Indianapolis}, 
       YEAR=1985, 
       NOTE="(Edited by E. Miller{,} essay first published 1758.)"} 
 
 
 
@INCOLLECTION{Hume48, 
     AUTHOR="Hume, D.", 
   BOOKTITLE="Essays Moral{,} Political and Literary", 
      TITLE="Of the Original Contract", 
  PUBLISHER={Liberty Classics}, 
    ADDRESS={Indianapolis}, 
       YEAR=1985, 
       NOTE="(Edited by E. Miller{,} essay first published 1748.)"} 
 
@INCOLLECTION{HumeX, 
     AUTHOR="Hume, D.", 
   BOOKTITLE="Essays Moral{,} Political and Literary", 
      TITLE="Of Suicide", 
  PUBLISHER={Liberty Classics}, 
    ADDRESS={Indianapolis}, 
       YEAR=1985} 
 
@BOOK{Hume77, 
     AUTHOR="Hume, D.", 
      TITLE="Enquiries Concerning Human Understanding and 
Concerning 
the Principles of Morals. {\rm 3rd edition}", 
  PUBLISHER={Clarendon Press}, 
    ADDRESS={Oxford}, 
       YEAR=1975, 
       NOTE="(Edited by L. A. Selby-Bigge{,} revised by 
              P. Nidditch{,} first published 1777.)"} 
 
 
%Hunt 
 
@BOOK{Hunt90, 
     AUTHOR="Hunt, M.", 
      TITLE="The Compassionate Beast", 
  PUBLISHER={William Morrow}, 
    ADDRESS={New York}, 
       YEAR=1990} 
 
%I 
 
%Isbell 
 
@ARTICLE{Isbell, 
    AUTHOR="Isbell, J.", 
     TITLE="A Modification of Harsanyi's Bargaining Model", 
   JOURNAL="Bulletin of the American Mathematical Society", 
      YEAR=1960, 
    VOLUME=66, 
     PAGES="70-73"} 
 
%J 
 
%Jervis 
 
@ARTICLE{Jervis78, 
    AUTHOR="Jervis, R.", 
     TITLE="Cooperation under the Security Dilemma", 
   JOURNAL="World Politics", 
      YEAR=1978, 
    VOLUME=30, 
     PAGES="167-214"} 
 
%Jordan 
 
@UNPUBLISHED{Jordan89, 
    AUTHOR="Jordan, J.", 
     TITLE="Bayesian Learning in Normal-Form Games", 
      NOTE={forthcoming in {\em Games and Economic Behavior}}, 
      YEAR="1989"} 
 
@UNPUBLISHED{Jordan90, 
    AUTHOR="Jordan, J.", 
     TITLE="The Exponential Rate of Learning in Repeated Games", 
      NOTE={Working paper{,} University of Minnesota}, 
      YEAR="1990"} 
 
%K 
 
%Kadane and Larkey 
 
@ARTICLE{KandL, 
    AUTHOR="Kadane, J. and P. Larkey", 
     TITLE="Subjective Probability and the Theory of Games", 
   JOURNAL="Management Science", 
      YEAR=1982, 
    VOLUME=28, 
     PAGES="113-120"} 
 
%Kahneman and Tversky 
 
@ARTICLE{KandT79, 
    AUTHOR="Kahneman, D. and A. Tversky", 
     TITLE="Prospect Theory: {A}n Analysis of Decision under Risk", 
   JOURNAL="Econometrica", 
      YEAR=1979, 
    VOLUME=47, 
     PAGES="263-291"} 
 
@INCOLLECTION{KandT87, 
     AUTHOR="Kahneman, D. and A. Tversky", 
   BOOKTITLE="Decision Making", 
      TITLE="Rational Choice and the Framing of Decisions", 
  PUBLISHER={Cambridge University Press}, 
    ADDRESS={Cambridge}, 
       YEAR=1988} 
 
 
%Kalai 
 
@ARTICLE{Kalai77b, 
    AUTHOR="Kalai, E.", 
     TITLE="Solutions to Bargaining Situations: Interpersonal 
Utility Comparisons", 
   JOURNAL="Econometrica", 
      YEAR=1977, 
    VOLUME=45, 
     PAGES="1623-1630"} 
 
@INPROCEEDINGS{Kalai88, 
    AUTHOR="Kalai, E.", 
     TITLE="Bounded Rationality and Strategic Complexity in 
Repeated Games", 
 BOOKTITLE="Game Theory and Applications", 
  PUBLISHER="Academic Press", 
      YEAR="(forthcoming 1990)"} 
 
%Kalai and Lehrer 
 
@UNPUBLISHED{KandL90, 
      AUTHOR="Kalai, E. and E. Lehrer", 
       TITLE="Rational Learning Leads to {N}ash Equilibrium", 
        YEAR="1990", 
        NOTE="Discussion Paper 895{,} {Northwestern University}"} 
 
%Kalai and Neme 
 
@UNPUBLISHED{KandN89, 
      AUTHOR="Kalai, E. and A. Neme", 
       TITLE="The Strength of a Little Perfection", 
        YEAR="1989", 
        NOTE="Discussion Paper 858{,} {Northwestern University}"} 
 
%Kalai and Smorodinsky 
 
@ARTICLE{KandS, 
    AUTHOR="Kalai, E. and Smorodinsky, M.", 
     TITLE="Other Solutions to {N}ash's Bargaining Problem", 
   JOURNAL="Econometrica", 
      YEAR=1975, 
    VOLUME=45, 
     PAGES="1623-1630"} 
 
%Kalai and Stanford 
 
@ARTICLE{KandS89, 
      AUTHOR="Kalai, E. and W. Stanford", 
       TITLE="Finite Rationality and Interpersonal Complexity in 
Repeated Games", 
     JOURNAL="Econometrica", 
        YEAR="1989", 
      VOLUME="56", 
       PAGES="397-410"} 
 
%Kandori, Mailath and Rob 
 
@ARTICLE{KMandR93, 
    AUTHOR="Kandori, M. and G. Mailath and R. Rob", 
     TITLE="Learning{,} Mutation{,} and Long Run Equilibria in Games", 
   JOURNAL="Econometrica",       
      YEAR=1993, 
    VOLUME="61", 
     PAGES="29-56"} 
 
%Kaneko 
 
@UNPUBLISHED{Kaneko, 
    AUTHOR="Kaneko, M.", 
     TITLE="Structural Common Knowledge and Factual Common 
Knowledge", 
      YEAR=1987, 
      NOTE="{RUEE} Working Paper 87-27"} 
 
%Kaneko and Nagashima 
 
@UNPUBLISHED{KandN, 
    AUTHOR="Kaneko, M. and T. Nagashima", 
     TITLE="Game Logic {I} and {II}", 
      YEAR=1990, 
      NOTE="Discussion Papers {EPO}-3-1 and 2{,} {Virginia 
Polytechnic Institute}"} 
 
@UNPUBLISHED{KandN88, 
      AUTHOR="Kaneko, M. and T. Nagashima", 
       TITLE="Players' Deductions and Deductive Knowledge and 
Common Knowledge as Theories", 
        YEAR="1988", 
        NOTE="Discussion Paper {E}88-02-01{,} {Virginia Polytechnic 
Institute}"} 
 
%Kant 
 
@BOOK{Kant49, 
    AUTHOR="Kant, I.", 
     TITLE="The Philosophy of Kant", 
 PUBLISHER={Random House}, 
   ADDRESS={New York}, 
      YEAR=1949, 
      NOTE="(Edited by C. Friedrich.)"} 
 
@INCOLLECTION{Kant84, 
     AUTHOR="Kant, I.", 
  BOOKTITLE="The Philosophy of Kant", 
      TITLE="Idea of a Universal History ", 
  PUBLISHER={Random House}, 
    ADDRESS={New York}, 
       YEAR=1949, 
       NOTE="(Edited by C. Friedrich; first published 1788.)"} 
 
 
@BOOK{Kant85, 
    AUTHOR="Kant, I.", 
     TITLE="Groundwork of the Metaphysic of Morals", 
 PUBLISHER={Harper Torchbooks}, 
   ADDRESS={New York}, 
      YEAR=1964, 
      NOTE="(Translated and analyzed by H. Paton. First published 
1785.)"} 
 
@BOOK{Kant88, 
    AUTHOR="Kant, I.", 
     TITLE="Critique of Practical Reason", 
 PUBLISHER={Macmillan}, 
   ADDRESS={New York}, 
      YEAR=1989, 
      NOTE="(Translated by L. Beck. First published 1788.)"} 
 
@INCOLLECTION{Kant88a, 
     AUTHOR="Kant, I.", 
  BOOKTITLE="The Philosophy of Kant", 
      TITLE="Critique of Judgment", 
  PUBLISHER={Random House}, 
    ADDRESS={New York}, 
       YEAR=1949, 
       NOTE="(Edited by C. Friedrich; first published 1788.)"} 
 
@INCOLLECTION{Kant93, 
     AUTHOR="Kant, I.", 
  BOOKTITLE="The Philosophy of Kant", 
      TITLE="Theory and Practice", 
  PUBLISHER={Random House}, 
    ADDRESS={New York}, 
       YEAR=1949, 
       NOTE="(Edited by C. Friedrich. First published 1793.)"} 
 
%Kavka 
 
@BOOK{Kavka86, 
    AUTHOR="Kavka, G.", 
     TITLE="Hobbesian Moral and Political Theory", 
 PUBLISHER={Princeton University Press}, 
   ADDRESS={Princeton}, 
      YEAR=1986} 
 
@ARTICLE{Kavka83, 
      AUTHOR="Kavka, G.", 
       TITLE="Hobbes' War of All against All", 
     JOURNAL="Ethics", 
        YEAR="1983", 
      VOLUME="93", 
       PAGES="291-310"} 
 
%Keynes 
 
@BOOK{Keynes, 
     AUTHOR="Keynes, J. M.", 
      TITLE="The General Theory of Employment{,} Interest and Money", 
  PUBLISHER={Macmillan}, 
    ADDRESS={London}, 
       YEAR=1937} 
 
@BOOK{Keynes21, 
     AUTHOR="Keynes, J. M.", 
      TITLE="A Treatise on Probability", 
  PUBLISHER={Macmillan}, 
    ADDRESS={London}, 
       YEAR=1921} 
 
%Kim 
 
@UNPUBLISHED{Kim89, 
      AUTHOR="Kim, Y.", 
       TITLE="Evolutionary Stable Strategies in the Repeated 
{P}risoners{'} {D}ilemma", 
        YEAR="1989", 
        NOTE="Working Paper{,} U{.} C{.} at San Diego"} 
 
%Kimura 
 
@BOOK{Kimura83, 
     AUTHOR="Kimura, M.", 
      TITLE="The Neutral Theory of Molecular Evolution", 
  PUBLISHER={Cambridge University Press}, 
    ADDRESS={Cambridge}, 
       YEAR=1983} 
 
 
%Kirk 
 
@BOOK{Kirk67, 
    AUTHOR="Kirk, R.", 
     TITLE="Edmund Burke: A Genius Reconsidered", 
 PUBLISHER={Arlington House}, 
   ADDRESS={New York}, 
      YEAR=1967} 
 
%Kohlberg 
 
@UNPUBLISHED{Kohlberg89, 
      AUTHOR="Kohlberg, E.", 
       TITLE="Refinements of {N}ash Equilibria:  The Main Ideas", 
        YEAR="1989", 
        NOTE="Working Paper 89-073{,} Harvard Business School"} 
 
%Kohlberg and Mertens 
 
@ARTICLE{KandM86, 
      AUTHOR="Kohlberg, E. and J. Mertens", 
       TITLE="On the Strategic Stability of Equilibria", 
     JOURNAL="Econometrica", 
        YEAR="1986", 
      VOLUME="54", 
       PAGES="1003-1037"} 
 
 
%Kolmogorov 
 
@BOOK{Kolmogorov50, 
     AUTHOR="Kolmogorov, A.", 
      TITLE="Foundations of the Theory of Probability", 
  PUBLISHER="Chelsea", 
    ADDRESS="New York", 
       YEAR="1950"} 
 
%Konyndyk 
 
@BOOK{Konyndyk86, 
     AUTHOR="Konyndyk, K.", 
      TITLE="Introduction to Modal Logic", 
  PUBLISHER="University of Notre Dame", 
    ADDRESS="South Bend", 
       YEAR="1986"} 
 
%Kreps 
 
@BOOK{Kreps88, 
     AUTHOR="Kreps, D.", 
      TITLE="Notes on the Theory of Choice", 
  PUBLISHER="Westview Press", 
    ADDRESS="London and Boulder, Colo.", 
       YEAR="1988"} 
 
 
 
%Kreps, Milgrom, Roberts and Wilson 
 
@ARTICLE{KMRandW, 
      AUTHOR="Kreps, D. and P. Milgrom and J. Roberts and R. Wilson", 
       TITLE="Rational Cooperation in the Finitely Repeated 
Prisoners' Dilemma", 
     JOURNAL="Journal of Economic Theory", 
        YEAR="1982", 
      VOLUME="27", 
       PAGES="245-252"} 
 
 
 
%Kreps and Wilson 
 
@ARTICLE{KandW82a, 
      AUTHOR="Kreps, D. and R. Wilson", 
       TITLE="Sequential Equilibria", 
     JOURNAL="Econometrica", 
        YEAR="1982", 
      VOLUME="50", 
       PAGES="863-894"} 
 
@ARTICLE{KandW82b, 
      AUTHOR="Kreps, D. and R. Wilson", 
       TITLE="Reputation and Imperfect Information", 
     JOURNAL="Journal of Economic Theory", 
        YEAR="1982", 
      VOLUME="27", 
       PAGES="253-279"} 
 
 
%Kripke 
 
%Kukathas and Pettit 
 
@BOOK{KandP90, 
    AUTHOR="Kukathas, C. and P. Pettit", 
    TITLE="Rawls: A Theory of Justice and its Critics", 
   PUBLISHER={Polity Press with Basil Blackwell}, 
     ADDRESS={Oxford}, 
        YEAR=1990} 
 
%L 
 
%La Mettrie 
 
@BOOK{LaMettrie48, 
    AUTHOR="Mettrie, J. de la", 
    TITLE="Man a Machine", 
   PUBLISHER={Open Court}, 
     ADDRESS={La Salle, Ill.}, 
        YEAR=1988, 
        NOTE="(First published 1748.)"} 
 
%Land 
 
@ARTICLE{Land59, 
    AUTHOR="Land, E.", 
     TITLE="Experiments in Color Vision", 
   JOURNAL="Scientific American", 
      YEAR=1959, 
    VOLUME=May, 
     PAGES="84-90"} 
 
%Laplace 
 
@BOOK{Laplace, 
    AUTHOR="{Laplace}, P.-S. de", 
    TITLE="A Philosophical Essay on Probabilities", 
   PUBLISHER={Wiley}, 
     ADDRESS={New York}, 
        YEAR=1902, 
        NOTE={(Translated by  R. Brambrough)}} 
 
%Ledyard 
 
@UNPUBLISHED{Ledyard92, 
      AUTHOR="Ledyard, J.", 
       TITLE="Public Goods{:} {A} Survey of Experimental Research", 
        YEAR="1992", 
        NOTE="Working Paper{,} California Institute of Technology"} 
 
%Lehrer 
 
@ARTICLE{Lehrer87, 
      AUTHOR="Lehrer, E.", 
       TITLE="Repeated Games with Stationary Bounded Recall Strategies", 
     JOURNAL="Journal of Economic Theory", 
        YEAR="1988", 
      VOLUME="46", 
       PAGES="130-144"} 
 
%Lerner 
 
@BOOK{Lerner44, 
      AUTHOR="Lerner, A.", 
       TITLE="The Economics of Control", 
   PUBLISHER={Macmillan}, 
     ADDRESS={New York}, 
        YEAR=1944} 
 
 
 
 
%Levi 
 
@ARTICLE{Levi82, 
      AUTHOR="Levi, I.", 
       TITLE="A note on {N}ewcombmania", 
     JOURNAL="Journal of Philosophy", 
        YEAR="1982", 
      VOLUME="79", 
       PAGES="337-342"} 
 
%Lewis, A. 
 
@ARTICLE{Lewisa, 
    AUTHOR="Lewis, A.", 
     TITLE="On Effectively Computable Realizations of Choice 
Functions", 
   JOURNAL="Mathematical Social Sciences", 
      YEAR=1985, 
    VOLUME=10, 
     PAGES="43-80"} 
 
 
%Lewis, C. I. 
 
@BOOK{Lewis46, 
      AUTHOR="Lewis, C. I.", 
       TITLE="An Analysis of Knowledge and Valuation", 
   PUBLISHER={Open Court}, 
     ADDRESS={La Salle{,} Ill.}, 
        YEAR=1946} 
 
%Lewis, D. 
 
@BOOK{Lewis69, 
    AUTHOR="Lewis, D.", 
    TITLE="Conventions: A Philosophical Study", 
   PUBLISHER={Harvard University Press}, 
     ADDRESS={Cambridge, Mass.}, 
        YEAR=1969} 
 
 
@BOOK{Lewis76, 
      AUTHOR="Lewis, D.", 
       TITLE="Counterfactuals", 
   PUBLISHER="Basil Blackwell", 
     ADDRESS="Oxford", 
        YEAR="1976"} 
 
 
@ARTICLE{Lewis79, 
    AUTHOR="Lewis, D.", 
     TITLE="Prisoners' {D}ilemma as a {N}ewcomb problem", 
   JOURNAL="Philosophy and Public Affairs", 
      YEAR=1979, 
    VOLUME=8, 
     PAGES="235-240"} 
 
@ARTICLE{Lewis81, 
    AUTHOR="Lewis, D.", 
     TITLE="Causal Decision Theory", 
   JOURNAL="Australian Journal of Philosophy", 
      YEAR=1981, 
    VOLUME=51, 
     PAGES="5-30"} 
 
%Linhart, Radner and Schotter 
 
@UNPUBLISHED{LRandS89, 
      AUTHOR="Linhart, P. and R. Radner and A. Schotter", 
       TITLE="Behavior and Efficiency in the Sealed-Bid Mechanism", 
        YEAR="1989", 
        NOTE="Working Paper{,} New York University"} 
 
%Linster 
 
@PHDTHESIS{Linster90, 
      AUTHOR="Linster, B.", 
       TITLE="Essays on Cooperation and Competition", 
      SCHOOL="{U}niversity of {M}ichigan", 
        YEAR="1990"} 
 
%Lipman 
 
@UNPUBLISHED{Lipman, 
      AUTHOR="Lipman, B.", 
       TITLE="How to Decide How to Decide How to \ldots", 
        NOTE="Working Paper{,} Carnegie Mellon University", 
        YEAR="1989"} 
 
 
%Littlewood 
 
 
@BOOK{Littlewood, 
     AUTHOR="Littlewood, J.E.", 
      TITLE="Mathematical Miscellany", 
  PUBLISHER={Cambridge University Press}, 
    ADDRESS="London", 
       YEAR=1953, 
       NOTE="(Edited by {B}. {B}ollobas)"} 
 
%Locke 
 
@BOOK{Locke, 
    AUTHOR="Locke, J.", 
    TITLE="Two Treatises of Government", 
PUBLISHER={Cambridge University Press}, 
  ADDRESS={Cambridge}, 
     YEAR=1963, 
     NOTE="(First published 1690.)" } 
 
@BOOK{Locke75, 
      AUTHOR="Locke, J.", 
       TITLE="An Essay Concerning Human Understanding", 
   PUBLISHER="Clarendon Press", 
     ADDRESS="Oxford", 
        YEAR="1975", 
        NOTE="(Edited by P. Nidditch)"} 
 
%Luce and Raiffa 
 
@BOOK{LandR, 
    AUTHOR="Luce, R. and Raiffa, H.", 
    TITLE="Games and Decisions", 
   PUBLISHER={Wiley}, 
     ADDRESS={New York}, 
        YEAR=1957} 
 
%Lumsden and Wilson 
 
@BOOK{LandW, 
   AUTHOR="Lumsden C., and Wilson, E.", 
    TITLE="Genes{,} Mind and Culture", 
PUBLISHER={Harvard University Press}, 
  ADDRESS={Cambridge, Mass.}, 
     YEAR=1981} 
 
%M 
 
%Machina 
 
@ARTICLE{Machina87, 
      AUTHOR="Machina, M.", 
       TITLE="Choice under Uncertainty", 
     JOURNAL="Economic Perspectives", 
        YEAR="1987", 
      VOLUME="1", 
       PAGES="121-154"} 
 
%Macintyre 
 
@BOOK{Macintyre81, 
    AUTHOR="Macintyre, A.", 
    TITLE="After Virtue", 
   PUBLISHER={Duckworth}, 
     ADDRESS={London}, 
        YEAR=1981} 
 
@BOOK{Macintyre88, 
    AUTHOR="Macintyre, A.", 
    TITLE="Whose Justice? Which Rationality?", 
   PUBLISHER={Duckworth}, 
     ADDRESS={London}, 
        YEAR=1988} 
 
@BOOK{Macintyre90, 
    AUTHOR="Macintyre, A.", 
    TITLE="Three Rival Versions of Moral Enquiry", 
   PUBLISHER={Duckworth}, 
     ADDRESS={London}, 
        YEAR=1990} 
 
%Mackie 
 
@BOOK{Mackie77, 
    AUTHOR="Mackie, J.", 
    TITLE="Ethics{,} Inventing Right and Wrong", 
   PUBLISHER={Penguin Books}, 
     ADDRESS={London}, 
        YEAR=1977} 
 
@BOOK{Mackie80, 
    AUTHOR="Mackie, J.", 
    TITLE="Hume's Moral Theory", 
   PUBLISHER={Routledge and Kegan Paul}, 
     ADDRESS={London}, 
        YEAR=1980} 
 
@BOOK{Mackie76, 
    AUTHOR="Mackie, J.", 
    TITLE="Problems from Locke", 
   PUBLISHER={Clarendon Press}, 
     ADDRESS={Oxford}, 
        YEAR=1976} 
 
%MacMullen 
 
@BOOK{MacMullen88, 
    AUTHOR="Mac{M}ullen, R.", 
    TITLE="Corruption and the Decline of Rome", 
   PUBLISHER={Yale University Press}, 
     ADDRESS={New Haven}, 
        YEAR=1988} 
 
%Macpherson 
 
@ARTICLE{Macpherson54, 
    AUTHOR="Macpherson, C.", 
     TITLE="The Social Bearing of {L}ocke's Political Theory", 
   JOURNAL="Western Political Quarterly", 
      YEAR=19547, 
    VOLUME=7, 
     PAGES="1-22"} 
 
 
%Maskin 
 
@INCOLLECTION{Maskin85, 
    AUTHOR="Maskin, E.", 
     TITLE="The Theory of implementation in {N}ash equilbrium: a 
survey", 
   BOOKTITLE="Social Goals and Social Organization", 
     EDITOR="L. Hurwicz, D. Schmeidler and H. Sonnenschein", 
   PUBLISHER="Cambridge University Press", 
     ADDRESS="Cambridge", 
        YEAR="1985"} 
 
%Mandeville 
 
@BOOK{Mandeville14, 
     AUTHOR="Mandeville, B. de", 
      TITLE="The Fable of the Bees---or Private Vices, Publick 
Benefits", 
  PUBLISHER={Liberty Classics}, 
    ADDRESS={Indianapolis}, 
       YEAR=1988, 
       NOTE="(Edited by F. Kaye{,} first published 1714.)"} 
 
 
%Mansbridge 
 
@BOOK{Mansbridge90, 
    AUTHOR="Mansbridge, J.", 
    TITLE="Beyond Self-{I}nterest", 
   PUBLISHER={University of Chicago Press}, 
     ADDRESS={Chicago}, 
        YEAR=1990} 
 
%Margolis 
 
@BOOK{Margolis, 
    AUTHOR="Margolis, H.", 
    TITLE="Selfishness{,} Altruism{,} and Rationality", 
   PUBLISHER={Cambridge University Press}, 
     ADDRESS={Cambridge}, 
        YEAR=1982} 
 
%Martinez-Coll and Hirshleifer 
 
@ARTICLE{MCandH91, 
    AUTHOR="Martinez-Coll, J. and J. Hirshleifer", 
     TITLE="The Limits of Reciprocity", 
   JOURNAL="Rationality and Society", 
      YEAR="1991", 
    VOLUME="3", 
     PAGES="35-64"} 
 
%Maynard Smith 
 
@BOOK{MaynardSmith82, 
      AUTHOR="{Maynard Smith}, J.", 
       TITLE="Evolution and the Theory of Games", 
   PUBLISHER="Cambridge University Press", 
     ADDRESS="Cambridge", 
        YEAR="1982"} 
 
%Maynard Smith and Price 
 
@ARTICLE{MSandP, 
    AUTHOR="{Maynard Smith}, J. and G. Price", 
     TITLE="The Logic of Animal Conflict", 
   JOURNAL="Nature", 
      YEAR=1972, 
    VOLUME=246, 
     PAGES="15-18"} 
 
%Mayo 
 
@BOOK{Mayo86, 
      AUTHOR="Mayo, B.", 
       TITLE="The Philosophy of Right and Wrong", 
   PUBLISHER="Routledge and Kegan Paul", 
     ADDRESS="London and New York", 
        YEAR="1986"} 
 
%McClennen 
 
@BOOK{McClennen90, 
      AUTHOR="McClennen, E.", 
       TITLE="Rationality and Dynamic Choice", 
   PUBLISHER="Cambridge University Press", 
     ADDRESS="Cambridge", 
        YEAR="1990"} 
 
%McKelvey and Page 
 
@ARTICLE{MandP, 
    AUTHOR="McKelvey, R. and T. Page", 
     TITLE="Common Knowledge{,} Consensus{,} and Aggregate 
Information", 
   JOURNAL="Econometrica", 
      YEAR=1986, 
    VOLUME=54, 
     PAGES="109-127"} 
 
 
%McLean 
 
@ARTICLE{McLean, 
    AUTHOR="Mc{L}ean, I.", 
     TITLE="The Social Contract and the {P}risoners' {D}ilemma 
super-game", 
   JOURNAL="Political Studies", 
      YEAR=1981, 
    VOLUME=29, 
     PAGES="339-351"} 
 
%Meggido 
 
@TECHREPORT{Meggido, 
    AUTHOR="Meggido, N.", 
     TITLE="Approximating Common Knowledge by Common Beliefs", 
INSTITUTION="{I.B.M.}", 
      YEAR=1986, 
    NUMBER="Research Paper RJ5270"} 
 
@UNPUBLISHED{Meggido86, 
      AUTHOR="Meggido, N.", 
       TITLE="Remarks on Bounded Rationality", 
        YEAR="1986", 
        NOTE="{IBM} Research Paper RJ5270"} 
 
 
%Meggido and Wigderson 
 
@INCOLLECTION{MandW86, 
      AUTHOR="Meggido, N. and A. Wigderson", 
       TITLE="On Play by Means of Computing Machines", 
   BOOKTITLE="Theoretical Aspects of Reasoning about Knowledge", 
     EDITOR="J. Halpern", 
   PUBLISHER="Morgan Kaufmann", 
     ADDRESS="Los Altos", 
        YEAR="1986"} 
 
 
%Mendelson 
 
@BOOK{Mendelson, 
     AUTHOR="Mendelson, E.", 
      TITLE="Introduction to Mathematical Logic", 
  PUBLISHER={D. Van Nostrand}, 
    ADDRESS="New York", 
       YEAR=1964} 
 
 
%Mertens and Zamir 
 
@ARTICLE{MandZ, 
    AUTHOR="Mertens, J.F. and S. Zamir", 
     TITLE="Formulation of {B}ayesian Analysis for Games with 
Incomplete 
Information", 
   JOURNAL="International Journal of Game Theory", 
      YEAR=1985, 
    VOLUME="14", 
     PAGES="1-29"} 
 
%Milgrom 
 
@ARTICLE{Milgrom81, 
    AUTHOR="Milgrom, P.", 
     TITLE="An Axiomatic Characterization of Common Knowledge", 
   JOURNAL="Econometrica", 
      YEAR=1981, 
    VOLUME=49, 
     PAGES="219-222"} 
 
%Milgrom and Roberts 
 
@UNPUBLISHED{MandR90, 
      AUTHOR="Milgrom, P. and J. Roberts", 
       TITLE="Adaptive and Sophisticated Learning in Repeated 
Normal Form Games", 
        YEAR="1990", 
        NOTE="Working Paper{,} Stanford University"} 
 
 
%Milgrom and Stokey 
 
@ARTICLE{MandS, 
    AUTHOR="Milgrom, P. and N. Stokey", 
     TITLE="Information{,} Trade{,} and Common Knowledge", 
   JOURNAL="Journal of Economic Theory", 
      YEAR=1982, 
    VOLUME=26, 
     PAGES="177-27"} 
 
%Mill 
 
@INCOLLECTION{Mill63, 
     AUTHOR="Mill, J. S.", 
  BOOKTITLE="Utilitarianism and Other Essays", 
      TITLE="Utilitarianism", 
  PUBLISHER={Penguin Books}, 
    ADDRESS={Harmondsworth, Middlesex, U.K.}, 
       YEAR=1987, 
       NOTE="(Introduction by A. Ryan{;} essay first published 
1863.)"} 
 
%Milnor 
 
@INCOLLECTION{Milnor54, 
     AUTHOR="Milnor, J.", 
  BOOKTITLE="Decision Processes", 
      TITLE="Games against {N}ature", 
  PUBLISHER={Wiley}, 
    ADDRESS={New York}, 
       YEAR=1954, 
       NOTE="(Edited by R. Thrall, C. Coombs, and R. Davies.)"} 
 
@UNPUBLISHED{Milnor51, 
      AUTHOR="Milnor, J.", 
       TITLE="Games Against Nature", 
        YEAR="1951", 
        NOTE="{RAND} publication {RM}-679"} 
 
%Monderer and Samet 
 
@ARTICLE{MandSa, 
    AUTHOR="Monderer, D. and D. Samet", 
     TITLE="Approximating Common Knowledge with Common Beliefs", 
   JOURNAL="Games and Economic Behavior", 
      YEAR=1989, 
    VOLUME=1} 
 
%Montaigne 
 
@BOOK{Montaigne, 
    AUTHOR="{Montaigne}, M. de", 
     TITLE="Essays of Montaigne", 
 PUBLISHER={Edwards Brothers}, 
   ADDRESS={Ann Arbor}, 
      YEAR=1947} 
 
%Moore 
 
@BOOK{Moore02, 
    AUTHOR="Moore, G. E.", 
     TITLE="Principia Ethica", 
 PUBLISHER={Prometheus Books}, 
   ADDRESS={Buffalo, N.Y.}, 
      YEAR=1988, 
      NOTE="(First published 1902.)"} 
 
%Moulin 
 
@ARTICLE{MandV, 
    AUTHOR="Moulin, H. and J.P. Vial", 
     TITLE="Strategically Zero-Sum Games: The Class of Games 
Whose Completely Mixed Equilibria Cannot be Improved Upon", 
   JOURNAL="International Journal of Game Theory", 
      YEAR=1978, 
    VOLUME=7, 
     PAGES="201-221"} 
 
@ARTICLE{Moulin84, 
    AUTHOR="Moulin, H.", 
     TITLE="Implementing the {K}alai-{S}morodinsky Bargaining 
Solution", 
   JOURNAL="Journal of Economic Theory", 
      YEAR=1984, 
    VOLUME=33, 
     PAGES="32-45"} 
 
@BOOK{Moulin88, 
    AUTHOR="Moulin, H.", 
     TITLE="Axioms of Cooperative Decision-Making", 
 PUBLISHER={Cambridge University Press}, 
   ADDRESS={Cambridge}, 
      YEAR=1988} 
 
%Mulhall and Swift 
 
@BOOK{MandS92, 
    AUTHOR="Mulhall, S. and A. Swift", 
     TITLE="Liberals and Communitarians", 
 PUBLISHER={Blackwell}, 
   ADDRESS={Oxford}, 
      YEAR=1992} 
 
%Munro 
 
@BOOK{Munro69, 
    AUTHOR="Munro, D.", 
     TITLE="The Concept of Man in Early {C}hina", 
 PUBLISHER={Stanford University Press}, 
   ADDRESS={Stanford, Calif.}, 
      YEAR=1969} 
 
%Muthoo 
 
@UNPUBLISHED{Muthoo91, 
      AUTHOR="Muthoo, A.", 
       TITLE="Revocable Commitment and Sequential Bargaining", 
        YEAR="1991", 
        NOTE="To appear in Economic Journal"} 
 
%Myerson 
 
@ARTICLE{Myerson86, 
      AUTHOR="Myerson, R.", 
       TITLE="Multistage Games with Communication", 
     JOURNAL="Econometrica", 
        YEAR="1986", 
      VOLUME="54", 
       PAGES="323-358"} 
 
@BOOK{Myerson91, 
   AUTHOR="Myerson, R.", 
    TITLE="Game Theory: Analysis of Conflict", 
PUBLISHER={Harvard University Press}, 
  ADDRESS={Cambridge, Mass.}, 
     YEAR=1991} 
 
@ARTICLE{Myerson77, 
    AUTHOR="Myerson, R.", 
     TITLE="Two-person Bargaining and Comparable Utility", 
   JOURNAL="Econometrica", 
      YEAR=1977, 
    VOLUME=45, 
     PAGES="1631-1637"} 
 
%Myerson and Satterthwaite 
 
@ARTICLE{MandS83, 
    AUTHOR="Myerson, R. and Satterthwaite, M.", 
     TITLE="Efficient Mechanisms for Bilateral Trading", 
   JOURNAL="Journal of Economic Theory", 
      YEAR=1983, 
    VOLUME=29, 
     PAGES="265-281"} 
 
%N 
 
%Nachbar 
 
@UNPUBLISHED{Nachbar89, 
      AUTHOR="Nachbar, J.", 
       TITLE="The Evolution of Cooperation Revisited", 
        YEAR="1989", 
        NOTE="Working Paper{,} {R}and Cooperation"} 
 
%Nagel 
 
@BOOK{Nagel70, 
   AUTHOR="Nagel, T.", 
    TITLE="The Possibility of Altruism", 
PUBLISHER={Clarendon Press}, 
  ADDRESS={Oxford}, 
     YEAR=1970} 
 
%Nash 
 
@ARTICLE{Nash50, 
    AUTHOR="Nash, J.", 
     TITLE="The Bargaining Problem", 
   JOURNAL="Econometrica", 
      YEAR=1950, 
    VOLUME=18, 
     PAGES="155-162"} 
 
@ARTICLE{Nash51, 
    AUTHOR="Nash, J.", 
     TITLE="Non-Cooperative Games", 
   JOURNAL="Annals of Mathematics", 
      YEAR=1951, 
    VOLUME=54, 
     PAGES="286-295"} 
 
@ARTICLE{Nash53, 
    AUTHOR="Nash, J.", 
     TITLE="Two-Person Cooperative Games", 
   JOURNAL="Econometrica", 
      YEAR=1953, 
    VOLUME=21, 
     PAGES="128-140"} 
 
 
%Nau 
 
@TECHREPORT{Nau89, 
      AUTHOR="Nau, R.", 
       TITLE="Joint Coherence in Games of Incomplete Information", 
 INSTITUTION="Duke University", 
        YEAR="1989", 
        NOTE="Fuqua School discussion paper 8929"} 
 
 
%Nau and McCardle 
 
@ARTICLE{NandM90, 
      AUTHOR="Nau, R. and K. Mc{C}ardle", 
       TITLE="Coherent Behavior in Noncooperative Games", 
     JOURNAL="Journal of Economic Theory", 
        YEAR="1990", 
      VOLUME="50", 
       PAGES="424-444"} 
 
%Neyman 
 
@ARTICLE{Neyman86, 
      AUTHOR="Neyman, A.", 
       TITLE="Bounded Complexity Justifies Cooperation in the 
Finitely Repeated Prisoners' Dilemma", 
     JOURNAL="Economic Letters", 
        YEAR="1986", 
      VOLUME="19", 
       PAGES="227-229"} 
 
 
 
%Nielsen 
 
@ARTICLE{Nielsen, 
    AUTHOR="Nielsen, L.", 
     TITLE="Common Knowledge{,} Communication{,} and Convergence of 
Beliefs", 
   JOURNAL="Mathematical Social Sciences", 
      YEAR=1984, 
    VOLUME=8, 
     PAGES="1-14"} 
 
 
%Nietzsche 
 
@INCOLLECTION{Nietzsche64, 
     AUTHOR="Nietzsche, F.", 
      TITLE="Human{,} All Too Human", 
  BOOKTITLE="The Complete Works of Friedrich Nietzsche", 
  PUBLISHER={Russell and Russell}, 
    ADDRESS={New York}, 
     EDITOR="O. Levy", 
       YEAR=1964} 
 
@BOOK{Nietzsche86, 
   AUTHOR="Nietzsche, F.", 
    TITLE="Beyond Good and Evil", 
PUBLISHER={Penguin Books}, 
  ADDRESS={Harmondsworth, UK}, 
     YEAR=1973, 
     NOTE="(Translated by {R}. {H}ollingdale{,} first published 
1886.)"} 
 
@BOOK{Nietzsche56, 
   AUTHOR="Nietzsche, F.", 
    TITLE="The Birth of Tragedy and the Geneology of Morals", 
PUBLISHER={Doubleday}, 
  ADDRESS={New York}, 
     YEAR=1956} 
 
@BOOK{Nietzsche10, 
   AUTHOR="Nietzsche, F.", 
    TITLE="The Will to Power, {\rm Volume II}", 
PUBLISHER={Foulis}, 
  ADDRESS={Edinburgh}, 
     YEAR=1910, 
    NOTES={(Translated by A. Ludovicu)}} 
 
%Nozick 
 
@INCOLLECTION{Nozick69, 
     AUTHOR="Nozick, R.", 
      TITLE="Newcomb's problem and two principles of choice", 
  BOOKTITLE="Essays in Honor of {C}arl {G}{.} {H}empel", 
  PUBLISHER={Reidel}, 
    ADDRESS={Dordrecht, Netherlands}, 
     EDITOR="N. Rescher", 
       YEAR=1969} 
 
@BOOK{Nozick74, 
    AUTHOR="Nozick, R.", 
    TITLE="Anarchy{,} State, and Utopia", 
   PUBLISHER={Basic Books}, 
     ADDRESS={New York}, 
        YEAR=1974} 
 
 
%O 
 
%Olson 
 
@BOOK{Olson, 
    AUTHOR="Olson, M.", 
    TITLE="The Logic of Collective Action", 
   PUBLISHER={Harvard University Press}, 
     ADDRESS={Cambridge, Mass}, 
        YEAR=1965} 
 
@UNPUBLISHED{Olson86, 
    AUTHOR="Olson, M.", 
     TITLE="A New Approach to the Ethics of Income Distribution", 
      YEAR="1986", 
      NOTE="Manville American Enterprise Lecture, University of 
Notre Dame"} 
 
%Orwell 
 
@BOOK{Orwell, 
      AUTHOR="Orwell, G.", 
       TITLE="Animal Farm", 
   PUBLISHER={New American Library{,} Signet Classic}, 
     ADDRESS={New York}, 
        YEAR=1960, 
        NOTE="(First published 1946.)"} 
 
 
@INCOLLECTION{Orwell80, 
     AUTHOR="Orwell, G.", 
      TITLE="Writers and Leviathan", 
  BOOKTITLE="Selections from Essays and Journalism: 1931-1949", 
  PUBLISHER={Martin, Secker and Warburg}, 
    ADDRESS={London}, 
     EDITOR="H. Kuhn and A. Tucker", 
       YEAR=1980} 
 
%Owen 
 
@BOOK{Owen, 
    AUTHOR="Owen, G.", 
    TITLE="Game Theory. {\rm 2nd edition}", 
   PUBLISHER={Academic Press}, 
     ADDRESS={New York}, 
        YEAR=1982} 
 
%P 
 
%Pareto 
 
@BOOK{Pareto, 
    AUTHOR="Pareto, V.", 
    TITLE="Manuale di Economia Politica", 
   PUBLISHER={Societa Editrice Libraria}, 
     ADDRESS={Milan}, 
        YEAR=1906} 
 
 
%Pascal 
 
@BOOK{Pascal, 
    AUTHOR="Pascal, B.", 
    TITLE="Pascal's Pens\'{e}es", 
   PUBLISHER={Pantheon}, 
     ADDRESS={New York}, 
        YEAR=1950, 
        NOTE="(Translated by H. F. Stewart)"} 
 
 
%Pattanaik and Suzumura 
 
@UNPUBLISHED{PandS90, 
    AUTHOR="Pattanaik, P. and K. Suzumura", 
     TITLE="Professor {S}en on Minimal Liberty", 
      YEAR="1990", 
      NOTE="Discussion Paper A231{,} Institute of Economic 
Research, 
Hitotsubashi University, Tokyo"} 
 
%Pearce 
 
@ARTICLE{Pearce84, 
    AUTHOR="Pearce, D.", 
     TITLE="Rationalizable Strategic Behavior and the Problem 
of Perfection", 
   JOURNAL="Econometrica", 
      YEAR=1984, 
    VOLUME=52, 
     PAGES="1029-1050"} 
 
%Penrose 
 
@BOOK{Penrose89, 
      AUTHOR="Penrose, R.", 
       TITLE="The Emperor's New Mind", 
   PUBLISHER="Oxford University Press", 
     ADDRESS="Oxford", 
        YEAR="1989"} 
 
%Peters 
 
@PHDTHESIS{Peters86, 
      AUTHOR="Peters, H.", 
       TITLE="Bargaining Game Theory", 
      SCHOOL="{P}roefschritt Universitat Nijmegen", 
        YEAR="1986"} 
 
%Pettit and Sugden 
 
@UNPUBLISHED{PandS, 
      AUTHOR="Pettit, P. and R. Sugden", 
       TITLE="The Backwards Induction Paradox", 
        YEAR="1989", 
        NOTE="Working Paper{,} University of East Anglia{,} {UK}"} 
 
 
%Potryagin, Andropov and Vitt 
 
@INCOLLECTION{PAandV89, 
      AUTHOR="Pontryagin, L. and A. Andronov and A. Vitt", 
       TITLE="On the Statistical Treatment of Dynamical Systems", 
   BOOKTITLE="Noise in Nonlinear Dynamical Systems", 
      EDITOR="F. Moss and P. McClintock", 
   PUBLISHER="Cambridge University Press", 
     ADDRESS="Cambridge", 
        YEAR="1989"} 
 
%Poundstone 
 
@BOOK{Poundstone88, 
    AUTHOR="Poundstone, W.", 
    TITLE="Labyrinths of Reason", 
   PUBLISHER={Doubleday}, 
     ADDRESS={New York}, 
        YEAR=1988} 
 
%Probst 
 
@UNPUBLISHED{Probst92, 
      AUTHOR="Probst, D.", 
       TITLE="Evolution in the Repeated {P}risoners' {D}ilemma", 
       YEAR="1992", 
        NOTE="Working Paper (in preparation){,} University of 
Bonn"} 
 
 
@UNPUBLISHED{Probst93, 
      AUTHOR="Probst, D.", 
       TITLE="Evolution, Automata and the Repeated {P}risoners' 
{D}ilemma", 
       YEAR="1993", 
        NOTE="Working Paper{,} London Scool of Economics"} 
 
%Q 
 
@BOOK{Quine66, 
      AUTHOR="Quine, W.", 
       TITLE="The Ways of Paradox and Other Essays", 
   PUBLISHER="Harvard University Press", 
     ADDRESS="Cambridge, Mass.", 
        YEAR="1966"} 
 
%R 
 
%Raiffa 
 
@INCOLLECTION{Raiffa53, 
     AUTHOR="Raiffa, H.", 
      TITLE="Arbitration Schemes for Generalized Two-Person Games", 
  BOOKTITLE="Contributions to the Theory of Games {II}", 
  PUBLISHER={Princeton University Press}, 
    ADDRESS={Princeton}, 
     EDITOR="H. Kuhn and A. Tucker", 
       YEAR=1953} 
 
%Rapoport 
 
@BOOK{Rapoport66, 
      AUTHOR="Rapoport, A.", 
       TITLE="Two-Person Game Theory", 
   PUBLISHER="University of Michigan Press", 
     ADDRESS="Ann Arbor, Mich.", 
        YEAR="1966"} 
 
%Rawls 
 
@ARTICLE{Rawls58, 
    AUTHOR="Rawls, J.", 
     TITLE="Justice as Fairness", 
   JOURNAL="Philosophical Review", 
      YEAR=1958, 
    VOLUME=57, 
     PAGES="185-187"} 
 
@ARTICLE{Rawls68, 
    AUTHOR="Rawls, J.", 
     TITLE="Distributive Justice: {S}ome Addenda", 
   JOURNAL="Natural Law Forum", 
      YEAR=1968, 
    VOLUME=13} 
 
@BOOK{Rawls72, 
    AUTHOR="Rawls, J.", 
    TITLE="A Theory of Justice", 
   PUBLISHER={Oxford University Press}, 
     ADDRESS={Oxford}, 
        YEAR=1972} 
 
@ARTICLE{Rawls74, 
    AUTHOR="Rawls, J.", 
     TITLE="Some Reasons for the Maximin Criterion", 
   JOURNAL="American Economic Review", 
      YEAR=1974, 
    VOLUME="64 (papers and proceedings)", 
     PAGES="141-146"} 
 
@ARTICLE{Rawls80, 
    AUTHOR="Rawls, J.", 
     TITLE="Kantian Constructivism in Moral Theory", 
   JOURNAL="Journal of Philosophy", 
      YEAR=1980, 
    VOLUME="88", 
     PAGES="575-582"} 
 
 
 
@INCOLLECTION{Rawls82, 
     AUTHOR="Rawls, J.", 
      TITLE="Social Unity and Primary Goods", 
  BOOKTITLE="Utilitarianism and Beyond", 
  PUBLISHER={Cambridge University Press}, 
    ADDRESS={Cambridge}, 
       YEAR=1982, 
     EDITOR="A. Sen and B. Williams"} 
 
@BOOK{Rawls93, 
      AUTHOR="Rawls, J.", 
       TITLE="Political Liberalism", 
   PUBLISHER="Columbia University Press", 
     ADDRESS="New York", 
        YEAR="1993"} 
 
 
 
%Reny 
 
@UNPUBLISHED{Reny85, 
    AUTHOR="Reny, P.", 
     TITLE="Rationality{,} Common Knowledge{,} and the Theory of 
Games", 
      YEAR="1985", 
      NOTE="Working Paper{,} University of Western Ontario"} 
 
 
%Redondi 
 
@BOOK{Redondi87, 
      AUTHOR="Redondi, P.", 
       TITLE="Galileo: Heretic", 
   PUBLISHER="Princeton University Press", 
     ADDRESS="Princeton", 
        YEAR="1987", 
        NOTE="(Translated by R. Rosenthal)"} 
 
%Riker 
 
@BOOK{Riker82, 
      AUTHOR="Riker, W.", 
       TITLE="Liberalism against Populism", 
   PUBLISHER="Waveland Press", 
     ADDRESS="Prospect Heights, Ill.", 
        YEAR="1982"} 
 
%Risken 
 
@BOOK{Risken84, 
      AUTHOR="Risken, H.", 
       TITLE="The Fokker-Planck Equation", 
   PUBLISHER="Springer-Verlag", 
     ADDRESS="Berlin", 
        YEAR="1984"} 
 
 
%Robbins 
 
@BOOK{Robbins35, 
      AUTHOR="Robbins, L.", 
       TITLE="An Essay on the Nature and Significance of Economic 
Science", 
   PUBLISHER="Macmillan", 
     ADDRESS="London", 
        YEAR="1935"} 
 
@ARTICLE{Robbins38, 
    AUTHOR="Robbins, L.", 
     TITLE="Inter-personal comparisons of utility", 
   JOURNAL="Economic Journal", 
      YEAR=1938, 
    VOLUME=48, 
     PAGES="635-641"} 
 
%Robson 
 
@ARTICLE{Robson89, 
    AUTHOR="Robson, A.", 
     TITLE="Efficiency in Evolutionary Games{:} {D}arwin{,} 
{N}ash{,} 
and the Secret Handshake", 
   JOURNAL="Journal of Theoretical Biology", 
      YEAR=1990, 
    VOLUME=144, 
     PAGES="379-396"} 
 
 
%Rodriguez 
 
@ARTICLE{Rodriguez, 
    AUTHOR="Rodriguez, A.", 
     TITLE="Rawls' Maximin Criterion and Time Consistency: {A} 
Generalisation", 
   JOURNAL="Review of Economic Studies", 
      YEAR=1981, 
    VOLUME="48", 
     PAGES="599-605"} 
 
%Roemer 
 
@BOOK{Roemer82, 
      AUTHOR="Roemer, J.", 
       TITLE="A General Theory of Exploitation and Class", 
   PUBLISHER="Harvard University Press", 
     ADDRESS="Cambridge, Mass.", 
        YEAR="1982"} 
 
@BOOK{Roemer88, 
      AUTHOR="Roemer, J.", 
       TITLE="Free to Lose", 
   PUBLISHER="Radius", 
     ADDRESS="London", 
        YEAR="1988"} 
 
@BOOK{Roemer81, 
      AUTHOR="Roemer, J.", 
       TITLE="Analytic Foundations of Marxian Economic Theory", 
   PUBLISHER="Cambridge University Press", 
     ADDRESS="Cambridge", 
        YEAR="1981"} 
 
%Rosenthal 
 
@ARTICLE{Rosenthal81, 
    AUTHOR="Rosenthal, R.", 
     TITLE="Games of Perfect Information{,} Predatory Pricing{,} 
and 
Chain-Store Paradox", 
   JOURNAL="Journal of Economic Theory", 
      YEAR=1981, 
    VOLUME=25, 
     PAGES="92-100"} 
 
%Roth 
 
@BOOK{Roth79, 
     AUTHOR="Roth, A.", 
      TITLE="Axiomatic Models of Bargaining", 
  PUBLISHER={Springer-Verlag}, 
    ADDRESS={Berlin}, 
       YEAR=1979} 
 
@INCOLLECTION{Roth91, 
     AUTHOR = "Roth, A.", 
     TITLE = "Bargaining Experiments", 
     BOOKTITLE = "Handbook of Experimental Economics", 
     PUBLISHER = "Princeton University Press", 
     YEAR = 1991, 
     EDITOR = "J. Kagel and A. Roth", 
     NOTE = "Forthcoming"} 
 
%Roth and Erev 
 
@UNPUBLISHED{RandE93, 
      AUTHOR="Roth, A. and I. Erev", 
       TITLE="Learning in Extensive-Form Games: Experimental Data 
and Simple 
Dynamic Models in the Medium Term", 
        YEAR="1993", 
        NOTE="Working Paper{,} UNiversity of Pittsburgh"} 
 
%Rousseau 
 
@INCOLLECTION{Rousseau62a, 
      AUTHOR="Rousseau, J.-J.", 
       TITLE="The Social Contract", 
   BOOKTITLE="Rousseau's Social Contract and Discourses", 
     EDITOR="G. Cole", 
   PUBLISHER="J. M. Dent", 
     ADDRESS="London", 
        YEAR="1913", 
       PAGES="5-123", 
        NOTE={(First published 1762.)}} 
 
 
@INCOLLECTION{Rousseau55a, 
      AUTHOR="Rousseau, J.-J.", 
       TITLE="The Inequality of Man", 
   BOOKTITLE="Rousseau's Social Contract and Discourses", 
     EDITOR="G. Cole", 
   PUBLISHER="J. M. Dent", 
     ADDRESS="London", 
        YEAR="1913", 
       PAGES="157-246", 
        NOTE={(First published 1755.)}} 
 
@INCOLLECTION{Rousseau55b, 
      AUTHOR="Rousseau, J.-J.", 
       TITLE="A Discourse on Political Economy", 
   BOOKTITLE="Rousseau's Social Contract and Discourses", 
     EDITOR="G. Cole", 
   PUBLISHER="J. M. Dent", 
     ADDRESS="London", 
        YEAR="1913", 
       PAGES="249-287", 
        NOTE={(First published 1755.)}} 
 
@BOOK{Rousseau62b, 
     AUTHOR="Rousseau, J.-J.", 
      TITLE="Emile", 
  PUBLISHER={J. M. Dent}, 
    ADDRESS={London}, 
       YEAR=1908, 
       NOTE={(First published 1762.}} 
 
%Rubinstein 
 
@ARTICLE{Rubinstein82, 
      AUTHOR="Rubinstein, A.", 
       TITLE="Perfect Equilibrium in a Bargaining Model", 
     JOURNAL="Econometrica", 
        YEAR="1982", 
      VOLUME="50", 
       PAGES="97-109"} 
 
@ARTICLE{Rubinstein85a, 
      AUTHOR="Rubinstein, A.", 
       TITLE="A Bargaining Model with Incomplete Information about 
Time Preferences", 
     JOURNAL="Econometrica", 
        YEAR="1985", 
      VOLUME="53", 
       PAGES="1151-1172"} 
 
 
@ARTICLE{Rubinstein86, 
      AUTHOR="Rubinstein, A.", 
       TITLE="Finite Automata Play the Repeated Prisoners' 
Dilemma", 
     JOURNAL="Journal of Economic Theory", 
        YEAR="1986", 
      VOLUME="39", 
       PAGES="83-96"} 
 
@INCOLLECTION{Rubinstein87, 
      AUTHOR="Rubinstein, A.", 
       TITLE="The Complexity of Strategies and the Resolution of 
Conflict:  An Introduction", 
   BOOKTITLE="Global Macroeconomics:  Policy Conflict and 
Cooperation", 
     EDITOR="Bryant and Portes", 
   PUBLISHER="Macmillan Press", 
     ADDRESS="New York", 
        YEAR="1987", 
       PAGES="17-32"} 
 
 
@ARTICLE{Rubinstein89, 
    AUTHOR="Rubinstein, A.", 
     TITLE="The Electronic Mail Game:  Strategic Behavior Under 
``Almost Common Knowledge''", 
   JOURNAL="American Economic Review", 
      YEAR=1989, 
    VOLUME={70}, 
     PAGES="{385-391}"} 
 
%Rubinstein and Wolinsky 
 
@ARTICLE{RandW85, 
    AUTHOR="Rubinstein, A. and Wolinsky, A.", 
     TITLE="Equilibrium in a Market with Sequential Bargaining", 
   JOURNAL="Econometrica", 
      YEAR=1985, 
    VOLUME={53}, 
     PAGES="{1133-1150}"} 
 
@ARTICLE{RandW87, 
    AUTHOR="Rubinstein, A. and Wolinsky, A.", 
     TITLE="Middlemen", 
   JOURNAL="Quarterly Journal of Economics", 
      YEAR=1987, 
    VOLUME={102}, 
     PAGES="{581-593}"} 
 
@UNPUBLISHED{RandW, 
    AUTHOR="Rubinstein, A. and Wolinsky, A.", 
     TITLE="A Comment on the Logic of `Agreeing to Disagree' 
Results", 
      NOTE="{F}orthcoming in {\it Journal of Economic Theory}", 
      YEAR=1988} 
 
@BOOK{Ryle49, 
     AUTHOR="Ryle, G.", 
      TITLE="The Concept of Mind", 
  PUBLISHER={Hutchinson}, 
    ADDRESS={London}, 
       YEAR=1949} 
 
%S 
 
%Samet 
 
@TECHREPORT{Samet87, 
     AUTHOR="Samet, D.", 
      TITLE="Ignoring Ignorance and Agreeing to Disagree", 
INSTITUTION="Northwestern University", 
       YEAR=1987, 
     NUMBER="{M}.{E}.{D}.{S} Discussion Paper No.~749{,} {KGSM}", 
       NOTE="{F}orthcoming in {\it {J}ournal of {E}conomic 
{T}heory}"} 
 
%Samuelson 
 
@UNPUBLISHED{Samuelson90, 
      AUTHOR="Samuelson, L.", 
       TITLE="Evolutionary Foundations of Solution Concepts for 
Finite, Two-Player, Normal-Form Games", 
        YEAR="1990", 
        NOTE="Working Paper{,} State University of Pennsylvania"} 
 
@UNPUBLISHED{Samuelson89, 
      AUTHOR="Samuelson, L.", 
       TITLE="Evolutionary Stability in Asymmetric Games", 
        YEAR="1989", 
        NOTE="Working Paper{,} State University of Pennsylvania"} 
 
@UNPUBLISHED{Samuelson93a, 
     AUTHOR="Samuelson, L.", 
      TITLE="Does Evolution Eliminate Dominated Strategies?", 
       YEAR="1993", 
       NOTE="Working Paper{,} University of Wisconsin"} 
 
@UNPUBLISHED{Samuelson93b, 
     AUTHOR="Samuelson, L.", 
      TITLE="Stochastic Stability in Games with Alternative Best 
Replies", 
       YEAR="1993", 
       NOTE="To appear in {\em Journal of Economic Theory}."} 
 
%Samuelson and Zhang 
 
@ARTICLE{SandZ92, 
     AUTHOR = "Samuelson, L. and J. Zhang", 
     TITLE = "Evolutionary Stability in Asymmetric Games", 
     JOURNAL = "Journal of Economic Theory", 
      VOLUME ="57", 
      PAGES="364-391", 
       YEAR = 1992} 
 
%Sandel 
 
@ARTICLE{Sandel84, 
    AUTHOR="Sandel, M.", 
     TITLE="The Procedural Republic and the Unencumbered Self", 
   JOURNAL="Political Theory", 
      YEAR=1984, 
    VOLUME=12, 
     PAGES="81-96"} 
 
@BOOK{Sandel82, 
     AUTHOR="Sandel, M.", 
      TITLE="Liberalism and the Limits of Justice", 
  PUBLISHER={Cambridge University Press}, 
    ADDRESS={Cambridge}, 
       YEAR=1982} 
 
 
%Sandford 
 
@BOOK{Sandford89, 
     AUTHOR="Sandford, D.", 
      TITLE="If P{,} Then Q", 
  PUBLISHER={Routledge}, 
    ADDRESS={London}, 
       YEAR=1989} 
 
%Savage 
 
@BOOK{Savage, 
     AUTHOR="Savage, L.", 
      TITLE="The Foundations of Statistics", 
  PUBLISHER={Wiley}, 
    ADDRESS={New York}, 
       YEAR=1951} 
 
%Schelling 
 
@BOOK{Schelling, 
     AUTHOR="Schelling, T.", 
      TITLE="The Strategy of Conflict", 
  PUBLISHER={Harvard University Press}, 
    ADDRESS={Cambridge, Mass.}, 
       YEAR=1960} 
 
%Schopenhauer 
 
@BOOK{Schopenhauer19, 
     AUTHOR="Schopenhauer, A.", 
      TITLE="The World as Will and Representation", 
  PUBLISHER={Dover}, 
    ADDRESS={New York}, 
       YEAR=1969, 
       NOTE="(Translated by E. Payne, first published 1819.)"} 
 
%Schotter 
 
@BOOK{Schotter81, 
     AUTHOR="Schotter, A.", 
      TITLE="The Economic Theory of Social Institutions", 
  PUBLISHER={Cambridge University Press}, 
    ADDRESS={Cambridge}, 
       YEAR=1981} 
 
@BOOK{Schotter90, 
     AUTHOR="Schotter, A.", 
      TITLE="Free Market Economics. {\rm 2nd edition}", 
  PUBLISHER={Basil Blackwell}, 
    ADDRESS={Oxford}, 
       YEAR=1990} 
 
%Searle 
 
@INCOLLECTION{Searle81, 
      AUTHOR="Searle, J.", 
       TITLE="Minds, Brains and Programs", 
   BOOKTITLE="The Mind's I", 
      EDITOR="D. Hofstadter and D. Dunnett", 
   PUBLISHER="Basic Books", 
     ADDRESS="Harmondsworth, UK", 
        YEAR="1981"} 
 
@ARTICLE{Searle, 
    AUTHOR="Searle, J. R.", 
     TITLE="How to Derive ``Ought'' from ``Is''", 
   JOURNAL="Philosophical Review", 
      YEAR=1964, 
    VOLUME=73, 
     PAGES="43-58"} 
 
%Selten 
 
@ARTICLE{Selten75, 
    AUTHOR="Selten, R.", 
     TITLE="Reexamination of the Perfectness Concept for 
Equilibrium Points in Extensive-Games", 
   JOURNAL="International Journal of Game Theory", 
      YEAR=1975, 
    VOLUME=4, 
     PAGES="25-55"} 
 
@ARTICLE{Selten78, 
      AUTHOR="Selten, R.", 
       TITLE="The Chain-Store Paradox", 
     JOURNAL="Theory and Decision", 
        YEAR="1978", 
      VOLUME="9", 
       PAGES="127-159"} 
 
@UNPUBLISHED{Selten83, 
      AUTHOR="Selten, R.", 
       TITLE="Evolutionary Stability in Extensive 2-person Games", 
        YEAR="1983", 
        NOTE="Bielefeld Working Papers 121 and 122"} 
 
@UNPUBLISHED{Selten88, 
    AUTHOR="Selten, R.", 
     TITLE="Anticipatory Learning in Two-Person Games", 
      NOTE="{Friedrich-Wilhelms-Universit\"{a}t Bonn}{,} working 
paper B-93", 
      YEAR="1988"} 
 
@BOOKLET{Selten89, 
    AUTHOR="Selten, R.", 
     TITLE="Evolution, Learning and Economic Behavior", 
      NOTE="{N}ancy {L}. {S}chwarz Memorial Lecture", 
   ADDRESS="Kellogg Graduate School, Northwestern University", 
      YEAR="1989"} 
 
 
%Selten and Harsanyi 
 
@BOOK{SandH88, 
      AUTHOR="Selten, R. and J. Harsanyi", 
       TITLE="A General Theory of Equilibrium Selection in Games", 
   PUBLISHER="MIT Press", 
     ADDRESS="Cambridge", 
        YEAR="1988"} 
 
%Selten and Leopold 
 
@INCOLLECTION{SandL82, 
      AUTHOR="Selten, R. and U. Leopold", 
       TITLE="Subjunctive Conditionals in Decision Theory and Game 
Theory", 
   BOOKTITLE="Studies in Economics", 
        NOTE="{\em Philosophy of Economics}{,} {V}ol. 2", 
      EDITOR="Stegmuller/Balzer/Spohn", 
   PUBLISHER="Springer-Verlag", 
     ADDRESS="Berlin", 
        YEAR="1982"} 
 
%Sen 
 
@BOOK{Sen70, 
     AUTHOR="Sen, A.", 
      TITLE="Collective Choice and Social Welfare", 
  PUBLISHER={Holden Day}, 
    ADDRESS={San Francisco}, 
       YEAR=1970} 
 
@ARTICLE{Sen70b, 
    AUTHOR="Sen, A.", 
     TITLE="The Impossibility of a {P}aretian Liberal", 
   JOURNAL="Journal of Political Economy", 
      YEAR=1970, 
    VOLUME=78, 
     PAGES="152-157"} 
 
@ARTICLE{Sen71, 
    AUTHOR="Sen, A.", 
     TITLE="Choice Functions and Revealed Preference", 
   JOURNAL="Review of Economic Studies", 
      YEAR=1971, 
    VOLUME=38, 
     PAGES="307-317"} 
 
@ARTICLE{Sen73, 
    AUTHOR="Sen, A.", 
     TITLE="Behaviour and the Concept of Preference", 
   JOURNAL="Economica", 
      YEAR=1973, 
    VOLUME=40, 
     PAGES="241-259"} 
 
@ARTICLE{Sen73a, 
    AUTHOR="Sen, A.", 
     TITLE="On Ignorance and Equal Distribution", 
   JOURNAL="American Economic Review", 
      YEAR=1973, 
    VOLUME=63, 
     PAGES="1022-1024"} 
 
@INCOLLECTION{Sen74a, 
     AUTHOR="Sen, A.", 
  BOOKTITLE="Practical Reason", 
      TITLE="Choice Orderings and Morality", 
  PUBLISHER={Basil Blackwell}, 
    ADDRESS={Oxford}, 
       YEAR=1974, 
       NOTE="(Edited by S. Korner)"} 
 
@ARTICLE{Sen76, 
    AUTHOR="Sen, A.", 
     TITLE="Welfare Inequalities and {R}awlsian Axiomatics", 
   JOURNAL="Theory and Decision", 
      YEAR=1976, 
    VOLUME=7, 
     PAGES="243-262"} 
 
@ARTICLE{Sen77, 
    AUTHOR="Sen, A.", 
     TITLE="Rational Fools: {A} Critique of the Behavioral 
Foundations of Economic Theory", 
   JOURNAL="Philosophy and Public Affairs", 
      YEAR=1977, 
    VOLUME=6, 
     PAGES="317-344"} 
 
@BOOK{Sen87, 
     AUTHOR="Sen, A.", 
      TITLE="On Ethics and Economics", 
  PUBLISHER={Basil Blackwell}, 
    ADDRESS={Oxford}, 
       YEAR=1987} 
 
 
@ARTICLE{Sen92, 
    AUTHOR="Sen, A.", 
     TITLE="Minimal Liberty", 
   JOURNAL="Economica", 
      YEAR=1992, 
    VOLUME=59, 
     PAGES="139-160"} 
 
%Seymour 
 
@UNPUBLISHED{Seymour93, 
      AUTHOR="Seymour, R.", 
       TITLE="Continuous-Time Models of Evolutionary Games I: 
Populations of Fixed Size", 
        YEAR="1993", 
        NOTE="Mathematics Working Paper{,} University College London"} 
 
%Sidgwick 
 
@BOOK{Sidgwick, 
     AUTHOR="Sidgwick, H.", 
      TITLE="The Methods of Ethics", 
  PUBLISHER={Hackett}, 
    ADDRESS={Indianapolis}, 
       YEAR=1981, 
       NOTE={(Seventh edition of 1907.)}} 
 
%Shin 
 
@BOOK{Shin87, 
     AUTHOR="Shin, H.", 
      TITLE="Logical Structure of Common Knowledge{,} I and II", 
  PUBLISHER="Nuffield College", 
    ADDRESS="Oxford", 
       YEAR=1987} 
 
@UNPUBLISHED{Shin88, 
      AUTHOR="Shin, H.", 
       TITLE="A Comment on {A}umann's Definition of 
{B}ayes-rationality", 
        YEAR="1988", 
        NOTE="Working Paper{,} Nuffield College"} 
 
 
%Shubik 
 
@BOOK{Shubik82, 
     AUTHOR="Shubik, M.", 
      TITLE="Game Theory in the Social Sciences", 
  PUBLISHER={MIT Press}, 
    ADDRESS={Cambridge, Mass}, 
       YEAR=1982} 
 
@BOOK{Shubik84, 
     AUTHOR="Shubik, M.", 
      TITLE="A Game-Theoretic Approach to Political Economy", 
  PUBLISHER={MIT Press}, 
    ADDRESS={Cambridge, Mass}, 
       YEAR=1984} 
 
%Simon 
 
@BOOK{Simon57, 
      AUTHOR="Simon, H.", 
       TITLE="Models of Man", 
   PUBLISHER="Wiley", 
     ADDRESS="New York", 
        YEAR="1957"} 
 
@INCOLLECTION{Simon76, 
      AUTHOR="Simon, H.", 
       TITLE="From Substantive to Procedural Rationality", 
   BOOKTITLE="Method and Appraisal in Economics", 
      EDITOR="S. Latsis", 
   PUBLISHER="Cambridge University Press", 
     ADDRESS="Cambridge", 
        YEAR="1976"} 
 
 
 
%Singer 
 
@BOOK{Singer80, 
     AUTHOR="Singer, P.", 
      TITLE="The Expanding Circle: Ethics and Sociobiology", 
  PUBLISHER={Farrar, Strauss and Giroux}, 
    ADDRESS={New York}, 
       YEAR=1980} 
 
%Skyrms 
 
@INCOLLECTION{Skyrms90a, 
      AUTHOR="Skyrms, B.", 
       TITLE="Dynamic Models of Deliberation and the Theory of 
Games", 
   BOOKTITLE="Reasoning About Knowledge", 
      EDITOR="R. Parikh", 
   PUBLISHER="Morgan Kaufman", 
     ADDRESS="San Mateo, CA", 
        YEAR="1990"} 
 
@BOOK{Skyrms90, 
     AUTHOR="Skyrms, B.", 
      TITLE="The Dynamics of Rational Deliberation", 
  PUBLISHER={Harvard University Press}, 
    ADDRESS={Cambridge, MA}, 
       YEAR="1990"} 
 
%Smith, A. 
 
@BOOK{Smith59, 
     AUTHOR="Smith, A.", 
      TITLE="The Theory of Moral Sentiments", 
  PUBLISHER={Clarendon Press}, 
    ADDRESS={Oxford}, 
       YEAR=1975, 
       NOTE="(Edited by D. Raphael and A. Macfie{,} first published 
1759.)"} 
 
@ARTICLE{VernonSmith76, 
    AUTHOR="Smith, V.", 
     TITLE="Experimental Economics: {I}nduced Value Theory", 
   JOURNAL="American Economic Review", 
      YEAR=1976, 
    VOLUME=66, 
     PAGES="274-279"} 
 
%Solovay 
 
@ARTICLE{Solovay76, 
    AUTHOR="Solovay, R.", 
     TITLE="Provability Interpretation of Modal Logic", 
   JOURNAL="Israel Journal of Mathematics", 
      YEAR=1976, 
    VOLUME=25, 
     PAGES="287-304"} 
 
%Solow 
 
@ARTICLE{Solow74, 
    AUTHOR="Solow, R.", 
     TITLE="Intergenerational Equity and Exhaustible Resources", 
   JOURNAL="Review of Economic Studies", 
      YEAR=1974, 
    VOLUME=41, 
     PAGES="29-46"} 
 
%Spinoza 
 
@INCOLLECTION{Spinoza77, 
     AUTHOR="{Spinoza}, B. de", 
  BOOKTITLE="Collected Works of {S}pinoza{,} Volume {I} ", 
     TITLE="Ethics", 
  PUBLISHER={Princeton University Press}, 
    ADDRESS={Princeton}, 
       YEAR=1985, 
       NOTE="(Edited by E. Curley{,} first published 1677.)"} 
 
@INCOLLECTION{Spinoza74, 
     AUTHOR="{Spinoza}, B. de", 
  BOOKTITLE="Chief Works of {B}enedict de {S}pinoza{,} {\rm Volume 
I}", 
     TITLE="Tractatus Politicus", 
  PUBLISHER={Bell}, 
    ADDRESS={London}, 
       YEAR=1909, 
       NOTE="(Translated by R. Elwes{,} first published 1674.)"} 
 
%Stahl 
 
@BOOK{Stahl72, 
     AUTHOR="Stahl, I.", 
      TITLE="Bargaining Theory", 
  PUBLISHER={Economics Research Institute}, 
    ADDRESS={Stockholm}, 
       YEAR=1972} 
 
%Stanford 
 
@UNPUBLISHED{Stanford90, 
      AUTHOR="Stanford  W.", 
       TITLE="Pre-Stable Strategies in Discounted Duopoly Games", 
        YEAR="1990", 
        NOTE="forthcoming in {\em Games and Economic Behavior}"} 
 
%Stearns 
 
@UNPUBLISHED{Stearns89, 
      AUTHOR="Stearns, R.", 
       TITLE="Memory-Bounded{,} Game-Playing Computing Machines", 
        YEAR="1989", 
        NOTE="Working Paper{,} SUNY at Albany"} 
 
 
 
%Stigler 
 
@INCOLLECTION{Stigler81, 
     AUTHOR="Stigler, G. J.", 
      TITLE="Economics or Ethics{?}", 
  BOOKTITLE="Tanner Lectures on Human Values", 
  PUBLISHER={Cambridge University Press}, 
    ADDRESS={Cambridge}, 
       YEAR=1981, 
     EDITOR="S. Mc{M}urrin"} 
 
 
%Stinchcombe 
 
@UNPUBLISHED{Stinchcombe88, 
    AUTHOR="Stinchcombe, M.", 
     TITLE="Approximate Common Knowledge", 
      NOTE="University of California at San Diego", 
      YEAR=1988} 
 
%Sugden 
 
@ARTICLE{Sugden85, 
    AUTHOR="Sugden, R.", 
     TITLE="Liberty, Preference and Choice", 
   JOURNAL="Economics and Philosophy", 
      YEAR=1985, 
    VOLUME=1, 
     PAGES="213-219"} 
 
 
 
@BOOK{Sugden86, 
     AUTHOR="Sugden, R.", 
      TITLE="The Economics of Rights{,} Cooperation and Welfare", 
  PUBLISHER={Basil Blackwell}, 
    ADDRESS={Oxford}, 
       YEAR=1986} 
 
@ARTICLE{Sugden91, 
    AUTHOR="Sugden, R.", 
     TITLE="Rational Choice{:} A Survey of Contributions from 
Economics 
and Philosophy", 
   JOURNAL="Economic Journal", 
      YEAR=1991, 
    VOLUME=101, 
     PAGES="751-785"} 
 
@INCOLLECTION{Sugden93, 
     AUTHOR="Sugden, R.", 
      TITLE="Rationality and Impartiality: Is the Contractarian 
Enterprise Possible?", 
  BOOKTITLE="Rationality, Justice and the Social Contract: Themes 
from 
`Morals by Agreement'", 
  PUBLISHER="Simon and Schuster", 
    ADDRESS="Hemel Hempstead, U.K.", 
       YEAR=1992, 
     EDITOR="R. Sugden"} 
 
 
%Sumner 
 
@BOOK{Sumner, 
     AUTHOR="Sumner, W.", 
      TITLE="Essays of William Graham Sumner, Volume II", 
  PUBLISHER={Yale University Press}, 
    ADDRESS={Newhaven}, 
       YEAR=1934, 
       NOTE="(edited by A. Keller and M. Davie)"} 
 
%Suppes 
 
@ARTICLE{Suppes66, 
    AUTHOR="Suppes, P.", 
     TITLE="Some Formal Models of Grading Principles", 
   JOURNAL="Synth\`{e}se", 
      YEAR=1966, 
    VOLUME=6, 
     PAGES="284-306"} 
 
 
%T 
 
%Tan and Werlang 
 
@ARTICLE{TandW84, 
      AUTHOR="Tan, T. and S. Werlang", 
       TITLE="The {B}ayesian Foundations of Rationalizable 
Strategic Behavior and {N}ash Equilibrium Behavior", 
     JOURNAL="Journal of Economic Theory", 
        YEAR="1984", 
      VOLUME="45", 
       PAGES="370-391"} 
 
 
 
@TECHREPORT{TandW85, 
    AUTHOR="Tan, T. and S. Werlang", 
     TITLE="On {A}umann's Notion of Common Knowledge: An 
Alternative 
Approach", 
INSTITUTION="University of Chicago", 
      YEAR=1985, 
    NUMBER="WP 85-26"} 
 
%Taylor 
 
@BOOK{Taylor, 
     AUTHOR="Taylor, M.", 
      TITLE="The Possibility of Cooperation", 
  PUBLISHER={Cambridge University Press}, 
    ADDRESS={Cambridge}, 
       YEAR=1987} 
 
@BOOK{Taylor90, 
     AUTHOR="Taylor, C.", 
      TITLE="Sources of the Self", 
  PUBLISHER={Cambridge University Press}, 
    ADDRESS={Cambridge}, 
       YEAR=1990} 
 
 
%Taylor and Jonker 
 
@ARTICLE{TandJ78, 
      AUTHOR="Taylor, P. and L. Jonker", 
       TITLE="Evolutionary Stable Strategies and Game Dynamics", 
     JOURNAL="Mathematical Biosciences", 
        YEAR="1978", 
      VOLUME="40", 
       PAGES="145-156"} 
 
%THaler 
 
@ARTICLE{Thaler88, 
      AUTHOR="Thaler, R.", 
       TITLE="Anomalies: {T}he Ultimatum Game", 
     JOURNAL="Journal of Economic Perspectives", 
        YEAR="1988", 
      VOLUME="2", 
       PAGES="195-206"} 
 
 
%Thorpe 
 
@BOOK{Thorpe, 
     AUTHOR="Thorpe, E.", 
      TITLE="Beat the Dealer", 
  PUBLISHER={Random House}, 
    ADDRESS={New York}, 
       YEAR=1966} 
 
%Thomson 
 
@BOOK{Thomson92, 
     AUTHOR="Thomson, W.", 
      TITLE="Bargaining Theory: The Axiomatic Approach", 
  PUBLISHER={Cambridge University Press}, 
    ADDRESS={Cambridge}, 
       YEAR=1992} 
 
%U 
 
@BOOK{Ulmann-Margalit, 
     AUTHOR="Ulmann-Margalit, E.", 
      TITLE="The Emergence of Norms", 
  PUBLISHER={Oxford University Press}, 
    ADDRESS={New York}, 
       YEAR=1977} 
 
%V 
 
%Vanberg 
 
@ARTICLE{VandC91, 
      AUTHOR="Vanberg, V. and R. Congleton", 
       TITLE="Rationality, Morality and Exit", 
     JOURNAL="American Political Science Review", 
        YEAR="1992", 
      VOLUME="86", 
       PAGES="418-431"} 
         
%Van Damme 
 
@BOOK{VanDamme83, 
     AUTHOR="{Van Damme}, E.", 
      TITLE="Refinements of the {N}ash Equilibrium Concept", 
   PUBLISHER={Springer}, 
    ADDRESS={Berlin}, 
       YEAR=1983} 
 
@BOOK{VanDamme87, 
     AUTHOR="{Van Damme}, E.", 
      TITLE="Stability and Perfection of {N}ash Equilibria", 
  PUBLISHER={Springer}, 
    ADDRESS={Berlin}, 
       YEAR=1987} 
 
@UNPUBLISHED{VanDamme90, 
    AUTHOR="{Van Damme}, E.", 
     TITLE="Refinements of Equilibrium", 
      NOTE="Paper delivered at Barcelona World Congress", 
      YEAR="1990"} 
 
%Van Huyck, Battalio and Beil 
 
@ARTICLE{VHBandB91, 
    AUTHOR="Van Huyck, J. and R. Battalio and R. Beil", 
     TITLE="Strategic Uncertainty, Equilibrium Selection Principles 
and 
Coordination Failure", 
   JOURNAL="American Economic Review", 
      YEAR=1991, 
    VOLUME=80, 
     PAGES="234-238"} 
 
%Van Huyck, Battalio, Mathur, Ortmann and Van Huyck 
 
@UNPUBLISHED{VHBMOandVH, 
    AUTHOR="Van Huyck, J. and R. Battalio and S. Mathur and A. 
Ortmann and 
P. Van Huyck", 
     TITLE="On the Origin of Convention: Evidence from 
Symmetric Bargaining Games", 
      YEAR=1991, 
      NOTE="Working Paper{,} Economics Department, Texas A\&M 
University"} 
 
%Vassilakis and Zamir 
 
@UNPUBLISHED{VandZ90, 
    AUTHOR="Vassilakis, S. and S. Zamir", 
     TITLE="Common Beliefs and Common Knowledge", 
      NOTE="{U}niversity of {P}ittsburgh Working Paper 259", 
      YEAR="1990"} 
 
 
%Varian 
 
@BOOK{Varian92, 
     AUTHOR="Varian, H.", 
      TITLE="Microeconomic Analysis. {\rm 3rd edition}", 
  PUBLISHER={Norton}, 
    ADDRESS={New York}, 
       YEAR=1992} 
 
@ARTICLE{Varian74, 
    AUTHOR="Varian, H.", 
     TITLE="Equity{,} Envy and Efficiency", 
   JOURNAL="Journal of Economic Theory", 
      YEAR=1974, 
    VOLUME=9, 
     PAGES="63-91"} 
 
@ARTICLE{Varian75, 
    AUTHOR="Varian, H.", 
     TITLE="Distributive Justice{,} Welfare Economics and the 
Theory of 
Fairness", 
   JOURNAL="Philosophy and Public Affairs", 
      YEAR=1975, 
    VOLUME=13, 
     PAGES="319-333"} 
 
@TECHREPORT{Varian87, 
    AUTHOR="Varian, H.", 
     TITLE="Differences of Opinion in Financial Markets", 
INSTITUTION={University of Michigan}, 
      YEAR=1987, 
    NUMBER="CREST 87-23"} 
 
 
%Veblen 
 
@INCOLLECTION{Veblen99, 
      AUTHOR="Veblen, T.", 
       TITLE="The Theory of the Leisure Class", 
   BOOKTITLE="The Portable Veblen", 
      EDITOR="M. Lerner", 
   PUBLISHER="Viking Press", 
     ADDRESS="New York", 
        YEAR="1958", 
        NOTE="(First published 1899.)"} 
 
%Von Clausewitz 
 
@BOOK{VonClausewitz, 
      AUTHOR="{Clausewitz}, C. von", 
       TITLE="On War", 
   PUBLISHER="Princeton University Press", 
     ADDRESS="Princeton", 
        YEAR="1976", 
        NOTE="(First published 1832.)"} 
 
 
 
%Vickrey 
 
@ARTICLE{Vickrey45, 
    AUTHOR="Vickrey, W. S.", 
     TITLE="Measuring Marginal Utility by Reactions to Risk", 
   JOURNAL="Econometrica", 
      YEAR=1945, 
    VOLUME=13, 
     PAGES="319-333"} 
 
%Von Neumann 
 
 
@ARTICLE{VN28, 
    AUTHOR="Von {N}eumann, J.", 
     TITLE="Zur {T}heorie der {G}esellschaftsspiele", 
   JOURNAL="Mathematische Annalen", 
      YEAR=1928, 
    VOLUME=100, 
     PAGES="295-320"} 
 
%Von Neumann and Morgenstern 
 
@BOOK{VNandM, 
     AUTHOR="{Von Neumann}, J. and Morgenstern, O.", 
      TITLE="The Theory of Games and Economic Behavior", 
  PUBLISHER={Princeton University Press}, 
    ADDRESS={Princeton}, 
       YEAR=1944} 
 
 
 
%W 
 
%Walliser 
 
@TECHREPORT{Walliser90, 
      AUTHOR="Walliser, B.", 
       TITLE="Epistemic Logic and Game Theory", 
 INSTITUTION="ENPC-CERAS Discussion Paper", 
        YEAR="1990"} 
 
 
%Walzer 
 
@BOOK{Walzer83, 
    AUTHOR="Walzer, M.", 
    TITLE="Spheres of Justice", 
   PUBLISHER={Basic Books}, 
     ADDRESS={New York}, 
        YEAR=1983} 
 
%Weymark 
 
@INCOLLECTION{Weymark91, 
      AUTHOR="Weymark, J.", 
       TITLE="A Reconsideration of the {H}arsanyi-{S}en Debate 
on Utilitarianism", 
   BOOKTITLE="Interpersonal Comparisons of Well-Being", 
      EDITOR="J. Elster and J. Roemer", 
   PUBLISHER="Cambridge University Press", 
     ADDRESS="Cambridge", 
        YEAR="1991"} 
 
%Williams, Bernard 
 
@BOOK{Williams72, 
     AUTHOR="Williams, B.", 
      TITLE="Morality: An Introduction to Ethics", 
   PUBLISHER={Cambridge University Press}, 
    ADDRESS={Cambridge}, 
       YEAR=1972} 
 
@BOOK{Williams85, 
     AUTHOR="Williams, B.", 
      TITLE="Ethics and the Limits of Philosophy", 
   PUBLISHER={Fontana Press}, 
    ADDRESS={London}, 
       YEAR=1985} 
 
%Williams, J. 
 
@BOOK{Williams,  
     AUTHOR="Williams, J.", 
      TITLE="The Compleat Strategyst", 
   PUBLISHER={Mc{G}raw Hill}, 
    ADDRESS={New York}, 
       YEAR=1954} 
 
@BOOK{WilsonE75, 
     AUTHOR="Wilson, E.", 
      TITLE="Sociobiology: The New Synthesis", 
   PUBLISHER={Harvard University Press}, 
    ADDRESS={Cambridge, Mass.}, 
       YEAR=1975} 
 
 
%Wolff 
 
@ARTICLE{Wolff66, 
    AUTHOR="Wolff, R.", 
     TITLE="A Refutation of {R}awls' Theorem on Justice", 
   JOURNAL="Journal of Philosophy", 
      YEAR=1966, 
    VOLUME=63, 
     PAGES="170-190"} 
 
@BOOK{Wolff77, 
     AUTHOR="Wolff, R.", 
      TITLE="Understanding Rawls", 
  PUBLISHER={Princeton University Press}, 
    ADDRESS={Princeton}, 
       YEAR=1977} 
 
%X 
 
%Y 
 
%Yaari 
 
@ARTICLE{Yaari81, 
    AUTHOR="Yaari, M.", 
     TITLE="{R}awls, {E}dgeworth, {S}hapley, {N}ash: Theories of 
Distributive Justice Re-Examined", 
   JOURNAL="Journal of Economic Theory", 
      YEAR=1981, 
    VOLUME=24, 
     PAGES="1-39"} 
 
%Yardley 
 
@BOOK{Yardley, 
     AUTHOR="Yardley, H.", 
      TITLE="The Education of a Poker Player", 
  PUBLISHER={Fontana/Collins}, 
    ADDRESS={London}, 
       YEAR=1980} 
 
%Young 
 
@TECHREPORT{Young89, 
      AUTHOR="Young, P.", 
       TITLE="The Evolution of Bargaining Conventions", 
 INSTITUTION="University of Maryland", 
        YEAR="1989"} 
 
 
@ARTICLE{Young93, 
    AUTHOR="Young, P.", 
     TITLE="The Evolution of Conventions", 
   JOURNAL="Econometrica", 
      YEAR=1993, 
    VOLUME=61, 
     PAGES="57-84"} 
 
%Z 
 

%%%%%%%%%%%%%%%%muddling.tex%%%%%%%%%%%%%%%%%%%%%%%%

\documentstyle[11pt,titlepage]{article} 
 
\begin{document} 
\bibliographystyle{plain} 
\input{kbmacros} 
 
\title{MUDDLING THROUGH:  NOISY EQUILIBRIUM SELECTION\thanks{First 
draft August 26, 1992. 
Financial support from the National Science Foundation and the 
Deutsche Forschungsgemeinschaft, Sonderforschungsbereich 303 at the 
University of Bonn, is gratefully acknowledged.  Part of this work 
was done while the authors were visiting the University of Bonn, 
for whose hospitality we are grateful.  We thank Drew Fudenberg, 
Reinhard Selten, Avner Shaked and 
Richard Vaughan for helpful discussions.}} 
 
\author{Ken Binmore\\ 
Department of Economics\\ 
University College London\\ 
Gower Street\\ 
London  WC1E 6BT  England 
\and 
Larry Samuelson\\ 
Department of Economics\\ 
University of Wisconsin\\ 
1180 Observatory Drive\\ 
Madison, Wisconsin  53706 USA} 
\date{\today} 
 
\maketitle 
 
 
\begin{abstract} 
We examine an evolutionary model in which the primary source of ``noise'' that

moves the model between equilibria is not random, arbitrarily improbable
mutations 
but mistakes in learning. We find conditions under which the payoff-dominant 
equilibrium in a $2 \times 2$ game is selected by the model as well as
conditions 
under 
which the 
risk-dominant equilibrium is selected.  The relevant risk-dominance 
considerations, however, arise not in the original game but in a ``fitness
game'' 
derived from the process by which payoffs in the original game are translated
into 
evolutionary fitnesses.  We also find that waiting times until the limiting 
distribution is reached can be shorter than in a mutation-driven model.  To 
explore the robustness of the results to the specification of the model, we 
present a number of comparative static results as well as a ``two-tiered'' 
evolutionary model in which the rules by which agents learn 
to play the game are themselves subject to evolutionary pressure. 
 
\medskip 
\noindent {\em Journal of Economic Literature} Classification 
Number C70. 
\end{abstract} 
 
 
\newtheorem{theorem}{Theorem} 
\newtheorem{proposition}{Proposition} 
\newtheorem{definition}{Definition} 
\newtheorem{lemma}{Lemma} 
\newtheorem{conjecture}{Conjecture} 
\newtheorem{assumption}{Assumption} 
\newtheorem{corollary}{Corollary}      
 
 
 
 
\begin{center} 
{\LARGE 
{\bf MUDDLING THROUGH:\\[.9ex] 
NOISY EQUILIBRIUM SELECTION}}\\ 
 
\vspace{.5cm} 
 
 
by Ken Binmore and Larry Samuelson 
 
 
\vspace{.5cm} 
 
 
\parbox{6.5cm}{{\small \noindent 
Commonsense is a method of arriving at workable 
conclusions from false premisses by nonsensical reasoning. \hfill 
Schumpeter}} 
\end{center} 
 
%\vspace{1cm} 
 
\section{Introduction} 
 
Which equilibrium should be selected in a game with 
multiple equilibria? This paper pursues an evolutionary approach 
to 
equilibrium selection in which the equilibriating process 
or ``libration'' is explicitly modeled.  A boundedly rational 
player is identified with a learning rule and attention is 
centered on the interactive dynamics that result when pairs of 
players are repeatedly drawn at random from a given 
population to play a given game. 
 
A more orthodox approach to the equilibrium selection problem 
is to invent refinements of the Nash equilibrium concept. In the 
same spirit, 
numerous refinements of the notion of an evolutionarily stable 
strategy 
have been proposed.  From this perspective, the learning rules 
studied in a dynamic treatment may seem overly-specific or even 
arbitrary, and it may be troubling that 
the equilibrium selected by a dynamic model often 
depends on the fine details of the modeling, or on the initial 
conditions 
prevailing at the time the process began. But to criticize a 
dynamic model 
for such reasons is to miss an important point. The very 
fact that varying the details in a dynamic model can alter the 
equilibrium 
selected shows that the institutional environment in which a game is learned
and 
played 
can 
matter for equilibrium selection.  Theories of equilibrium selection 
therefore cannot neglect such details. 
 
At the same time, dynamic models are unlikely to yield much insight into which

aspects 
of a game's environment are 
significant 
in equilibrium selection if the mathematical properties of the 
learning 
rules studied are simply plucked from the air. For this reason, we consider the

question 
of {\bf  microfoundations} to be crucial: 
it is important to derive the model's equations of motion from 
explicitly stated assumptions about the manner in which individual learning 
is 
postulated to proceed.  To facilitate this task in the current paper, a study
of 
the more technical issues has been relegated to Binmore, Samuelson and Vaughan

\cite{BinmoreSamuelsonandVaughan1993}, in which the model is motivated by a
story 
of a competition for survival between stylized rabbits.  The present paper 
provides socio-economic microfoundations for the same model and uses the model
to 
explore the following issues: 
 
\paragraph{(a) Time:} 
We have distinguished elsewhere 
(\cite{BinmoreandSamuelson1993Saar,BinmoreSamuelsonandVaughan1993}) 
between 
four relevant time periods: the short run, the medium run, the long 
run, 
and the ultralong run. In the short run, the system will have 
little chance 
to stray from its initial condition. In the medium run, the system 
will begin to respond to evolutionary pressures. In the long run, 
the system may find its way to an equilibrium, with this equilibrium being 
determined by 
the initial conditions of the system.  Only in the ultralong run, 
history can be neglected for the purposes of equilibrium selection. 
In the ultralong run, random shocks will repeatedly bounce 
the system out of the basin of attraction of one equilibrium 
into the basin of attraction of another. The system may then 
eventually settle into a steady state in which each 
equilibrium is visited with a well-defined frequency. If we are 
lucky, all but one of these frequencies 
may be negligible. If so, then an equilibrium has been selected in 
the ultralong run.% 
% 
% 
\footnote{Kandori, Mailath and Rob \cite{KandoriMailathandRob1992} call such an

equilibrium a ``long-run equilibrium'', while for us it is an ``ultralong- 
run equilibrium''.} 
 
The appropriate technique of analysis depends upon the time period of interest.

Our concern in this paper is with equilibrium selection 
in the ultralong run. 
We are aware that the interest of such results depends on the waiting time
before 
we can expect an ultralong-run prediction to be realized.  Such waiting times 
depend upon the manner in which noise enters the model, prompting the next
issue. 
 
\paragraph{(b) Noise:} 
For ultralong-run phenomena to be meaningful, it is necessary 
to model the noise that perturbs the learning process and drives the selection
of 
equilibria 
in the ultra-long run. The pioneers 
in this regard are Young \cite{Young1992} and Kandori, Mailath and Rob 
\cite{KandoriMailathandRob1992}. In their models, agents normally choose a best

response given their information, and hence act as {\em maximizers}. 
However, after agents have decided on an action, there is a small 
probability 
$\lambda>0$ that they will switch their choice to some suboptimal 
alternative. Such switches are said to be {\em mutations}. The 
ultralong-run 
distribution over population states is then studied in the limit 
as 
$\lambda \rightarrow0$. 
 
We proceed in the same manner except that our agents are {\em 
muddlers\/} 
rather than maximizers. For muddlers, the learning process is 
itself 
noisy, in that agents do not always choose best responses. We think it
necessary 
to model such ``selection noise'' explicitly when mutation rates are allowed to

become very 
small because selection noise will then be the major source of randomness in
the 
system. 
Such an innovation has 
the important consequence that 
the expected waiting time before the ultralong-run predictions of 
the model become relevant is greatly reduced. To see why, consider 
the possibility that a population of agents has found its way, 
in the long run, to an equilibrium that is not selected in the 
ultralong 
run. In the maximizing models of Young \cite{Young1992} and 
Kandori, Mailath and Rob \cite{KandoriMailathandRob1992}, a large number of
{\em 
simultaneous\/} 
mutations are now necessary for the system to escape from its basin 
of attraction. In contrast, our muddling model requires only one 
mutation 
to step away from the equilibrium, after which the agents may 
{\em muddle} 
their way out of its basin of attraction. 
 
Incorporating noisy learning into the model has implications for equilibrium 
selection as 
well as waiting times:  muddling models do not always select 
the same equilibria as maximizing models. In the symmetric 
$2\times2$ 
games studied in this paper, maximizing models always choose between two strict

Nash 
equilibria by selecting the 
risk-dominant 
equilibrium. When risk-dominance and payoff-dominance conflict, our muddling
model 
sometimes selects the 
payoff-dominant 
equilibrium. There are 
therefore grounds for directing suspicion at risk-dominance as a 
refinement 
of Nash equilibrium even in symmetric $2\times2$ games. 
 
 
\paragraph{(c) Payoffs:} 
In conventional game theory, the payoffs are Von Neumann and 
Morgenstern 
utilities. In economic applications, the players are often assumed 
to be 
risk-neutral so that their utilities can be identified with 
physical {\em rewards}. For example, firms are usually assumed to 
maximize expected profit. In contrast, payoffs in biological models are taken
to 
be {\em fitnesses\/}, generally measured in units like the expected number of 
offspring, 
rather than physical rewards (such as food or square meters of territory).
When 
maximizing 
models are used, 
the distinction between a fitness and a reward is seldom important. 
In our muddling model, however, we find it essential to translate rewards into

fitnesses and 
to work with the latter.  For example, we find that the relevant risk-dominance

considerations in our model require the use of fitnesses rather than rewards. 
Except 
in special cases, there 
is no straightforward relation 
between risk-dominance in the reward game and in the fitness game.  Even in
cases 
where 
risk-dominance considerations in fitnesses and rewards coincide, 
one must beware of treating 
the payoffs in the fitness game like Von Neumann and Morgenstern 
payoffs. 
In particular, equilibrium selection in a muddling model is not invariant to 
strictly affine transformations of the 
fitnesses. 
 
\paragraph{(d) Robustness} 
Why study our model of muddled learning rather than one of 
the many alternatives that might be proposed? As in most studies, our choice of

model was 
constrained by the need to keep the mathematics 
simple and by a desire for our results to be comparable with 
those obtained by others.% 
% 
% 
\footnote{For example, Binmore, Samuelson and 
Vaughan \cite{BinmoreSamuelsonandVaughan1993}, studying the long run, show that

the muddling model is approximated by the replicator dynamics when 
only long-run considerations are relevant.} 
% 
% 
Operating under such constraints, one is inevitably led to make questionable 
modeling assumptions.  However, we hope ultimately to dispense with the need to

make  arbitrary choices in the construction of the 
model by 
treating the learning process itself as being 
determined by 
evolutionary  processes. 
 
To illustrate this proposed methodology, 
we conclude this paper with an examination of 
the evolutionary stability of the learning rules within 
the narrow class of rules considered in this paper. 
We ask whether a population using a certain 
learning rule, and hence receiving the payoffs associated with 
the corresponding ultralong-run distribution over population states, 
can be invaded by a mutant learning rule from the same class. If 
it 
can, then we have grounds for questioning its robustness. If it cannot be
invaded, 
then 
we say that the learning rule is itself evolutionarily stable.  We find
conditions 
under which 
evolutionarily stable learning rules in our muddling model select the 
risk-dominant 
equilibrium in the reward game for symmetric $2\times2$ 
games, thus matching the results of  maximizing models. In 
other cases, however, the models continue to give differing results. 
 
 
\medskip 
 
The paper is organized as follows. Section 2 describes the muddling 
model 
and introduces the distinction between a reward and a fitness. 
Section 3 reviews the dynamics of the resulting equations of 
motion and  takes up the problem of expected 
waiting times. Section 4 discusses ultralong-run equilibrium 
selection for the muddling model. Section 5 is devoted to 
comparative 
statics. Section 6 considers the evolutionary stability 
of the learning rules studied. 
Section 7 briefly discusses our conclusions. 
 
 
\section{A Muddling Model} 
 
\paragraph{The Reward Game.} \ We begin with the symmetric 
$2\Times2$ 
game ${\cal R}$ of Figure 1. The payoffs in this game, taken to 
be the familiar von Neumann and 
Morgenstern utilities of conventional game theory, are called expected 
{\em rewards\/} to distinguish them from the {\em fitnesses\/} that 
will appear  soon.  The randomness that motivates the label ``expected'' will
be 
introduced shortly. 
 
 
\begin{figure}[htb] 
\[ 
\begin{tabular}{c|@{}c@{}|@{}c@{}|} 
\MC{1}{c}{}&\MC{1}{c}{$X$}&\MC{1}{c}{$Y$}\\ 
\cline{2-3} 
\rule[-5mm]{0mm}{12mm}$X$&\Cell{$A$}{$A$}&\Cell{$C$}{$B$}\\ 
\cline{2-3} 
\rule[-5mm]{0mm}{12mm}$Y$&\Cell{$B$}{$C$}&\Cell{$D$}{$D$}\\ 
\cline{2-3} 
%\MC{3}{c}{\rule{0em}{1ex}}\\ 
%\MC{3}{c}{\ (a) Reward Game ${\cal R}$}\\ 
\end{tabular} 
\] 
\caption{The Reward Game ${\cal R}$} 
\end{figure} 
 
We assume that there is a single population containing $N$ agents. 
Time is divided into discrete intervals of length $\tau$.  In each 
time 
period, an agent is characterized by the strategy $X$ or $Y$ that 
he 
is programmed to use in that period. In each period of length 
$\tau$, 
pairs of agents are randomly (independently and with replacement) 
drawn to play the 
game. Such 
draws occur sufficiently frequently that the probability of 
each agent playing at least one game in each period can be taken 
to 
be unity.  Given that agents are drawn randomly with 
replacement to play the game, this implies that each agent will have played an

infinite number of games with a distribution of opponents that 
accurately reflects the distribution of strategies in the 
population.% 
% 
% 
\footnote{Since 
we shall 
consider the case when $\tau\rightarrow0$, this assumption has 
the effect of requiring that the game be played arbitrarily 
rapidly. We view this 
as an approximation of the case 
when play is frequent relative to strategy revision, 
which we consider the natural 
setting for evolutionary models. 
Similar hypotheses about rapid play are common in the literature, 
although 
they are sometimes hidden in the assumption that learning agents 
always switch to the current best reply. In particular, an agent 
cannot know the current best 
reply 
unless she can observe what the other agents are currently playing. 
Kandori, Mailath and Rob \cite{KandoriMailathandRob1992} are explicit 
in 
assuming that 
agents play an infinite number of times in each period. 
N\"{o}ldeke and Samuelson \cite{NoldekeandSamuelson1993} 
assume a round-robin tournament in each period. Young's model 
\cite{Young1992} 
is less demanding in this respect, though agents still have access 
to the result of each game as soon as it is played.} 
% 
% 
An agent playing $X$, for example, will receive an expected reward of $A$ in a

population in which all agents play $X$ and an ``average'' expected reward of
$qA + 
(1-q)C$ when proportion $q$ of his opponents play $X$ and proportion $(1-q)$
play 
$Y$.  In some cases, 
noted below, the model is formally identical to one in which 
each agent plays only once in each period. 
 
\paragraph{Noisy learning.} 
We consider a learning model that couples an aspiration-based rule 
for abandoning 
existing strategies with an imitation process for choosing new 
ones.% 
% 
% 
\footnote{Aspiration-based learning rules are examined by Bendor, 
Mookherjee and 
Ray 
\cite{BendorMookherjeeandRay1991} and Gilboa and Schmeidler 
\cite{GilboaandSchmeidler1992,GilboaandSchmeidler1993a,% 
GilboaandSchmeidler1993b}.} 
 
 
At the end of 
each period of length $\tau$, each agent experiences a Bernoulli 
trial which rings a mental bell with probability $\bbb\tau$. 
(Without loss of generality, we subsequently take $\bbb=1$.) 
If the bell does not ring, the agent does not consider changing 
her strategy. If the bell rings, an event that we refer to as 
``receiving the 
learn draw'', then the agent recalls her 
average realized reward in the last period 
and compares 
this with a fixed {\em aspiration level} $\Delta$. The average realized reward
in 
each period is random, 
being given by the sum of the average expected reward,
denoted $\rho$ and 
calculated 
from the Reward Game, and the 
realization $R$ of an idiosyncratic random variable $\tilde{R}$ with 
cumulative distribution $F$.% 
% 
% 
\footnote{The somewhat awkward phrases ``average expected reward'' and
``average
realized reward'' arise because the payoffs in the Reward Game are expected
payoffs, to
which $R$ must be added to get realized payoffs; and because the relevant
expected
payoff is the {\em average} of the expected payoffs received from games played
in the
repeated matches with the various members of the population.  The random
variable
$\tilde{R}$ 
represents an independent, identically distributed 
individual factor that 
that is independent of the current state and strategy and that 
yields a shock 
common to {\em each} payoff received by that agent in the given period. 
It would be 
interesting, but more complicated, to allow the distribution $F$ 
to 
depend either on the current 
state or strategy.  It would also be 
interesting 
to study cases in 
which this source of noise is correlated across individuals, 
perhaps as a result  of environmental factors that impose a common 
risk on all agents. Papers 
in which this type of uncertainty appears include Fudenberg and 
Harris 
\cite{FudenbergandHarris1992} and Robson \cite{Robson1992}.} 
% 
% 
If the average realized reward exceeds the aspiration level ($\rho + R >
\Delta$), 
then 
the agent makes no change in strategy. 
If instead the average realized reward falls short of the 
aspiration level ($\rho + R < \Delta$), then the agent loses faith in 
her current strategy and abandons that strategy. 
We assume that $F$ is log-concave, i.e.,  that $\ln F(z)$ is concave in $z$.% 
% 
% 
\footnote{See Bagnoli and Bergstrom \cite{BandB89} for a discussion 
of 
log-concavity 
and its implications.  We actually need log-concavity only on an open set
containing
$\{\Delta-\rho:\rho \in \{A,B,C,D\}\}$, and some of our examples will satisfy
only this weaker
assumption.} 
% 
% 
 
If agent $i$ has abandoned her strategy, then she 
must now choose a new strategy. We assume that she randomly selects 
a 
member $j$ of the population.  With probability $1-2\lambda$, $i$ 
imitates 
$j$'s strategy.% 
\footnote{She may thereby end up playing the strategy with 
which she began, having perhaps had her faith in it restored by 
seeing it played by the person she chose to copy.} 
With probability $2\lambda$, $i$ is a ``mutant'' who either does not observe
$j$'s 
strategy 
or ignores this observation, instead simply choosing a strategy randomly, with

equal 
probability of choosing each strategy.  We introduce mutations at this point in

the story, 
rather than in numerous other alternative places,  because we 
believe mistakes are most likely to occur when one agent copies 
the strategy of another.  Overall, $i$ then imitates $j$ with probability 
$1-\lambda$ and 
 adopts instead the strategy that $j$ is {\em not\/} 
playing 
with probability $\lambda$.  We hereafter find it convenient to refer to $i$ as

a mutant only 
in the case in which $i$ adopts the strategy that $j$ is not playing, and hence

to refer to 
$\lambda$ as the mutation rate. 
 
 
 
 
 
\paragraph{Aspiration Levels} 
 
The fact that we are free to specify the 
aspiration level $\Delta$ and the distribution $F$ allows 
several familiar 
formulations to appear as special 
cases. 
For example, suppose that the rewards $A$ and $D$ each exceed $B$ and $C$, so 
that the 
game has two strict Nash equilibria.  If we choose $F$ to put a 
probability mass of one on the value zero and take $\Delta$ to be 
the payoff 
of the mixed strategy 
equilibrium of the game, then we have random--best--reply dynamics, 
with agents who are chosen to learn switching strategies only if 
their current strategy is 
not a best reply.% 
% 
% 
\footnote{It may appear counterintuitive to speak 
of best-reply dynamics when agents are choosing strategies by 
simply imitating others, but a model in which agents abandon only 
inferior replies but choose strategies by imitation is analogous 
to 
a model in which agents are randomly chosen to switch to best 
replies.} 
 
 
 
 
\paragraph{Fitness Game ${\cal F}$.} 
The rewards that the players receive are 
not necessarily directly relevant to the dynamics that control 
the evolution of a population. Instead, it is important to 
translate these 
rewards $\rrr$ into {\em fitnesses\/} 
$\pi$. 
In general, how rewards are translated into fitnesses will depend on 
the dynamic process by which evolution proceeds. 
 
In our model, 
the relevant 
fitnesses are the probabilities that a player who has received the learn draw
will 
lose faith 
in and abandon his current strategy.  We follow 
\cite{BinmoreSamuelsonandVaughan1993} 
by referring to these as {\em death probabilities}. 
For each expected reward $\rho$, the corresponding death probability is 
given by $\pi(\rho) =\prob(\rho+R< \Delta) = F(\Delta -\rho)$; we say that 
the function $\pi\!:\!\IR\rightarrow [0,1]$ is the death probability {\em
induced} by the
distribution $F$.  We 
refer to the 
process by which 
rewards are transformed into fitnesses (death probabilities) as the 
{\em fitness 
process}. 
 
 
An interesting special case is that of a {\em 
uniform 
threshold rule with background fitness\/}. 
In this case, the random payoff shock $\tilde{R}$ takes on a value 
less than $\Delta - 
\overline{\rho}$, where $\overline{\rho} = 
\max_{\rho \in \{A,B,C,D\}}$, 
with probability $\theta$ 
and with probability $1-\theta$ is 
drawn from a uniform distribution on an interval 
$[\,-\omega,\omega\,]$, where $\{A,B,C,D\} \subset 
[\Delta-\omega,\Delta+\omega]$. 
The parameter $\theta$ 
is the background fitness probability, with which a learning agent 
abandons a 
current strategy 
regardless of payoffs, and 
upon which we expand in Section 6. 
Death probabilities are linear in expected rewards in this case. 
We can then 
represent them in terms of a {\em Fitness Game}. 
In particular, let $\alpha = \prob(A+R < \Delta) = F(\Delta - A)$, with similar

definitions for 
$\beta$, $\gamma$, and $\delta$.  Then the Fitness Game is given 
by Figure 2.% 
% 
% 
% 
\begin{figure}[htb] 
\[ 
\begin{tabular}{c|@{}c@{}|@{}c@{}|} 
\MC{1}{c}{}&\MC{1}{c}{$X$}&\MC{1}{c}{$Y$}\\ 
\cline{2-3} 
\rule[-5mm]{0mm}{12mm}$X$&\Cell{$\aaa$}{$\aaa$}&\Cell{$\ccc$} 
{$\bbb$}\\ 
\cline{2-3} 
\rule[-5mm]{0mm}{12mm}$Y$&\Cell{$\bbb$}{$\ccc$}&\Cell{$\ddd$} 
{$\ddd$}\\ 
\cline{2-3} 
%\MC{3}{c}{\rule{0em}{1ex}}\\ 
%\MC{3}{c}{\ (b) Fitness Game ${\cal F}$}\\ 
\end{tabular} 
\] 
\caption{The Fitness Game} 
\end{figure} 
% 
% 
% 
\noindent The probability that strategy $X$ is abandoned when $x$ 
of $N-1$ 
opponents play strategy $X$ is given by 
$\alpha x/(N-1)+ \gamma (N-x)/(N-1)$, which 
is the probability that the reward from strategy $X$ falls below 
the aspiration 
level $\Delta$. 
Notice that the Reward and Fitness Games have the same 
best-reply correspondence 
(recognizing that death is a ``bad'' rather than a ``good'', so 
that one wants to minimize 
payoffs in the Fitness Game). They therefore have the same payoff-dominant and

risk-dominant 
equilibria.  In this case, the model is equivalent to one in which 
each agent plays only once in each period. 
 
For other distributions 
of $\tilde{R}$, death probabilities will be nonlinear in 
expected rewards, so we cannot represent them with 
a simple $2 \times 2$ 
fitness game. 
Nevertheless, a notion of risk dominance is still useful in such cases. 
We say that strategy $Y$ is 
{\em risk dominant} in the fitness process given distribution $F$ 
if: 
% 
% 
\begin{displaymath} 
\int_0^1 \pi(kB + (1-k)D)dk  < 
\int_0^1 \pi(kA + (1-k)C)dk. 
\end{displaymath} 
% 
% 
Hence, strategy $Y$ is risk dominant if the expected probability 
of death for 
strategy $X$ exceeds that of $Y$.% 
% 
% 
\footnote{The expectation is taken over all (equally weighted) 
proportions of the 
population 
that might be playing strategy $X$, so that the expected 
probability of death for 
strategy 
$X$ is $\int_0^1 
(\pi(kA + (1-k)C)dk$.} 
% 
% 
If the fitness process is represented by the 
fitness game, then this is the usual notion of risk 
dominance.  We will 
say that the distribution $F$ {\em preserves dominance} if the same 
strategy is 
risk 
dominant in the Reward Game and in the fitness process. 
Dominance is clearly preserved if the fitness process can be 
represented by a 
fitness game. 
 
 
 
\paragraph{Limits.} 
In Binmore, Samuelson and Vaughan \cite{BinmoreSamuelsonandVaughan1993} 
we argue that 
much of the work in constructing an evolutionary model occurs when 
one chooses the order in which to take the various limits involved 
in the model.  Let $t$ be the time at which the system is to be studied.  Then
we 
shall take 
four limits in this 
model, in the following order: 
$t\rightarrow\infty$, $\tau\rightarrow0$, $N\rightarrow\infty$, 
and $\lambda\rightarrow0$. 
 
In letting $t\rightarrow\infty$ first, we are 
examining the stationary distribution reached by the 
system in the {\em ultralong} run. In particular, letting $t$ go to infinity
first 
ensures that 
we examine the stationary distribution without allowing other limiting
operations, 
most 
notably that of $N \rightarrow \infty$, to obscure the random shocks that drive

the 
ultralong-run selection results.% 
% 
% 
\footnote{Binmore, Samuelson and Vaughan 
\cite{BinmoreSamuelsonandVaughan1993} show 
that  if we first let 
$\tau$ approach zero and then $N$ approach infinity, we obtain 
a 
continuous 
time, deterministic version of the replicator dynamics that 
provides a useful approximation of the {\em long-run} behavior 
of our muddling model. 
The replicator dynamics but do not provide 
an adequate model of the {\em ultralong} run.  Instead, by taking the limit $N

\rightarrow 
\infty$ before $t \rightarrow \infty$, the replicator dynamics ``smooth out'' 
random perturbations that are 
very unlikely to occur in the long run, and hence are not important to a long
run 
analysis, 
but are crucial to ultralong-run considerations.} 
% 
% 
In letting $\tau\rightarrow0$, we are 
approximating 
the case of continuous time.  We do this because we think it 
appropriate to 
study the case in which births occur at random intervals, as in an 
overlapping 
generations model. This leads us to concentrate on  the case when 
$\tau$ is 
small compared with both $N$ and $\lll$, or $\tau \rightarrow 0$. 
 
 
 
In letting $N\rightarrow\infty$ and,  $\lambda\rightarrow0$, 
we 
plan 
to study the case of large populations with small mutation rates. 
 We do not see obvious principles 
guiding the choice of the order in which to take these limits, provided that
they 
are taken 
after $t \rightarrow \infty$ and $\tau \rightarrow 0$.  Fortunately it
sometimes 
does not 
matter to 
the equilibrium selected, as in the case of a symmetric $2\Times2$ 
game 
with two strict Nash equilibria. When it does matter, 
we allow $N\rightarrow\infty$  before $\lambda\rightarrow0$. 
The 
reverse order yields a system that can get ``stuck'' in a state 
where one strategy has 
accidentally become extinct and cannot reappear. 
 
\section{Dynamics} 
 
\paragraph{Stationary Distribution.}  For a fixed set of values of the
parameters 
$\tau$, 
$\lambda$, and $N$, we have a 
homogeneous Markov process on a finite state space. 
Let $x \in \{0,1, \ldots N\}$ specify the 
current state of the system, where $x$ denotes 
the number of agents who are currently playing 
strategy 
$X$.    Given any state $x  \in \{0,1, \ldots N\}$, there is a positive 
probability both that the 
Markov process moves to the state $x+1$ (if $x < N$), in which the number of 
agents 
playing $X$ is increased by one; and that the process moves to the state $x-1$
(if 
$x > 0$), 
in which the number of agents playing $X$ is decreased by one.  The transition

matrix for 
the Markov process is thus irreducible. The following result is then 
standard: 
 
\begin{proposition} 
The Markov process has a unique stationary distribution.  The 
expected frequencies of the states along any realization of the 
Markov process converge to this distribution; and the distribution 
of states at a given time $t$ converges to this distribution, as 
$t$ 
approaches infinity, from any initial condition. 
\end{proposition} 
 
\noindent {\bf Proof.~~}  Kemeny and Snell 
\cite{KemenyandSnell1960}, Theorems 4.1.4, 4.1.6, and 4.2.1. 
\hspace{\fill}  {\bf 
$\Box$} 
 
\medskip 
 
We study the stationary distribution in the limit as $\tau 
\rightarrow 0$. 
Binmore, Samuelson and Vaughan 
\cite{BinmoreSamuelsonandVaughan1993} consider 
the 
details of this limiting analysis and provide the details of the proof of 
Proposition \ref{FW} 
below.  The relevant implication for our analysis is that we can work with 
arbitrarily short 
time periods, in the sense that in a single period, there will be either no
agents 
who receive 
the learn draw or one agent who receives the learn draw.   The event that more

than one 
agent receives the learn draw occurs with negligible probability and can be 
ignored. 
Hence, given that the current state is $x$, the only transitions with which we

need be 
concerned are to states $x$, $x+1$, and $x-1$. 
 
 
Given 
state $x$, we 
let $r(x)$  be the 
probability 
that the system 
moves to state $x+1$, i.e., moves one state to the right.  Similarly, $\ell(x
)$ 
is the 
probability 
of moving to state $x-1$, or one state to the left. 
Consider $r(x )$.  For the 
number of agents playing $X$ to increase, four events must occur:  (1)  An
agent 
must 
receive 
the learn 
draw.  Let $\Lambda$ be the probability with which this occurs.    (2) The
agent 
who 
receives the learn draw must be playing strategy $Y$, since 
the number of agents playing $X$ can increase only if one agent abandons
strategy 
$Y$. 
If $x$ agents are currently playing strategy $X$, then the probability that an

agent drawn 
to learn is playing 
strategy $Y$ 
is given by 
$(N-x)/N$. (3) The learning agent must abandon his current 
strategy.  Because the average payoff of an agent playing strategy $Y$ is $(xB
+ 
(N-x- 
1)D)/(N-1)$, this occurs with 
probability 
$\pi((x B + (N-x-1)D)/(N-1))$. 
(4) The learning agent must choose $X$ for his new strategy.  This occurs with

probability 
$((1-\lambda) x + \lambda (N-x-1))/(N-1)$ since with probability
$(1-\lambda)x/(N- 
1)$, the learning agent chooses to imitate an agent playing $X$ and does so 
without 
mutation, and with probability $\lambda (N-x-1)/(N-1)$ the learning agent
chooses 
to imitate 
an agent playing $Y$ but is a mutant and chooses strategy $X$. Putting these 
probabilities 
together, we have: 
% 
% 
% 
\begin{equation}  \label{upx} 
r(x) = \Lambda  \frac{N-x}{N}~\pi \left(\frac{x B + 
(N-x-1)D}{N-1}\right)~\frac{(1-\lambda) x + \lambda 
(N-x-1)}{N-1}. 
\end{equation} 
% 
% 
\noindent The value of $\ell(x)$ is defined analogously. 
 
 
 
 
 
The basic tool used to characterize the stationary distribution is 
the following:% 
% 
% 
\footnote{From (\ref{upx}), we see that the probability $\Lambda$ does not
appear 
in 
(\ref{setlim}), so that we need not discuss the details of the determination of

$\Lambda$ 
here.} 
% 
% 
 
 
 
\begin{proposition}   \label{FW} 
Consider states $x$ and $x+1$. 
Let $\sigma(x)$ be the probability 
attached to state $x$ by the stationary distribution of the 
Markov 
process.  Then: 
% 
\begin{equation}  \label{setlim} 
\frac{\sigma(x+1)}{\sigma(x)} = 
\frac{r(x)}{\ell(x+1)}. 
\end{equation} 
\end{proposition} 
% 
\medskip 
 
\noindent {\bf Proof Sketch.~~}  From Freidlin and Wentzell 
(\cite{FreidlinandWentzell1984}, Lemma 3.1 on page 177), 
$\sigma(x+1)/\sigma(x)$ is given by the ratio of the sum of 
the 
products of the transition probabilities attached to all 
``$x+1$-trees'' to the similar calculation for ``$x$-trees''. 
Because only a single agent can receive the learn draw in each period, however,

there is 
only one such 
tree for each of states $x+1$ and $x$, consisting of a 
transition from each state other than $x+1$ (or $x$) to the 
immediate neighbor that lies closest to $x+1$ ($x$).% 
% 
% 
\footnote{The complete argument here would retain the very small probabilities
of 
multiple 
learn draws in each period, but would then observe that in the limit as $\tau$

becomes 
small, the only trees that are 
relevant are those that involve no transitions that occur with 
probability $\tau^2$ or less, i.e., involve only transitions from 
a state to one of its immediate neighbors.} 
% 
% 
It is 
then apparent that these two trees differ only in one probability: 
The $x+1$-tree contains the probability $r(x)$ while the 
$x$-tree contains $\ell(x+1)$, giving (\ref{setlim}). 
\hspace{\fill}  {\bf $\Box$} 
 
 
 
 
\medskip 
 
To interpret this result, notice that 
if $\lambda$ is very small and $N$ very large 
(we will see that the order of limits is interchangeable here), 
(\ref{setlim}) gives: 
% 
\begin{equation} \label{phi} 
\frac{\sigma(x+1)}{\sigma(x)}  \approx  \frac{\pi_Y(x/N)} 
{\pi_X(x/N)}  , 
\end{equation} 
% 
% 
\noindent where $\pi_X \!:\! [0,1] \rightarrow \IR$  and 
$\pi_Y \!:\! 
[0,1] 
\rightarrow \IR$ 
are the death probabilities of strategies $X$ and $Y$, 
in the limit as $N$ 
gets large, given that proportion $x/N$ of the population is 
playing strategy 
$X$.% 
% 
% 
\footnote{Hence,  $\pi_X (x/N) = \pi((x/N)A + ((N - x)/N)C)$, 
$\pi_Y (x/N) 
= \pi((x/N)B + 
((N - x)/N)D)$.  If it seems 
paradoxical 
that it is the average payoff to strategy $Y$ that appears in the 
numerator of 
(\ref{phi}), 
recall that 
these payoffs represent death probabilities and are hence ``bads'' 
rather than ``goods''.} 
% 
% 
Now consider a 
game with two strict Nash equilibria.  Let $x^*/N$ be the 
probability attached to $X$ by the mixed-strategy, Nash equilibrium of the
game. 
(Note that 
$x^*$ need 
not be an 
integer.)   Then $\sigma(x+1) > 
\sigma(x)$      whenever 
 $x > x^*$ (because strategy $X$ must 
be a best 
reply here, if the game is to have two strict Nash equilibria, and 
hence the ratio of death probabilities satisfies 
$\pi_Y/\pi_X   > 1$).  The stationary distribution $\sigma$ 
must then 
increase on 
$[x^*,N]$.  Similarly, $\sigma(x+1) < 
\sigma(x)$, and $\sigma(x)$ must decrease, on 
$[0,x^*]$.  The graph of 
$\sigma$ then has an ``inverted bowl'' shape, reaching its 
highest points 
at the 
endpoints of the state space, where we have the strict Nash equilibria in which

either 
all agents 
play $X$ or all 
agents play 
$Y$,  and its minimum at $x^*$. 
 
 
 
 
 
\paragraph{Convergence.} 
How long is the ultralong run?  We first provide a 
comparison of the convergence properties, for small mutation probabilities, 
of our muddling model and the model of  Kandori, Mailath and Rob 
\cite{KandoriMailathandRob1992}. 
We consider the case of a game with two strict Nash equilibria. 
We 
fix the value of 
$N$ and examine the performance of the system 
for 
very small values of $\lambda$. 
 
 
Let $\sigma_{KMR}^*(\lambda)$ be the stationary distribution of the 
Kandori, Mailath and Rob 
model given mutation rate $\lambda$. 
Consider the following measure, which is examined by Ellison 
\cite{Ellison1991}: 
% 
\begin{displaymath} 
\sup_{\sigma_0 }\limsup_{t \rightarrow \infty}  \| \sigma_0 
[\Gamma_{KMR}(\lambda)]^t - \sigma_{KMR}^*(\lambda) 
\|^{\frac{1}{t}}, 
\end{displaymath} 
% 
\noindent where $\sigma_0$ is the initial distribution and 
$\Gamma_{KMR}(\lambda)$ 
is the 
transition 
matrix of the Kandori, Mailath and Rob Markov process given mutation 
rate $\lambda$. 
This 
is then a 
measure of 
the distance between the distribution at 
time $t$ (given by 
$\sigma_0[\Gamma_{KMR}(\lambda)]^t$) and the stationary 
distribution given 
$\lambda$ (given 
by 
$\sigma_{KMR}^*(\lambda)$).  Ellison shows that there exists a function 
$\Theta_{KMR}(\lambda^z)\!:\!\IR \rightarrow \IR$ such that 
% 
\begin{equation} \label{conthem} 
1 - \sup_{\sigma_0 }\limsup_{t \rightarrow \infty}  \| \sigma_0 
[\Gamma_{KMR}(\lambda)]^t - \sigma_{KMR}^*(\lambda) 
\|^{\frac{1}{t}}    = \Theta_{KMR}(\lambda^z) 
\sim \lambda^z, 
\end{equation} 
% 
\noindent where $z$ is the minimum number of an agent's opponents that must 
play the risk-dominant 
equilibrium strategy in order for the latter to be a best reply for 
the agent in 
question.  (We say that the functions $f(\lambda)$ and $g(\lambda)$ are 
comparable, and 
write 
$f \sim 
g$, 
if there exist constants $c$ and $C$ 
such that for 
all 
sufficiently small $\lambda$, $c|g(\lambda)| \leq |f(\lambda) | 
\leq 
C|g(\lambda)|$).  The result (\ref{conthem}) then gives us a measure of how
fast 
the 
Kandori, Mailath and Rob 
model converges. 
 
 
 
We can provide some intuition for why a term of order 
$\lambda^z$ appears 
when 
considering convergence in the Kandori, Mailath and Rob model. 
Let 
all agents initially play the strategy that has the 
smaller basin of attraction.\footnote{It may seem 
misleading to 
take such an extreme initial condition.  Notice, however, that the 
best--response learning scheme 
immediately moves the system to this state from any initial 
condition in the basin 
of 
attraction of this equilibrium, so that this initial condition is 
relevant for a 
potentially large 
proportion of the state space.}    Then the probability of 
switching 
from 
the state 
with the small basin of attraction to the state with the large basin of 
attraction in a single period is given by the probability of at 
least $z$ 
mutations, or: 
% 
\begin{equation}  \label{KMRrate} 
\sum_{h=z}^{N}    \left(\!\! \begin{array}{c} 
N\\ 
h 
\end{array} \!\!\right) 
\lambda ^{h} (1- 
\lambda)^{N-h}. 
\end{equation} 
% 
\noindent A lower bound on the speed of convergence of 
the Kandori, Mailath and Rob 
model is then given by the fact that it cannot be very close to its 
stationary distribution until sufficient time has elapsed for an 
event to have occurred whose probability in any given period is on 
the order of $\lambda^z$. 
 
 
 
We now seek an analogous measure of the rate at which our muddling 
model 
converges.  First, fix the unit in which time is to be measured.  This unit of

measurement 
will remain 
constant throughout the analysis, even as we allow the length of the time
periods 
between 
learn draws in our muddling model to shrink. 
Our basic question then concerns how much time, measured in terms of the fixed

unit, 
must 
pass before the probability 
measure describing the expected state of the relevant dynamic process is 
sufficiently close 
to its stationary distribution.  To make the discrete Kandori, Mailath and Rob

model 
and our 
continuous model comparable, we choose the units in which time is measured so
that 
in 
the Kandori, Mailath and Rob model every agent learns at each of the discrete 
times 
$\{1,2,\ldots,  \}$.  We then let $\beta$, the probability of a birth per unit

time in our model, 
equal one, so that, in the limit as our time period length $\tau$ shrinks, the

expected 
number of times in an interval of time of length one (which will contain very
many 
of our 
very short 
time periods) that an agent in our model learns is one, matching the 
Kandori, Mailath and Rob 
model. 
 
 
Let $\Gamma_{M}(\lambda,\tau)$ be the transition 
matrix for the 
Markov process of 
our 
muddling model given mutation rate $\lambda$ and period 
length $\tau$. 
The quantity $\Gamma_{M}(\lambda,\tau)$ 
depends on $\tau$ because the probability of an agent receiving the 
learn draw depends on the period length.  Notice also that as 
$\tau$ decreases, the number $t /\tau$ of periods that occur by time $t$ 
increases. 
 
\begin{proposition}       \label{conprop} 
There exists a function $\Theta_{M}$ such that 
\begin{equation}  \label{conus} 
\lim_{\tau \rightarrow 0} \left( 1 - \sup_{\sigma_0} \limsup_{t 
 \rightarrow \infty} \|\sigma_0 
[\Gamma_{M} 
(\lambda,\tau)]^{\frac{t}{\tau}} - 
\sigma^*_{M}(\lambda) \|^{\frac{1}{t-1}} \right) \leq \Theta_{M}(\lambda) \sim

\lambda. 
\end{equation} 
\end{proposition} 
 
\medskip 
 
The proof is contained in the Appendix.  Together, 
(\ref{conthem}) and (\ref{conus}) imply that for very 
small values of $\lambda$, the muddling 
model converges 
much 
faster than does the Kandori, Mailath and Rob model.  In particular, let 
$T_{KMR}(\eta)$ 
be 
the length of time required for the Kandori, Mailath and Rob model to 
be 
within $\eta$ 
of its 
stationary 
distribution.  Let $T_{M}(\eta)$ be similarly defined for our 
muddling model. 
Then from 
(\ref{conthem}) and (\ref{conus}), we have $\eta = 
(1-\Theta_{KMR}(\lambda^z))^{T_{KMR}(\eta)}$ and $\eta \geq  (1- 
\Theta_{M}(\lambda))^{T_{M}(\eta)-1}$, giving, for small values of $\lambda$, 
% 
% 
\begin{equation}  \label{comrate} 
\frac{T_{KMR}(\eta)}{T_{M}(\eta)-1}  \geq  \frac{ln 
(1-\Theta_{KMR}(\lambda^z))}{ln 
(1-\Theta_{M}(\lambda))} \approx 
\frac{\Theta_{M}(\lambda)}{\Theta_{KMR}(\lambda^z)} 
\sim     \frac{1}{\lambda^{z-1}}. 
\end{equation} 
% 
% 
If, for 
example, 
$N= 100$ and $z = 33$, so that $1/3$ of one's opponents must play 
the risk-dominant 
strategy in order for it to be a best reply, then it will take 
$1/\lambda^{32}$ 
times as long 
for the Kandori, Mailath and Rob model to be within $\eta$ of its 
stationary 
distribution as it takes 
the muddling 
model. 
Ellison \cite{Ellison1991} obtains a similar 
comparison 
for the Kandori, Mailath and Rob model and his ``two-neighbor'' matching 
model.  Ellison 
notes that if 
$N = 
100$ and $z = 33$,  then halving the mutation rate causes 
 his 
two--neighbor matching model (and hence our muddling model) to take 
about twice 
as long 
to 
converge, 
while the Kandori, Mailath and Rob model will take $2^{33}$ ($> 8$ 
billion) times as 
long to converge. 
 
The difference in rates of convergence for these two models will be most
striking 
when the 
mutation rate is very small.  There remains the question of how the models 
might 
compare for less extreme rates of mutation.  In 
\cite{BinmoreSamuelsonandVaughan1993}, we 
present an example in which $N = 100$, $z = 33$, and $\lambda = .001$.  The 
expected 
waiting time in the Kandori, Mailath and Rob model is approximately 
$1.7~\times~10^{72}$, 
while that of the muddling model is approximately $3 ~\times~10^5$. 
 
 
This 
result appears because the Kandori, Mailath and Rob model relies upon 
mutations to 
accomplish 
its transitions between equilibria, so that convergence requires waiting 
until the 
number of 
simultaneous mutations required to move from one equilibrium to the basin of 
attraction of 
another becomes a reasonably likely event.  In contrast, the muddling model 
requires 
mutations 
only to escape boundary states.  Once a single mutation has allowed this
escape, 
then 
subsequent adjustments can be performed by the noisy learning dynamics.  When 
mutation 
rates are small, the learning dynamics proceed at a very much faster rate than

mutations 
occur, 
so that convergence requires waiting only long enough to make a single mutation

a 
reasonably likely event. 
 
 
We can provide an additional perspective by noting that an 
analogous argument 
establishes: 
 
\begin{proposition} 
For sufficiently small $\lambda$, 
% 
\begin{displaymath} 
\lim_{N \rightarrow \infty} T_{KMR}(\eta)/T_{M}(\eta) = \infty. 
\end{displaymath} 
% 
\end{proposition} 
 
 
 
\noindent  We thus have faster convergence for either small 
mutation 
rates or large population sizes. 
 
 
 
 
\section{Equilibrium Selection} 
 
 
We now consider equilibrium selection.  To do this, we examine the stationary 
distribution 
of the Markov process, in the limit as the population size gets large and the 
mutation rate 
small.  In particular, we begin with a stationary distribution and then study
the 
limits  $N 
\rightarrow \infty$ and $\lambda 
\rightarrow 0$ as an exercise in comparative statics.  The order in which these

two limits 
are taken is one of the issues to be examined. 
 
 
 
 
\paragraph{Two Strict Nash Equilibria.}  We first assume $A>B$ and $D > C$, so

that the 
Reward Game has two 
strict Nash equilibria . 
Let $\sigma_{(\lambda,N)}(x)$ 
denote the stationary distribution of the Markov 
process on $\{0,1, \ldots, N\}$ given the mutation rate $\lambda$ and 
population size 
$N$.  Extend this to a measure on $([0,1],\Im)$ (where $\Im$ is 
the Borel sets) by letting $\sigma_{\lambda,N}(A) = \sum_{(x/N) 
\in 
A} \sigma_{\lambda,N}(x)$. 
Then: 
 
\begin{proposition}         \label{twostrict} 
There exists a unique probability measure $\sigma^*$ on 
$([0,1],\Im)$ 
with 
$\lim_{N \rightarrow \infty} \lim_{\lambda \rightarrow 0} 
\sigma_{(\lambda,N)} = \lim_{\lambda \rightarrow 
0} 
\lim_{N 
\rightarrow 
\infty} \sigma_{(\lambda,N)} = \sigma^*$, 
where the limits refer to the weak convergence of probability measures.  In 
addition, 
$\sigma^*(\{0\}) + \sigma^*(\{1\}) = 1.$ 
\end{proposition} 
 
\noindent {\bf Proof.~~} 
We first construct a candidate for $\sigma^*$. 
Fix $N$. Then it must be the case that $\lim_{\lambda \rightarrow 0} 
\sigma_{(\lambda,N)}(0) + \lim_{\lambda \rightarrow 0} 
\sigma_{(\lambda,N)}(1) = 1$,   because% 
% 
% 
\footnote{To conserve on notation, we freely write ``$0$'' and ``$1$'' as 
arguments of 
measures, rather than ``$\{0\}$'' and ``$\{1\}$''.} 
% 
% 
% 
% 
% 
\begin{displaymath} 
\lim_{\lambda \rightarrow 0} \frac{r(0)}{\ell(1)} = \lim_{\lambda 
\rightarrow 0} 
\frac{\ell(N)}{r(N-1)} = 0, 
\end{displaymath} 
% 
\noindent and the result then follows from (\ref{setlim}) and the 
fact that 
$\lim_{\lambda 
\rightarrow 0} (r(x)/\ell(x+1))$ is nonzero and finite 
for 
every value $x 
\in \{1,2, 
\ldots, N-2\}$. 
This result states that as the mutation rate approaches zero, the system spends

an 
increasing amount of time ``stuck'' at its endpoints, so that in the limit all

probability must 
accumulate on these endpoints. 
 
Hence, we set $\sigma^*(0) + 
\sigma^*(1) = 1$, and the only remaining question concerns the ratio of these
two 
values. 
To fix this ratio, we note that 
for fixed $N$ and $\lambda$, we have: 
% 
\begin{displaymath} 
\frac{\sigma_{(\lambda,N)}(1)}{\sigma_{(\lambda,N)}(0)}~~~ =~~~ 
\prod_{x=0}^{N-1} 
\frac{r(x)}{\ell((x+1))} ~~~= 
\end{displaymath} 
% 
\begin{displaymath} 
\prod_{x=0}^{N-1} \frac{\left(x(1-\lambda) + 
\lambda 
(N-x-1)\right)  (N-x) 
 } 
{\left( 
(N-x -1)(1- \lambda) 
 + \lambda x \right) (x + 1) 
}  ~    \Xi ~~~ = ~~~ \Xi, 
\end{displaymath} 
% 
% 
\noindent where the last equality is obtained by pairing each term 
corresponding to $x$ 
in the 
numerator on the left side with the term in the denominator 
corresponding 
to $N- 
x-1$, and where 
% 
\begin{equation}       \label{prod} 
\Xi ~=~ 
\prod_{x=0}^{N-1} 
\pi \left( 
\frac{x B + (N - x - 1)D}{N-1} \right) 
\left(\pi \left( 
\frac{xA + (N-x-1)C}{N-x} \right) \right)^{-1}. 
\end{equation} 
% 
 
\noindent We then take logarithms to obtain: 
% 
% 
% 
\begin{displaymath} 
\ln \Xi = \sum_{x=0}^{N-1} 
\left\{ \ln \pi \left( 
\frac{x B + (N - x - 1)D} 
{N-1} 
\right)- \ln \pi \left( 
\frac{(x-1)A +  (N-x)C} 
{N-1} 
\right)\right\}. 
\end{displaymath} 
 
% 
% 
\noindent The  Lebesgue dominated convergence theorem then allows us to 
conclude that, for any value of $\lambda$: 
% 
% 
\begin{equation} \label{selecteq} 
\lim_{N \rightarrow \infty} 
%\lim_{\lambda 
%\rightarrow 0} 
\frac{1}{N}\ln \frac{\sigma_{(\lambda,N)} 
(1)}{\sigma_{(\lambda,N)} (0)} =  \int_0^1  ( \ln 
\pi_Y(k) - \ln \pi_X(k))dk. 
\end{equation} 
% 
\noindent  Our candidate for $\sigma^*$ thus gives (in addition to $\sigma^*(0)

+ 
\sigma^*(1) = 1$) that $\sigma^*(0) = 1$ if the right  side of (\ref{selecteq})

is negative and 
$\sigma^*(1) = 1$ if the right side of (\ref{selecteq}) is positive.% 
% 
% 
\footnote{If the right side of (\ref{selecteq}) equals zero, then both 
$\sigma^*(0)$ and $\sigma^*(1)$ may be positive.} 
% 
% 
 
Next, we argue that $\lim_{N \rightarrow \infty} \lim_{\lambda \rightarrow 0} 
\sigma_{\lambda,N} = \sigma^*$, where the limits are understood to represent 
weak 
convergence.  From Theorem 2.2 of Billingsley \cite{Billingsley1968}, it
suffices 
to show this 
for any relatively open subinterval of $[0,1]$.  Fix such an interval $A$.  If

$\{0,1\} \not \subseteq 
A$, then the result follows because 
$\lim_{\lambda \rightarrow 0}\sigma_{\lambda,N}(A) = 0 = \sigma^*(A)$.  If
either 
$0$ 
or $1$ is contained in $A$, then the result follows because (\ref{selecteq})
holds 
for all 
$\lambda$. 
 
 
 
 
 
It remains to show that $\lim_{\lambda \rightarrow 
0} 
\lim_{N 
\rightarrow 
\infty} \sigma_{(\lambda,N)} = \sigma^*$.  Toward this end, we note that,
because 
(\ref{selecteq}) holds for any $\lambda$, 
% 
\begin{displaymath} 
\frac{\lim_{\lambda \rightarrow 
0} 
\lim_{N 
\rightarrow 
\infty} \sigma_{(\lambda,N)}(1) } 
{\lim_{\lambda \rightarrow 
0} 
\lim_{N 
\rightarrow 
\infty} \sigma_{(\lambda,N)}(0) } 
=\frac{ \sigma^*(1)}{\sigma^*(0)}. 
\end{displaymath} 
% 
 
\noindent An analogous argument shows 
that this inequality holds when $\{1\}$ and $\{0\}$ are replaced by sets of the

form $(1- 
\theta,1]$ and $[0,\theta)$ for any 
$\theta < 1/2$. 
It then 
remains only to show that $(\lim_{\lambda \rightarrow 
0} 
\lim_{N 
\rightarrow 
\infty} \sigma_{(\lambda,N)}(\{0,1\}) = 1$.    For any integer $n$, consider
the 
sets $[0,1/n]$ and $[m/n,(m+1)/n]$, for $m \geq 2$ and $(m+1)/n < 
x^*/N$ (recall that $x^*/N$ is the probability attached to 
strategy $X$ by the 
mixed strategy equilibrium).  Then 
% 
% 
\begin{displaymath} 
\frac{\sigma_{(\lambda,N)}([0,\frac{1}{n}])}{\sigma_{(\lambda,N)} 
([\frac{m}{n},\frac{m+1}{n}])} \geq \prod_{x = 
N/n}^{Nm/n}\frac{r(x)}{\ell(x+1)}. 
\end{displaymath} 
% 
% 
For sufficiently small $\lambda$, every term in the product on the right side
of 
this 
inequality is less than one.  This ensures that, for sufficiently small 
$\lambda$,  $\lim_{N 
\rightarrow \infty} 
\sigma_{(\lambda,N)}([m/n,(m+1)/n]) = 0$.  A similar argument 
applies 
to subsets of $[x^*/N,1]$ and yields the result. 
\hspace{\fill}           {\bf $\Box$} 
 
\medskip 
 
We thus have that, in the limit as mutation probabilities get small 
and the 
population gets 
large (in any order), the stationary distribution of the Markov 
process attaches 
probability 
only to the two pure strategy equilibria.  In generic cases (those for which
the 
right side of 
(\ref{selecteq}) is nonzero), 
probability will be 
attached to only 
one of these equilibria, which we refer to  as the ``selected'' 
equilibrium.  From (\ref{selecteq}), we immediately have a criterion for which

equilibrium will 
be selected: 
 
\begin{proposition} 
The selected equilibrium will 
be $(X,X)$ $[(Y,Y)]$ if 
% 
% 
\begin{equation} \label{simsel} 
\int_0^1 \ln \pi_Y(k)dk 
{}~~>~~[<]~~ 
 \int_0^1 \ln \pi_X(k)) dk. 
\end{equation} 
\end{proposition} 
% 
 
A number of papers have recently 
addressed the 
problem of 
equilibrium selection in symmetric $2\Times2$ games.  A 
common finding is that the risk-dominant equilibrium in the Reward Game is 
selected.  Our process does not always select the risk-dominant 
equilibrium of either the Reward Game or the fitness process: 
 
\begin{corollary} \label{Cor} 
 
{\em (\ref{Cor}.1)}  If the payoff-dominant equilibrium in the Reward 
Game is also risk 
dominant (in the 
Reward Game), 
then 
the payoff-dominant equilibrium is selected. 
 
 
{\em (\ref{Cor}.2)}  The payoff-dominant equilibrium in the Reward Game can be

selected 
even if it fails to be risk dominant in both the Reward and Fitness Games. 
\end{corollary} 
 
 
\medskip 
 
\noindent {\bf Proof.~~}  (\ref{Cor}.1) Let $(X,X)$ and $(Y,Y)$ be 
risk-equivalent in the Reward Game, so that $A+C = B+D$, and let $A = D$.  Then

(\ref{simsel}) holds with equality.  Now make $(X,X)$ the payoff-dominant 
equilibrium by 
increasing $A$ and decreasing $C$ so as to 
preserve $A+C = B+D$ (and hence to preserve the risk-equivalence of the two 
strategies). 
This leaves $\int_0^1 \rho_X(k)dk$ 
unchanged, where $\rho_X$ is the expected reward to strategy $X$ 
given that the 
proportion of agents playing $X$ in the current state is $k$.  The 
log-concavity of $F$ then ensures that $\int_0^1 \ln 
(\pi_X(k))dk$ decreases.  The left side of (\ref{simsel}) then exceeds the
right 
side and the 
payoff-dominant equilibrium $(X,X)$ is selected.  Next, note 
that adding a constant to $A$ and $C$ or subtracting a constant from 
$D$ and $B$ so 
as to also 
make $(X,X)$ 
risk dominant increases the function $\ln \pi_Y(k) - \ln 
\pi_X(k)$ on 
$[0,1]$, which can only strengthen the inequality in 
(\ref{simsel}), and hence 
preserves the 
result that the payoff-dominant equilibrium is selected. 
 
 
 
 
(\ref{Cor}.2)  Let $F$ be a uniform distribution.  Then we can 
represent the fitness process with a Fitness Game, an example of which is given

in Figure 
3. 
% 
\begin{figure}[htb] 
\[ 
\begin{tabular}{c|@{}c@{}|@{}c@{}|} 
\MC{1}{c}{}&\MC{1}{c}{$X$}&\MC{1}{c}{$Y$}\\ 
\cline{2-3} 
\rule[-5mm]{0mm}{12mm}$X$&\Cell{$0$}{$0$}&\Cell{$.75$}{$.25$}\\ 
\cline{2-3} 
\rule[-5mm]{0mm}{12mm}$Y$&\Cell{$.25$}{$.75$}&\Cell{$.5$}{$.5$}\\ 
\cline{2-3} 
\end{tabular} 
\] 
\caption{Fitness Game} 
\end{figure} 
% 
\noindent Neither of the two pure strategy Nash equilibria, given 
by $(X,X)$ and $(Y,Y)$, risk-dominates the other in either the reward or
fitness 
games, but 
$(X,X)$ is 
the 
payoff-dominant 
equilibrium.     The proof of (\ref{Cor}.1) then indicates that $(X,X)$ will be

the selected 
equilibrium. 
We can confirm this by performing the integration in (\ref{selecteq}) 
to 
give: 
% 
% 
\begin{equation} \label{intsel} 
\frac{\sigma^*(1)}{\sigma^*(0)} = \lim_{N \rightarrow \infty} 
\left(\left(\frac{\delta^{\delta}}{\beta^{\beta}}\right)^{\frac{1 
}{\delta-\beta 
}}\left( 
\frac{\alpha^{\alpha}}{\gamma^{\gamma}}\right)^ 
{\frac{1}{\gamma-\alpha} 
}\right)^N  =  \lim_{N \rightarrow \infty} \left( \frac{4}{3}\right)^N. 
\end{equation} 
% 
% 
This 
game can now be perturbed so that $Y$ is the risk-dominant equilibrium (and so

that no 
death 
probabilities are zero); 
but with our model still selecting $(X,X)$ and the latter continuing to be 
payoff-dominant but not risk-dominant in either the Reward or Fitness Games. 
\hspace{\fill} {\bf $\Box$} 
 
 
\medskip 
 
We can provide some intuition as to why this result differs from 
that of Kandori, Mailath and Rob \cite{KandoriMailathandRob1992}. 
The Kandori, Mailath and Rob 
model 
selects the equilibrium with the larger basin of attraction under 
best-reply 
dynamics, namely the risk-dominant equilibrium. 
In the perturbed version of the game given in Figure 3 that we considered in
the 
previous proof, the equilibrium $(X,X)$ has a 
basin of 
attraction which 
is smaller than $(Y,Y)$'s, but in 
$(X,X)$'s basin the relative payoff of $X$ to 
$Y$ is very small, being nearly zero for states in which nearly all 
agents play 
$X$. 
Given that payoffs are (undesirable) death probabilities, 
this makes it very difficult to leave $(X,X)$'s basin, and yields 
a 
selection in which all agents play $X$.  Similar considerations 
appear in 
Fudenberg 
and 
Harris \cite{FudenbergandHarris1992}, who point out that the system of 
Kandori, Mailath, and Rob moves away 
from an equilibrium by ``jumping over'' the relevant basin of 
attraction, so that 
only the size 
of this basin matters; whereas in a continuous system such as ours, 
the process 
must 
``swim upstream'' through the basin of attraction, causing the 
``slope'' as well 
as size of the 
basin to matter. 
 
 
\paragraph{No Pure Strategy Equilibria.}  We can contrast these results with
the 
case of 
games in which $B 
> A$ and $C > D$, 
so that there is a unique, mixed-strategy Nash equilibrium.  Then 
an argument analogous to that of the previous two propositions 
gives: 
 
\begin{proposition}  \label{HD} 
Let $x^*/N$ be the probability attached to $X$ in the mixed 
strategy 
equilibrium.  Then 
$\lim_{\lambda \rightarrow 0} \lim_{N \rightarrow \infty}
\sigma_{(\lambda,N)}(A) 
= 0$ if 
$x^*/N \not \in 
A$. 
However, $\lim_{\lambda \rightarrow 0} 
\sigma_{(\lambda,N)}(0) \ + \lim_{\lambda \rightarrow 0} 
\sigma_{(\lambda,N)}(1) = 1$. 
\end{proposition} 
 
 
\noindent The order of limits makes a difference in this case.  If 
mutation rates 
are first 
allowed to approach zero, then the long-run dynamics are driven by 
the possibility 
of 
accidental extinction coupled with the impossibility of recovery, attaching 
probability only 
to the nonequilibrium states in which all agents play the same strategy. 
If the 
population size is first 
allowed to get large, then accidental extinctions are not a factor 
and the long 
run outcome 
is the mixed strategy equilibrium.  Our inclination here is to 
embrace the latter 
as the more 
useful model. 
 
 
 
 
 
\section{Comparative Statics} 
 
In this section, we establish several comparative static results 
for the case of games with 
two strict Nash equilibria. 
 
\paragraph{High Background Fitness.} 
Suppose that the random payoff variable $\tilde{R}$ is given by the 
value $\underline{R}$ with 
probability $\theta$, where 
$\underline{R} + \overline{\rho} < \Delta$.  (Recall $\overline{\rho} = 
max_{\rho \in 
\{A,B,C,D\}}\rho$).  With probability 
$1-\theta$, $\tilde{R}$ is 
given by a draw from a distribution 
$\hat{F}$.  Hence, with 
probability $\theta$, a learning agent simply abandons the current 
strategy 
regardless of its payoff.  We say that the distribution of 
$\tilde{R}$ (denoted 
$F$) has {\em background 
fitness level $\theta$} when it takes this form.  In this section, 
we hold 
$\hat{F}$ fixed and investigate the implications of varying levels 
of background 
fitness $\theta$. 
 
The probability $\theta$ is intended to capture background 
considerations 
that are independent of what happens in the game under study. Such 
notions of background fitness have played an important role in 
biological models of evolution.  If 
$\theta = 1$, then the process is driven {\em entirely\/} 
by background considerations, in 
that decisions to switch strategies have nothing to do with the 
rewards in the game ${\cal R}$.  If $\theta = 0$, then there are 
no 
background   considerations and only rewards in the game matter 
when  adjusting strategies.% 
% 
% 
\footnote{There are several 
alternative interpretations of the level of 
background fitness. For example, 
agents may be occasionally withdrawn from the game-playing 
population. 
Alternatively, bounded rationality might be at work, 
there being some probability that an agent's 
thought processes are currently congested with other 
considerations. In either 
case, it seems reasonable to assume that an agent's switching 
decision might then be determined by factors extraneous to the 
game. 
For example, it may be that the agent is actually participating in 
a large number of similar games using the same strategy for each, 
as in 
the model of Carlsson and Van Damme \cite{CarlssonandvanDamme1992}. 
Strategy revisions will then often be determined by what happens 
in games other than the game actually under study.} 
 
 
\begin{proposition}  \label{bigback} 
Let $F$ have background fitness $\theta$.  Then for sufficiently large
$\theta$, 
the 
risk-dominant equilibrium of the fitness process is selected.  If 
$F$ preserves 
dominance, then for sufficiently large $\theta$ the risk-dominant 
equilibrium of 
the Reward Game is therefore selected. 
\end{proposition} 
 
\noindent   Our model thus justifies selecting the risk-dominant 
equilibrium in economic games as long as one believes that the 
learning context 
in which 
the game is played is such that strategy choices have little 
to do with 
the 
game.% 
% 
% 
\footnote{Notice that whether strategy $Y$ is risk-dominant in the fitness 
process and hence 
whether dominance is preserved depends only on $\hat{F}$ and not 
on the 
level of background fitness, so that the proposition is well-defined 
in invoking risk-dominance 
notions independently of $\theta$.} 
% 
% 
 
 
 
 
 
\medskip 
 
\noindent {\bf Proof.~~}  Let 
$(X,X)$ and $(Y,Y)$ be Nash equilibria, 
with $(Y,Y)$ being risk dominant in the fitness process. 
{}From (\ref{simsel}), the selected equilibrium will be the 
risk-dominant 
equilibrium if: 
% 
\begin{equation}  \label{airback1} 
\int_0^1(\ln(\theta + (1-\theta)\pi_Y(k)) - \ln(\theta + (1- 
\theta)\pi_X(k))dk < 0, 
\end{equation} 
% 
where $\pi_Y$ and $\pi_X$ are the death probabilities induced by $\hat{F}$.  As

$\theta \rightarrow 1$, the left side of (\ref{airback1}) approaches zero.  We

accordingly examine the first derivative of the left side of 
(\ref{airback1}).  A positive derivative ensures that 
(\ref{airback1}) approaches zero from below, so that 
(\ref{airback1}) is negative  and hence the 
risk-dominant equilibrium is selected for $\theta$ near $1$. 
Evaluated at 
$\theta=1$, 
this derivative is: 
% 
\begin{equation} \label{airback2} 
\int_0^1 (\pi_X(k) - \pi_Y(k))dk. 
\end{equation} 
% 
Because $(Y,Y)$ is risk dominant in the fitness process, 
(\ref{airback2}) is 
positive. 
The selected equilibrium, for large background fitness, is thus the 
equilibrium 
that is 
risk dominant 
in the fitness process.  If $F$ preserves dominance, then this is 
also the 
equilibrium that is risk dominant in the Reward Game. 
\hspace{\fill}  {\bf $\Box$} 
 
\medskip 
 
To see the forces behind this result, consider the function 
$\pi_Y(k)/\pi_X(k) \equiv \Psi(k)$, where $\pi_Y$ and $\pi_X$ are induced by
$F$.  
{}From (\ref{phi}) and (\ref{selecteq}), this 
function holds 
the key to determining which equilibrium is selected. 
Recall that $\Psi(k)$ is increasing in $k$, and that a bias in favor of 
the payoff-dominant 
equilibrium 
is caused by the fact that values of $\Psi(k)$ are especially 
large for high 
values of $x$. 
The reason is that $(X,X)$ is the 
payoff-dominant 
equilibrium, so 
the death probability $\pi_X$ is small (and so $\Psi$ large) for high 
values of 
$x$. 
As $\theta$ 
approaches unity, however, $\Psi$ also converges to unity, because deaths are
now 
caused primarily by exogenous factors rather than the payoffs of the Reward
Game. 
The effect of 
having very 
large values of $\Psi$ for values of $x$ near one is then 
removed, and 
equilibrium 
selection is driven by the sizes of the basins of attraction.  This 
leads to 
selection of the risk-dominant equilibrium. 
 
 
 
In summary, we get the risk-dominant 
equilibrium in relatively 
unimportant games -- those whose payoffs have little to do with 
agents' 
behavior.   Notice 
that the relevant risk dominance notion arises in the fitness 
process and not in 
the 
Reward Game.  This is simply a reflection of the fact that the 
learning process is driven by fitnesses rather than rewards, and 
the common 
practice of 
treating rewards as fitnesses may not always be appropriate. 
 
 
\paragraph{Noisy Learning.} 
Another result can be interpreted as saying that risk-dominant 
equilibria are 
selected in 
unimportant games.  Say that the distribution $F$ {\em has noise 
level $\theta$} 
if 
$F$ attaches probability $\theta /2$ to each of the values 
$\underline{R}$ and 
$\overline{R}$, with $\underline{R} + \overline{\rho} < \Delta$ and
$\overline{R} 
+ 
\underline{\rho}  > \Delta$ (where 
 $\underline{\rho} = \min_{\rho \in \{A,B,C,D\}} \rho$) and if $F$ is given by

the distribution 
$\hat{F}$ with 
probability $1-\theta$.  Hence, with probability $\theta$, the 
agent simply chooses randomly whether to abandon or retain a strategy, with
equal 
probability of each outcome.  Then the proof of the previous 
proposition can be recycled to show: 
 
\begin{proposition}  \label{noiseback} 
Let $F$ have noise level $\theta$.  Then for sufficiently large $\theta$, the 
risk-dominant equilibrium of the fitness process is selected.  If 
$F$ preserves 
dominance, then for sufficiently large $\theta$ the risk-dominant 
equilibrium of 
the Reward Game is therefore selected. 
\end{proposition} 
 
 
 
\noindent We again have the risk-dominant outcome appearing in 
games where agents' 
strategy choices have little to do with the payoffs of the 
game, and hence 
have risk dominance in relatively unimportant games.% 
\footnote{Many variations on these last two propositions, involving 
various rules under which $\tilde{R}$ is chosen from a distribution 
$\hat{F}$ with probability $1-\theta$ and with probability $\theta$ is
determined 
by a 
random process exogenous to the game, can be similarly established.} 
 
 
 
\paragraph{Best-Response Dynamics.} 
We next ask how the selected equilibrium varies as the 
learning rule approaches best--response learning. 
Let $A > B$ and $D > C$, and let $x^*/N > 
1/2$;  so that there are two strict Nash equilibria, with $(Y,Y)$ being 
the risk-dominant equilibrium.    Let $\Delta$ equal 
the mixed-strategy equilibrium payoff, and let the distribution of 
$\tilde{R}$ 
put probability mass $\phi$ on zero and be otherwise distributed 
according to some function $\hat{F}$ with probability $1-\phi$. 
The death probability for strategy $Y$ at state $x$ is then 
given 
by $\phi + (1-\phi)\pi( (x B + (N-x-1)D)/(N-1) )$ if $X$ is 
a best reply and  by 
$(1- 
\phi)\pi((x B + (N-x-1)D)/(N-1))$ otherwise, where $\pi$ is induced by
$\hat{F}$.   As 
$\phi$ 
approaches unity, 
the learning rule 
approaches the best-reply dynamics. 
 
\begin{proposition} 
Let $\Delta$ equal the mixed strategy equilibrium payoff.  Fix a 
distribution $\hat{F}$ and let $F$ put a probability mass of 
$\phi \in (0,1)$ on zero and otherwise be given by 
$\hat{F}$. 
Then for values of $\phi$ sufficiently close to one, 
the 
selected equilibrium is the risk-dominant equilibrium in the Reward Game. 
\end{proposition} 
 
\noindent {\bf Proof.~~} 
Let $k^* \equiv x^*/N > 1/2$, so that $(Y,Y)$ is the risk-dominant equilibrium
in 
the Reward 
Game. 
{}From (\ref{simsel}), the selected equilibrium will be $(Y,Y)$ if: 
% 
\begin{displaymath} 
\int_0^{k'}\{ \ln((1-\phi)\pi_Y(k)) - \ln (\phi + (1- 
\phi)\pi_X(k))\}dk 
\end{displaymath} 
% 
\begin{displaymath} 
+~\int_{k'}^{k^*}\{\ln((1-\phi)\pi_Y(k)) - \ln (\phi + (1- 
\phi)\pi_X(k))\}dk 
\end{displaymath} 
% 
\begin{displaymath} 
+~\int_{k^*}^{1}\{ \ln(\phi +(1-\phi)\pi_Y(k)) - 
\ln ((1-\phi)\pi_X(k))\}dk    < 0, 
\end{displaymath} 
% 
% 
where $k' > 0$ satisfies $1-k^*= k^*-k'$.  Because $1-k^* = 
k^*-k'$, the sum of the second and third terms on the left approaches 
a finite 
number as $\phi$ approaches unity.  The first term approaches 
negative infinity, and hence the result. 
\hspace{\fill}  {\bf $\Box$} 
 
\medskip 
 
The forces that drive this result mimic those that appear in the 
 models of Young \cite{Young1992} and Kandori, Mailath and Rob 
\cite{KandoriMailathandRob1992}.  If $\phi$ is very close to 1, 
then the learning 
process almost certainly 
pushes the system in the direction of a best reply, and moving 
against 
this direction requires a mutation.  As the mutation probability 
gets arbitrarily small, the only relevant factor in equilibrium 
selection becomes the number of mutations required to leave each 
equilibrium.  This number is higher for the risk-dominant 
equilibrium in the Reward Game, and the latter is selected. 
 
 
 
 
\paragraph{Risk Dominance.} 
We now ask how the likelihood of choosing the risk-dominant 
equilibrium in the Reward Game varies with the rewards in that 
game.  To pose this 
question precisely, let $k^* = x^*/N$ be the probability attached to $X$ in the

mixed 
strategy equilibrium.    Let 
$(Y,Y)$ be the risk-dominant equilibrium, so $k^* > 1/2$, but let 
$(X,X)$ be 
payoff dominant. 
Let $\Upsilon$ be the payoff in the Reward Game from the mixed strategy 
equilibrium. 
We will fix 
$k^*$ and 
$\Upsilon$ and vary the rewards $A,B,C,D$.   In particular, let 
$C(A)$ and $B(D)$ 
solve 
$(1-k^*)C(A) + k^*A = (1-k^*)B(D) + k^*D = \Upsilon$.  We will then 
examine pairs 
of values 
$(A,D)$, and take the associated $B$ and $C$ to be given by $B(D)$ and 
$C(A)$, which ensures 
that 
$k^*$ and $\Upsilon$ (and hence the best-reply correspondence) are preserved as

rewards 
are varied.    Let $\Omega(k^*,\Upsilon) 
= 
\{(A,D):D \in (C(A),A)\}$; this is the set of possible values of 
$A$ and $D$ that are consistent with $(X,X)$ being the payoff-dominant 
equilibrium.  Let 
$\Omega^*(k^*,\Upsilon) 
\subset \Omega(k^*,\Upsilon)$ be the subset of values for which the 
payoff-dominant equilibrium 
in the 
Reward Game 
is 
selected.  Then we have: 
 
\begin{proposition}    \label{payvar} 
 
{\em (\ref{payvar}.1)}  Fix $k^*$ and $\Upsilon$.  If $(A,D) \in 
\Omega^*(k^*,\Upsilon)$ and $(A',D') \in \Omega(k^*,\Upsilon)$ with  $A' 
\geq A$ and $D' \leq D$, then $(A',D') \in \Omega^*(k^*,\Upsilon)$. 
 
{\em (\ref{payvar}.2)}  Fix $\Upsilon$.  Then there 
exist values of $k^*$ for which $\Omega^*(k^*,\Upsilon)$ is nonempty, and if 
$\Omega^*(k^{**},\Upsilon)$ is 
nonempty 
then $\Omega^*(k^*,\Upsilon)$ is nonempty for all $k^* \in (.5,k^{**})$. 
\end{proposition} 
 
\noindent This proposition tells us, from (\ref{payvar}.1), that movements in
the 
rewards 
to strategy $Y$ 
that 
increase $B$ and decrease $D$ make the payoff-dominant equilibrium 
more likely to 
be 
selected.  The best case for the payoff-dominant equilibrium is 
that in which $D 
< B$. 
Hence, the payoff-dominant equilibrium is favored by reducing the 
variation in 
rewards to 
strategy $Y$ or even ``inverting'' them, so that while $(Y,Y)$ is 
an equilibrium, 
the 
highest 
reward to strategy $Y$ is obtained if the opponent plays $X$.  On 
the other hand, 
the payoff-dominant equilibrium appears if $A$ is large, so that 
there is 
considerable 
variation in the 
rewards to strategy $X$.  Finally, from (\ref{payvar}.2), we have that the
larger 
is the 
basin of 
attraction of the risk-dominant equilibrium, the harder it is for 
the 
payoff-dominant equilibrium to be 
selected. 
 
\medskip 
 
\noindent {\bf Proof of Proposition \ref{payvar}:~~} 
(\ref{payvar}.1)  Fix $k^* > 1/2$.  Notice that $C(A)$ is linear. 
The second 
derivative of 
$\int_0^1(\ln(\pi_X(k))dk= 
\int_0^1(\ln F(\Delta - ((1-k)C(A) + kA)))dk$ with respect to $A$ is then given

by 
$\int_0^1(\zeta^2(F(\eta)F''(\eta) - (F'(\eta))^2)/(F(\eta))^2)dk$, 
where $\eta 
=\Delta -((1- 
k)C(A)+kA)$ and $\zeta = d \eta /dk$.  The concavity of $\ln F$ 
ensures that this 
is 
negative and hence  $\int_0^1(\ln(\pi_X(k))dk$ (and similarly 
$\int_0^1(\ln(\pi_Y(k))dk$) is 
concave.  Fix $A$ and hence $C(A)$.  It is clear that if we set 
$D=A$, then the 
equilibrium 
$(Y,Y)$ will be selected, since it is risk dominant and payoff 
undominated.  Now 
let $D$ 
decline.  From (\ref{simsel}), the concavity of 
$\int_0^1(\ln(\pi_Y(k))dk$ 
ensures that 
if there are any values of $D$ for which $(X,X)$ is selected, they 
form an 
interval of the form 
$(C(A), D')$ for some $D' < A$.  Similarly, fix $D$.  If $A = D$, 
then $(Y,Y)$ is 
selected. 
{}From (\ref{simsel}), the convexity of $-\int_0^1(\ln(\pi_X(k))dk$ 
ensures that if 
there is a set 
of values for which the payoff-dominant equilibrium is selected, 
it is of the form 
$(A',\infty)$ 
for some $A'$. 
 
(\ref{payvar}.2)  Let values $A$, $B$, $C$, and $D$ exist such that 
the 
payoff-dominant equilibrium is selected and the mixed strategy 
equilibrium is given 
by 
$k^{**}$, with payoff $\Upsilon$. 
Let $1/2 < k^* < k^{**}$.  Then we can find values $A' > A$, $C'> 
C$, $B'<B$, and 
$D'<D$ that 
(1) give mixed strategy equilibrium $k^*$; (2) increase all rewards 
to $X$ 
and decrease all rewards to $Y$;  and (3) preserve $\Upsilon$. 
These payoffs 
must then preserve the property that the    payoff-dominant 
equilibrium is selected, so $\Omega^*(k^*,\Upsilon)$ is nonempty. 
\hspace{\fill} {\bf $\Box$} 
 
\medskip 
 
 
Given this result, it is interesting to note that Straub \cite{Straub93} has 
conducted 
experiments to investigate the conditions under which risk-dominant and 
payoff-dominant equilibria are selected.  Straub found that the risk-dominant 
equilibrium was the most common equilibrium played in seven out of eight of the

experiments.  The exception, in which the payoff dominant equilibrium appeared,

was the only game in which $D < B$. 
 
 
\section{Endogenous Learning} 
 
Our model is driven by a particular learning process that is arbitrary in some

respects.   Why 
should we be interested in this process, or more generally, why should we be 
interested 
in any evolutionary model that is driven by an arbitrary learning rule that is
not 
derived from 
empirical studies? 
 
One response to this question emphasizes the need to investigate the 
implications of many learning rules of different kinds.  The analysis of the 
previous section 
is 
in this spirit.  A potentially more fruitful approach, in our view, is to 
recognize that learning rules themselves are likely to have been learned in an

evolutionary process.  In this section, 
we therefore expand our model so that the learning rules themselves become the

objects 
of evolutionary selection. 
 
An attempt to model the evolution of learning rules raises the specter of an 
infinite regress.  A model in which agents learn how to play games now becomes
a 
model in which agents learn how to learn to play games.  But then why not a
model 
in which agents learn how to learn how to learn, and so on?  Two considerations

arise.  First, the higher 
processes may well be biological and hence hard-wired into our cognitive 
apparatus. 
The second is that agents will learn only when there is something to learn
about. 
We can 
therefore 
hope to escape the infinite regress by showing that outcomes are not 
particularly sensitive to the nature of the learning rules that agents use in 
learning how to 
learn.  In particular, the 
results we establish below hold for any evolutionary process in which learning

rules that yield higher average payoffs fare better than those with lower
average 
payoffs. 
If this robustness had appeared at the first level, when examining how agents 
learn to play games, we would not have been prompted to seek a level higher.
If 
this robustness persists outside of our narrow model, then it may be
unnecessary 
to proceed further. 
 
In seeking to endogenize some aspects of the learning rule, we 
allow only the value of the aspiration level $\Delta$ to be subject to change.%
%%
% 
% 
\footnote{We view the information available to agents and the distribution of 
$\tilde{R}$  as 
being part of 
the 
technology of the game.  It 
would be interesting to consider models in which players might take 
steps, perhaps 
at a 
cost, to influence this latter distribution.} 
% 
% 
We therefore label a learning rule by the aspiration level $\Delta$ which it 
incorporates. 
 
 
 
 
\paragraph{Rational Expectations.} 
One might expect the aspiration level $\Delta$ to bear some relationship to the

payoffs earned  in the game.  We accordingly 
first investigate 
a simple 
``rational expectations'' property. 
We say that the learning process exhibits {\em rational 
expectations} 
if the aspiration level $\Delta$ is set equal to the average expected 
payoff of the 
selected equilibrium.% 
% 
% 
 
 
It may appear demanding for 
every agent 
to require 
at least an average payoff.  However, there is no logical inconsistency in 
holding an aspiration level that not only equals the average payoff but even 
equals a much higher 
value, though such an agent is likely to switch strategies frequently. 
Indeed, we are all familiar with 
colleagues who, driven by a quest for exceptionally high payoffs, 
change strategies (in this case, perhaps, jobs) frequently.  The use of such an

aspiration 
level should not be confused with the  inconsistent 
behavior of the 
vast 
majority of drivers who consider themselves to be better than average 
at driving, or the 
academic 
departments which demand that their members all receive above average 
teaching 
ratings.  Finally, similar impossibility results could be constructed that
require 
not that the 
aspiration level be equal to the average payoff but merely that it be
positively 
related 
to the average 
payoff. 
 
Appendix II proves: 
 
\begin{proposition}       \label{ratprop} 
There exist games in which it is impossible for the learning 
process to exhibit 
rational 
expectations. 
\end{proposition} 
 
 
 
 
\medskip 
 
To see why this impossibility appears, let $A$ be the payoff from the 
payoff-dominant equilibrium 
and $D$ the payoff from the risk-dominant equilibrium, so $A > D$. 
Setting the aspiration level equal to  $A$ yields relatively high death 
probabilities for both 
strategies.  However, relatively high death probabilities for both strategies
are 
the 
conditions under which the risk-dominant equilibrium is likely to be selected.

We have seen 
this in the results of Propositions \ref{bigback} and \ref{noiseback}, where
high 
death 
probabilities for both strategies, produced, by high background fitness levels
or 
noisy 
learning, 
drive the outcome to the 
risk-dominant equilibrium. 
Similarly, setting the aspiration level equal to $D$ yields relatively low
death 
probabilities 
for both strategies, which are the conditions under which the payoff-dominant 
equilibrium 
is likely to be selected.  It is then possible 
that when 
$\Delta = D$, death probabilities are low enough to support the 
payoff-dominant 
equilibrium 
but setting $\Delta =A$ raises death probabilities enough to tip 
the system to the 
risk-dominant equilibrium.  If so, it cannot be that $\Delta$ is 
set 
equal to the 
equilibrium payoff. 
 
 
\paragraph{Evolution of Learning Rules.} 
Proposition \ref{ratprop} prompts us to examine 
models in which the aspiration level $\Delta$ is set by evolutionary forces. 
Only games with two strict Nash equilibria are considered. 
Because we identify learning rules with their associated value of $\Delta$, we

refer to the 
evolution of $\Delta$ as the evolution of a learning rule. 
 
We capture the evolution of learning rules in a ``two--tiered'' 
model. 
We view the 
evolution of strategy choices, guided by a particular learning 
rule, as proceeding 
at a pace that 
is rapid compared to the evolution of the learning rule 
itself.  We 
take our 
existing model to represent the evolution of strategy choices given 
a learning 
rule.  We shall 
not offer an explicit dynamic model of the evolution of learning 
rules.  Instead, 
we seek conditions under which a learning rule will satisfy 
conditions analogous to those of evolutionary 
stability.% 
% 
% 
\footnote{See Harley 
\cite{Harley1981}, Maynard Smith \cite{MaynardSmith1982}, and 
Ellison and Fudenberg \cite{EllisonandFudenberg1992} for 
work in this vein.} 
% 
% 
These will ensure that our results hold for a broad class of evolutionary 
processes, namely those driven by differences in the average payoffs of various

learning rules. 
 
We find that the evolution of the 
aspiration 
level $\Delta$ depends upon the distribution of $\tilde{R}$, and especially 
on how 
dispersed is 
this distribution. We first prove a simple result: 
 
\begin{proposition}  \label{endo1} 
Consider an aspiration level $\Delta$.  Fix  $t 
> 0$.  Suppose 
that $\prob(R < \Delta-t)/ 
\prob(R<\Delta)$ is increasing in $\Delta$. Then for large enough 
$N$ and small enough $\lambda$ and for any $\Delta'$ with $\Delta' <  \Delta$,
the 
payoff 
to a 
player characterized by $\Delta'$ in any population consisting of 
these two rules 
exceeds that of $\Delta$. 
\end{proposition} 
 
\noindent To interpret this, consider the condition that $\prob(R < 
\Delta-t)/ 
\prob(R<\Delta)$ is increasing in $\Delta$.  This will hold for 
distributions that 
are not too dispersed, in the sense that the probability in the 
tails of the 
distribution falls 
off sufficiently rapidly as one moves out along the tail.  For 
example, this 
condition holds 
for the Normal distribution.  To see this, we note that for the 
Standard Normal, 
the condition 
is 
equivalent to the 
statement that $\prob(R>s+t)/\prob(R>s)$ is 
decreasing in $s$. 
We can then 
calculate: 
% 
\begin{displaymath} 
\frac{d}{ds}\frac{\prob(R > s+t)}{\prob(R>s)} = \frac{-e^{- 
(s+t)^2/2}\prob(R>s)+e^{-s^2/2}\prob(R>s+t)}{(\prob(R>s))^2}, 
\end{displaymath} 
% 
% 
and it suffices to show: 
% 
\begin{displaymath} 
\frac{\prob(R>s+t)}{\prob(R>s)} < e^{-st}e^{-t^2/2}. 
\end{displaymath} 
% 
To establish this inequality, we note the following: 
% 
% 
\begin{displaymath} 
\prob(R>s+t) = \int_{s+t}^{\infty} e^{-x^2/2}dx = 
\int_{s}^{\infty} 
e^{-(t+x)^2/2}dx ~ = 
\end{displaymath} 
% 
% 
\begin{equation} \label{normal} 
e^{- 
t^2/2}\!\int_{s}^{\infty}\! e^{-tx}e^{-x^2/2}dx \leq 
e^{-t^2/2}e^{-st}\!\int_{s}^{\infty}\! e^{-x^2/2}dx 
= e^{-t^2/2}e^{-st}\prob(R>s). 
\end{equation} 
% 
 
 
 
 
\medskip 
 
\noindent {\bf Proof of Proposition \ref{endo1}.~~} 
Let the agents in the population be distributed 
among aspiration levels $\Delta$ and $\Delta'$ with $\Delta' < 
\Delta$. 
For 
sufficiently large population size and small mutation rates, there 
exist numbers 
$p_X(N,\lambda)$, $p_Y(N,\lambda)$, 
$x_Y(N,\lambda)$ and 
$x_X(N,\lambda)$, with the sum of the first two numbers arbitrarily close to
one 
and the 
latter two numbers arbitrarily close to $0$ and $1$, such that 
the stationary distribution induced by the prevailing 
collection of learning 
rules spends a proportion of 
at least $p_Y(N,\lambda)$ of the time in states in 
which $x/N < x_Y(N,\lambda)$ (in which case $Y$ is a best reply) and 
$p_X(N,\lambda)$ 
of the time in states in which $x/N > x_X(N,\lambda)$ (in which case $X$ is a
best 
reply).  Call these sets of states 
$P_Y$ 
and $P_X$, and call the remaining states $P_D$. 
 
The difference between the 
payoffs to aspiration levels $\Delta'$ and $\Delta$ 
is given by 
% 
\begin{displaymath} 
p_Y(N,\lambda) \Pi(P_Y) + p_X(N,\lambda)\Pi(P_X) + (1- 
p_Y(N,\lambda) - p_X(N,\lambda)) \Pi(P_D), 
\end{displaymath} 
% 
\noindent where $\Pi(P_Y)$ is the expected payoff difference 
between aspiration 
levels 
$\Delta'$ and $\Delta$ conditional on the system being in set 
$P_Y$, and $\Pi(P_X)$ 
and 
$\Pi(P_D)$ are analogous.  Because the 
time spend in set $P_D$ can be made arbitrarily small by increasing 
$N$ and decreasing $\lambda$, it suffices 
to show that at least one of $p_Y(N,\lambda)\Pi(P_Y)$ 
or $p_X(N,\lambda)\Pi(P_X)$ is positive and bounded away from zero as $N$ 
increases and $\lambda$ decreases. 
 
At least one of $p_Y(N,\lambda)$ and $p_X(N,\lambda)$ must be bounded away from

zero.  Suppose $p_Y(N,\lambda)$ is, and consider $\Pi(P_Y)$.  This is given by

% 
\begin{equation}   \label{paycomp1} 
\sum_{k=0}^{\infty} 
\left( 
\frac{k p^k_Y(N,\lambda)} 
{\sum_{h=0}^{\infty}hp_Y^h(N,\lambda)} 
\right) 
\Pi^k(P_Y), 
\end{equation} 
% 
\noindent where $p^k_Y(N,\lambda)$ is the probability that a given 
episode 
in which the Markov chain is in the set $P_Y$ lasts $k$ periods, and 
$\Pi^k(P_Y)$ is the expected per-period payoff difference between learning 
rules 
$\Delta'$ and $\Delta$ during a collection of periods in which the system stays

in the set $P_Y$ for exactly 
$k$ 
periods.  Then for sufficiently small $\lambda$ we have: 
% 
\begin{equation} \label{paycomp3} 
\lim_{N \rightarrow \infty} \sum_{h=0}^{\infty} h 
p^h_Y(N,\lambda) = \infty, 
\end{equation} 
% 
\noindent  since, as $N$ gets large, the system 
spends an arbitrarily small proportion of its periods in $P_D$ and 
every stay in 
$P_Y$ 
must end with an entry into $P_D$. 
 
Next, let $\Pi^{\infty}(P_Y)$  be the difference in the 
payoffs to learning rules with aspiration levels $\Delta'$ and 
$\Delta$, conditional 
on the system 
staying in 
the set $P_Y$ an {\em infinite} number of periods. As the 
length of a stay 
in the set 
$P_Y$ 
increases, the difference in payoffs between aspiration levels 
$\Delta'$ and $\Delta$, contingent on such a stay, 
must approach $\Pi^{\infty}(P_Y)$.    Suppose 
$\Pi^{\infty}(P_Y) > 0$. 
Then for any 
$\epsilon > 0$, there is 
a $k' > 0$ such that for all $k 
> k'$, 
% 
\begin{equation} \label{paycomp2} 
\Pi^k(P_Y) > \Pi^{\infty}(P_Y) - \epsilon. 
\end{equation} 
% 
% 
\noindent 
Then (\ref{paycomp2}) and (\ref{paycomp3}) 
(along with $\Pi^{\infty}(P_Y) >0$) imply that (\ref{paycomp1}) is positive for

sufficiently large $N$ and small enough $\lambda$, and does not approach zero
as 
$N$ 
grows and $\lambda$ shrinks, giving the 
result. 
 
It then remains only to show that $\Pi^{\infty}(P_Y)$ is positive 
when $\Delta'< 
\Delta$.  For this, however, it suffices that 
$\prob(R < \Delta-t)/ 
\prob(R<\Delta)$ is increasing in $\Delta$.  In particular, within set $P_Y$,
the 
notions 
of inferior and best reply are 
unambiguous; 
and 
$P_Y$ 
can be made sufficiently small that the lowest payoff to 
a best reply over states 
in this set 
exceeds the highest payoff to an inferior reply.  We can then think of the
payoffs 
of each 
agent as being determined by the stationary distribution of a two-state Markov

process, with 
the two states being ``best reply'' and ``inferior reply'', and with the latter

giving a higher payoff than 
the former.  Call this Markov process $\Gamma^*$.  If $\prob(R < \Delta-t)/ 
\prob(R<\Delta)$ is increasing in $\Delta$ then 
the ratio of the probability of 
abandoning a best for an inferior reply to the probability of abandoning an 
inferior for a 
best reply 
is lower for 
aspiration level 
$\Delta'$ than for $\Delta$.  This in turn implies that the stationary 
distribution of 
$\Gamma^*$ must spend more time in the best-reply state for learning rule 
$\Delta'$ than for $\Delta$.  The former must then receive 
a higher 
payoff, yielding the result.  \hspace{\fill} {\bf $\Box$} 
 
 
\medskip 
 
The mechanism behind this result is straightforward.  In any 
stationary 
distribution, players spend long periods of time facing 
a mix of strategies that is concentrated on a particular 
strategy (say $Y$) but 
also 
includes other strategies.  The highest expected payoffs will be 
garnered by those 
agents 
whose learning rules cause them to spend the greatest proportion 
of 
the time 
playing the best reply $Y$.  These will be agents with learning 
rules that make 
them relatively unlikely to switch away from high payoff 
realizations and relatively likely to switch away from low payoff 
realizations.     In the case of distributions like the Normal, for which
$\prob(R 
< \Delta-t)/ 
\prob(R<\Delta)$ is increasing in $\Delta$, 
these learning 
rules 
involve smaller aspiration levels. 
 
 
 
 
 
If $\prob(R 
< \Delta-t)/ 
\prob(R<\Delta)$ is increasing in $\Delta$,  then 
the dynamics 
concerning aspiration levels are straightforward.   Let the 
aspiration level 
initially be given 
by $\Delta$.  Let the 
population be sufficiently large and mutation rate sufficiently 
small.  Then if an 
aspiration level $\Delta' < \Delta$ appears, it will displace $\Delta$.  This 
can in turn be 
displaced by a lower aspiration level $\Delta''$. Continuing in 
this fashion, the 
process will push the aspiration level ever lower.% 
% 
% 
\footnote{Fortunately, we do not have to worry about the 
possibility that lower 
values of $\Delta$ will vitiate the assumption $N$ is sufficiently 
large and 
$\lambda$ sufficiently small.  Decreasing $\Delta$ increases the 
pressure towards 
the ends of the state space, ensuring that $N$ will still be large 
enough and 
$\lambda$ small enough to yield a stationary distribution 
sufficiently 
concentrated near the ends of the state space.} 
% 
% 
Whereas we have presented this result in terms of only two 
coexisting aspiration 
levels, it generalizes in a straightforward fashion to any existing 
configuration 
of aspiration levels.  The highest such levels will always earn 
lower payoffs than 
at least 
some 
lower levels,  creating a downward pressure on 
aspiration levels. 
 
What are the implications of this process for the selected 
equilibrium?  Here we 
specialize 
to the Standard Normal distribution (the extension to other Normal 
distributions is immediate). 
 
\begin{proposition} 
Let F be the Standard Normal distribution.  Then for sufficiently small 
$\Delta$, the 
selected 
equilibrium is the equilibrium that is risk-dominant in the 
Reward Game. 
\end{proposition} 
 
 
\noindent {\bf Proof:~~} 
Let $A>C$ and $D>B$ with $A+C<B+D$, so that $(Y,Y)$ is risk 
dominant.  From 
(\ref{simsel})), we can calculate that in order to show 
that the 
risk-dominant equilibrium will be selected it suffices to show that $\ln 
F(\Delta-B) + \ln 
F(\Delta-D) - 
\ln F(\Delta -A) - \ln F(\Delta-C) <0$.  Let 
$N_{A} = 1-F(-(\Delta-A))$, and let 
$N_B$, $N_C$, and $n_D$ be analogous. 
Then it 
suffices to show that, for small values of $\Delta$, 
% 
\begin{displaymath} 
\frac{N_B N_D}{N_A N_C} < 1. 
\end{displaymath} 
% 
% 
For the Standard Normal distribution, l'Hopital's rule can be used to show 
that $\lim_{s 
\rightarrow 
\infty} N_{s+t}/N_s = e^{-t^2/2}e^{- 
st}$.  Hence, we need to show that, for small values of $\Delta$ (and hence
large 
values of $-\Delta$), 
% 
% 
\begin{displaymath} 
\frac{e^{-(D-C)^2/2}e^{-(D-C)(-\Delta+C)}}{e^{-(A-B)^2/2}e^{-(A-B 
)(-\Delta+B)}} 
< 
0. 
\end{displaymath} 
% 
% 
This in turn is equivalent to showing that 
$(A-B)[(A-B)+2(-\Delta+B)] - (D-C)[(D- 
C)+2(-\Delta+C)] < 0$.  As $-\Delta$ gets large, we need only 
examine the terms 
involving 
$-\Delta$, which gives $2(-\Delta) (A+C-B-D) < 0$.  This will hold 
for large 
$-\Delta$ 
because $(Y,Y)$ is risk dominant in the Reward Game, so that $A+C-B-D < 0$ and

hence 
the coefficient on $-\Delta$ is negative. 
\hspace{\fill} {\bf $\Box$} 
 
\medskip 
 
\noindent Evolution thus leads to learning rules that will select 
the 
risk-dominant equilibrium in the Reward Game. 
Three considerations arise in interpreting this result. 
 
First, 
there are widely dispersed 
distributions, such 
as the Cauchy distribution, which do not push $\Delta$ ever lower, 
and hence the 
payoff-dominant equilibrium may survive.  To see that $\Delta$ is not
relentlessly 
pushed lower under the Cauchy distribution, it 
it is sufficient to show that 
$\prob(R>s+t)/\prob(R>s)$ {\em increases} 
in $s$.  Here, we have $\prob(R>s) = \int_s^{\infty} 
(1/(\pi(1+x^2)))dx = (1/2)- 
(1/\pi)\arctan x$.  We then need that 
% 
\begin{displaymath} 
\frac{\pi/2-\arctan(s+t)}{\pi/2-\arctan s} 
\end{displaymath} 
% 
is increasing in $s$; and the argument is completed by noting that 
this ratio 
increases to unity as $s$ increases.   In this case, $\Delta$ will 
not be pushed 
ever-lower, 
since if $\Delta$ is sufficiently small, then further lowering 
$\Delta$ 
will increase 
$\prob(R < \Delta-t)/ 
\prob(R<\Delta)$, lowering payoffs.  At the same time, 
however, 
$\Delta$ will not be pushed arbitrarily high, since as $\Delta$ 
increases all 
death 
probabilities approach unity, eventually again increasing $\prob(R 
< \Delta-t)/ 
\prob(R<\Delta)$ and lowering payoffs.  There will then be a 
finite, 
payoff-maximizing value of $\Delta$ which may allow either the risk-dominant or

payoff-dominant equilibrium to survive. 
 
 
 
 
 
 
Second, in cases where 
evolution produces a learning rule that in turn produces the risk-dominant 
equilibrium, we must consider the length of time required to produce such an 
outcome.  We have described the evolution of learning rules as taking place at
a 
slower pace than the evolution of play in the game.  This suggests that we are

talking about a time span even longer than the ultralong run, which is likely
to 
be too long to be of relevance.  Notice, however, that the arguments in the
proof 
of Proposition \ref{endo1} require only that the system spend most of its time

close to either the risk-dominant or payoff-dominant equilibrium, and do not 
depend upon the relative amounts of time spent near these two equilibria.  This

is a condition that is satisfied in the long run, allowing the evolution of 
learning rules to proceed in the ultralong run alongside the evolutionary
process 
that produces our equilibrium selection results. 
There may then still be time for the evolution of learning rules to be
relevant. 
 
Finally, we have examined the evolution of a very narrow class of learning
rules, 
allowing only the aspiration level to adjust.  The task remains of examining
the 
implications of allowing evolution to affect other aspects of the model, such
as 
the distribution of $\tilde{R}$, which might capture changes in the ability to

identify the links between payoffs and actions.  The task also remains of 
investigating the evolution of learning rules in fundamentally different
learning 
models. 
 
 
 
 
 
 
\section{Conclusion} 
 
Evolutionary game theory offers the promise of progress 
on the problem of equilibrium selection.  At the same time, it is capable of 
reproducing 
the worst features of the equilibrium refinements literature, creating 
an ever-growing menagerie of conflicting and uninterpreted results.  To achieve

the former rather than the latter outcome, we think that evolutionary models
need 
to be 
provided with microfoundations which identify the links between the 
dynamics of the model and the underlying choice behavior.  In this paper, we 
explore four issues that arise in the course of constructing such 
microfoundations. 
 
First, it is important to identify the time-period of interest.  The techniques

relevant for a long-run analysis are not the same as those required for an 
ultralong run investigation.  In particular, perturbations of the learning
process 
that are irrelevant in the long run assume center stage when analyzing 
ultralong-run 
selection results. 
 
 
Second, the nature of these perturbations is important.  Our innovation here is

to examine a muddling rather than a maximizing model, with the primary source
of 
noise in our model being nonnegligible mistakes within the learning process 
itself. 
Introducing muddling behavior has implications both for equilibrium selection 
(where we find that the payoff-dominant equilibrium is sometimes selected) and

also for 
questions of timing.  In particular, we find that the length 
of time needed to reach the ultralong run may be shorter in a muddling than in
a 
maximizing model, making it more likely that the ultralong run will be of
interest 
in potential 
applications. 
 
The third issue brought to the fore when constructing an evolutionary model is
the 
difference between fitnesses and rewards.  The payoffs that drive an
evolutionary 
model are 
fitnesses.  We think it important to bear in mind that the relationship between

fitnesses 
and the rewards that conventionally 
appear in games depends on the nature of the selection process lying behind the

evolutionary model. 
 
The paper closes with a model in which the rules by which agents learn to play

games are themselves subject to evolutionary pressures.  Our work here is both

preliminary and incomplete, in that we have examined only a very narrow class
of 
learning rules.  But we believe this to be an important area for further work.

 
 
 
 
 
\section{Appendix I:  Proof of Proposition \ref{conprop}} 
 
 
 
 
 
Fix a time $t$ and a period length $\tau$, so that $t/\tau$ periods 
will have occurred by time $t$.  Let $\iota(k,z)$ be the 
probability that out of $z$ periods, there are exactly $k$ periods 
in 
which some individual receives the learn draw.  Then we have: 
% 
\begin{displaymath} 
\sigma_0 [\Gamma_{M}(\lambda,\tau)]^{\frac{t}{\tau}} = 
\sum_{k=0}^{t/\tau} 
\iota(k,\frac{t}{\tau})[A_{M}(\lambda)]^k, 
\end{displaymath} 
% 
\noindent where $[A_{M}(\lambda)]$ is the transition matrix 
contingent upon a 
learn draw having been received and we take $[A_{M}(\lambda)]^0 
= 
I$, the identity 
matrix.  Hence, it 
\noindent  suffices for (\ref{conus}) to show that for any 
$\sigma_0$, 
 
 
%\clearpage 
 
\begin{equation} \label{conus1} 
\lim_{\tau \rightarrow 0} 
\left(1- \|\sigma_0 \sum_{k=0}^{t/\tau} 
\iota(k,\frac{t}{\tau}) 
[A_M(\lambda)]^k - 
\sigma^*_M(\lambda) \| 
^{\frac{1}{t-1}} 
\right) 
\leq \Theta(\lambda) \sim \lambda. 
\end{equation} 
 
 
\noindent  Let 
$\tau_n = 1/n$.  Then 
$t/\tau_n = nt$, an equality we shall use repeatedly.  For each 
$n$, let $\{Z_{nk}, k \in \{1, 
\ldots, nt\} \}$ be a collection random variables, one 
for each of the periods that occur by time $t$, with each random 
variable taking the value one if a learn draw is 
received (with probability approaching $\tau_n N$) and zero if a learn draw is

not received (with probability approaching $1- \tau_n N$).  $\iota(k,nt)$ is 
then the probability that exactly $k$ of the random variables 
$\{Z_{nh}, h \in \{1, \ldots, nt\}\}$ take the value one.  Notice 
that 
for any $n$, we have 
$\sum_{k=1}^{nt} \tau_n N = nt\tau_nN =   tN$, 
so that for any $n$, the sum over the collection 
$\{Z_{nk}, k \in \{1, \ldots, nt\}\}$ of the probabilities of 
receiving 
the outcome one is finite and given by $tN$.  Coupled with the fact 
that $\tau_n$ and hence $\tau_n N$ approaches zero as $n$ gets 
large, this allows us to apply Theorem 23.2 of Billingsley 
\cite{Billingsley1986} to conclude that: 
% 
\begin{displaymath} 
\lim_{n \rightarrow \infty} \iota(k,\frac{t}{\tau_n}) = 
\frac{(Nt)^k}{e^{-Nt}k!}. 
\end{displaymath} 
% 
\noindent Hence, as $\tau_n$ gets small, $\iota(k, 
\frac{t}{\tau_n}) = 
\iota(k,nt)$ 
is given by a Poisson distribution with mean and variance $Nt$. 
It accordingly suffices for (\ref{conus1}) to show, for any 
$\sigma_0$, that 
% 
% 
% 
% 
% 
% 
 
\begin{equation} \label{conus3} 
\|\sigma_0 
\sum_{k=0}^{\infty} 
\frac{(Nt)^k}{e^{-Nt}k!}[A_{M}(\lambda)]^k - 
\sigma^*_{M}(\lambda) \| \leq (1-\Theta(\lambda))^{t-1}. 
\end{equation} 
 
% 
% 
% 
\noindent  We first observe that $\sigma^*_{M}(\lambda) 
\Gamma_{M}(\lambda) = 
\sigma^*_{M}((\lambda) (1-\tau 
N)I + \tau N A_{M}(\lambda)) = \sigma^*_{M}(\lambda)$, which we 
can 
solve for $\sigma^*_ 
{M}(\lambda)A_{M}(\lambda) = 
\sigma^*_{M}(\lambda) I = \sigma^*_{M}(\lambda)$. 
Then $\sigma^*_{M}(\lambda)$ is the (unique) stationary 
distribution of 
the matrix $[A_{M}(\lambda)]$, and we accordingly have: 
% 
\begin{displaymath} 
\lim_{k \rightarrow \infty} \sigma_0 [A_{M}(\lambda)]^k = 
\sigma^*_{M}(\lambda). 
\end{displaymath} 
% 
\noindent  The matrix $[A_{M}(\lambda)]$ has many zero elements, 
but the matrix 
$[A_{M}(\lambda)]^N$ is strictly positive.  Corollary 4.1.5 of 
Kemeny and 
Snell 
(\cite{KemenyandSnell1960},  page 71) can then be applied to show 
that 
% 
% 
\begin{equation} \label{KS1} 
\|\sigma_0 ([A_{M}(\lambda)]^N)^t - \sigma_{M}^*(\lambda)\| \leq 
(1-2S(\lambda))^{t - 1}, 
\end{equation} 
% 
% 
\noindent where $S(\lambda)$ is the smallest transition probability 
in $[A_{M}(\lambda)]^N$.   We must then examine the probability 
$S(\lambda)$. 
It 
is not immediately obvious what the least likely transition in the 
matrix $[A_{M}(\lambda)]^N$ is.  One possibility is that this is 
the transition 
from the state in which all agents play $Y$ ($x = 0$) to the 
state in which all agents play $X$ ($x = N$).  If so, then 
$S_{M}(\lambda)$ is given by 
% 
% 
% 
\begin{equation} \label{usrate} 
\prod_{x=0}^{N-1} 
\frac{N-x}{N} 
\pi\left(\frac{x B +(N-x-1)D }{N-1} 
\right) 
\left(\frac{x (1-\lambda)+\lambda(N-x-1)}{N-1} 
\right) 
\end{equation} 
% 
% 
% 
%\begin{displaymath} 
%=~~~ \lambda \left( \tau N \delta \prod_{k=1}^{N-1} \tau N 
%\frac{k}{N-1}\frac{N- 
%k}{N}\left(\frac{k}{N-1}\beta + \frac{N-k-1}{N-1}\delta \right) 
%+ 
%h(\lambda) 
%\right) 
%\end{displaymath} 
% 
\begin{displaymath} 
\equiv \lambda [c' + h(\lambda)], 
\end{displaymath} 
% 
\noindent where $c'$ does not depend on $\lambda$ and 
$\lim_{\lambda \rightarrow 
0} 
h(\lambda) = 0$.  Hence, for sufficiently small $\lambda$, we have 
$h(\lambda) < 
\epsilon$ for some $\epsilon > 0$.  We then let $c = c' - \epsilon$ 
and $C = c' 
+ \epsilon$ to obtain $S(\lambda)  \sim \lambda$.  A similar 
argument establishes that {\em any} transition within 
$[A_{M}(\lambda)]^N$ can be made with at most one step that 
requires a mutation, 
ensuring that $S(\lambda) \sim \lambda$. 
 
An argument analogous to the one giving (\ref{KS1}) gives, for any 
$\alpha >0$, 
% 
\begin{displaymath} 
\|\sigma_0 [A_{M}(\lambda)]^{\alpha N t} - 
\sigma_{M}^*(\lambda)\| 
\leq 
(1-2S(\lambda))^{\alpha t - 1}. 
\end{displaymath} 
% 
\noindent This would allow us to conclude that there exists a function 
$\Theta_M(\lambda) \sim \lambda$ such that 
% 
 
\begin{equation} \label{conus4} 
\|\sigma_0 
\sum_{k=0}^{\infty} 
\frac{(Nt)^k}{e^{-Nt}k!}[A_{M}(\lambda)]^k - 
\sigma^*_{M}(\lambda) \| \leq (1-\Theta_M(\lambda))^{\alpha t - 1} 
\end{equation} 
 
% 
\noindent holds for large $t$ and $\alpha \in (0,1)$ if we could 
conclude that 
% 
 
\begin{equation}  \label{smallk} 
\lim_{t \rightarrow \infty} \frac{(Nt)^k}{e^{-Nt}k 
!}[A_{M}(\lambda)]^k = 0 
\end{equation} 
% 
 
\noindent for any $k < \alpha N t$. 
This in turn follows from noting that as $t$ grows, the 
Poisson distribution with mean and variance $Nt$ approaches a 
Normal distribution 
with mean and variance 
$Nt$ (cf. Billingsley \cite{Billingsley1986}, problem 27.3 on page 
379).  (\ref{smallk}) is then 
$\lim_{t \rightarrow \infty} \prob [{\bf N}(Nt,Nt) < \alpha N t] 
= 
\lim_{t \rightarrow \infty} \prob  [ {\bf N} (0,1) <(\alpha 
- 
1) (Nt)^{\frac{1}{2}} ] = 0$, 
which follows from 
the fact that $(\alpha - 1) (NT)^{\frac{1}{2}}$ 
tends to negative infinity as $t$ grows. 
 
Finally, we note that because (\ref{conus4}) holds for any $\alpha 
\in (0,1)$, we must have (\ref{conus3}),  which is the desired 
result. 
\hspace{\fill} {\bf $\Box$} 
 
%\clearpage 
 
\section{Appendix II:  Proof of Proposition \ref{ratprop}} 
 
 
Consider the Reward Game given in Figure 4. 
% 
% 
\begin{figure}[p] 
\[ 
\begin{tabular}{c|@{}c@{}|@{}c@{}|} 
\MC{1}{c}{}&\MC{1}{c}{$X$}&\MC{1}{c}{$Y$}\\ 
\cline{2-3} 
\rule[-5mm]{0mm}{12mm}$X$&\Cell{$4$}{$4$}&\Cell{$0$}{$1.2$}\\ 
\cline{2-3} 
\rule[-5mm]{0mm}{12mm}$Y$&\Cell{$1.2$}{$0$}&\Cell{$3$}{$3$}\\ 
\cline{2-3} 
\end{tabular} 
\] 
\caption{Reward Game} 
\end{figure} 
% 
%\newpage 
% 
% 
% 
Then $(X,X)$ is the payoff-dominant equilibrium and 
$(Y,Y)$ is the risk-dominant 
equilibrium. 
If the risk-dominant equilibrium $Y$ is to be the selected 
outcome, and rational 
expectations are to prevail, then we would have $\Delta = 3$.  Let 
$\tilde{R}$ be 
uniformly 
distributed on the 
interval $[-4,4]$.  Coupled with $\Delta = 3$, this gives the fitness 
game of Figure 5. 
% 
%\clearpage 
%
\begin{figure}[htb] 
\[ 
\begin{tabular}{c|@{}c@{}|@{}c@{}|} 
\MC{1}{c}{}&\MC{1}{c}{$X$}&\MC{1}{c}{$Y$}\\ 
\cline{2-3} 
\rule[-5mm]{0mm}{12mm}$X$&\Cell{$.375$}{$.375$}&\Cell{$.875$}{$.725$}\\ 
\cline{2-3} 
\rule[-5mm]{0mm}{12mm}$Y$&\Cell{$.725$}{$.875$}&\Cell{$.5$}{$.5$}\\ 
\cline{2-3} 
\end{tabular} 
\] 
\caption{Fitness Game with $\Delta = 3$} 
\end{figure} 
% 
% 
% 
% 
% 
We can then calculate: 
% 
\begin{displaymath} 
\left(\frac{\delta^{\delta}}{\beta^{\beta}}\right)^{\frac{1} 
{\delta-\beta}}\left( 
\frac{\alpha^{\alpha}}{\gamma^{\gamma}}\right)^{\frac 
{1}{\gamma-\alpha}}~~=~~ 
1.0022. 
\end{displaymath} 
% 
\noindent From (\ref{intsel}), this ensures that the 
equilibrium 
outcome will 
consist entirely of the 
payoff-dominant equilibrium, contrary to the hypothesis of the 
appearance 
of the risk-dominant 
equilibrium. 
 
%\clearpage 
 
Alternatively suppose that the equilibrium outcome consists 
entirely of the payoff-dominant 
equilibrium, $(X,X)$.  Then if rational expectations are to 
prevail, we will have 
$\Delta = 4$. 
The induced Fitness Game is now given by Figure 6. 
% 
\begin{figure}[htb] 
\[ 
\begin{tabular}{c|@{}c@{}|@{}c@{}|} 
\MC{1}{c}{}&\MC{1}{c}{$X$}&\MC{1}{c}{$Y$}\\ 
\cline{2-3} 
\rule[-5mm]{0mm}{12mm}$X$&\Cell{$.5$}{$.5$}&\Cell{$1$}{$.85$}\\ 
\cline{2-3} 
\rule[-5mm]{0mm}{12mm}$Y$&\Cell{$.85$}{$1$}&\Cell{$.625$}{$.625$}\\ 
\cline{2-3} 
%\MC{3}{c}{\rule{0em}{1ex}}\\ 
%\MC{3}{c}{\ (a) Reward Game ${\cal R}$}\\ 
\end{tabular} 
\] 
\caption{Fitness Game with $\Delta = 4$} 
\end{figure} 
% 
% 
% 
% 
We can then calculate: 
% 
\begin{displaymath} 
\left(\frac{\delta^{\delta}}{\beta^{\beta}}\right)^{\frac{1} 
{\delta-\beta}}\left( 
\frac{\alpha^{\alpha}}{\gamma^{\gamma}}\right)^{\frac 
{1}{\gamma-\alpha}}~~=~~ 
.9985. 
\end{displaymath} 
% 
\noindent From (\ref{intsel}), this ensures that the equilibrium 
outcome will 
consist 
entirely of the 
risk-dominant 
equilibrium.  The learning process cannot exhibit rational 
expectations. 
\hspace{\fill} {\bf $\Box$} 
 
 
 
 
 
%\begin{figure}[htb] 
%\[ 
%\begin{tabular}{c|@{}c@{}|@{}c@{}|} 
%\MC{1}{c}{}&\MC{1}{c}{$X$}&\MC{1}{c}{$Y$}\\ 
%\cline{2-3} 
%\rule[-5mm]{0mm}{12mm}$X$&\Cell{$A$}{$A$}&\Cell{$C$}{$B$}\\ 
%\cline{2-3} 
%\rule[-5mm]{0mm}{12mm}$Y$&\Cell{$B$}{$C$}&\Cell{$D$}{$D$}\\ 
%\cline{2-3} 
%\MC{3}{c}{\rule{0em}{1ex}}\\ 
%\MC{3}{c}{\ (a) Reward Game ${\cal R}$}\\ 
%\end{tabular} 
%\quad 
%\begin{tabular}{c|@{}c@{}|@{}c@{}|} 
%\MC{1}{c}{}&\MC{1}{c}{$X$}&\MC{1}{c}{$Y$}\\ 
%\cline{2-3} 
%\rule[-5mm]{0mm}{12mm}$X$&\Cell{$\aaa$}{$\aaa$}&\Cell{$\ccc$} 
%{$\bbb$}\\ 
%\cline{2-3} 
%\rule[-5mm]{0mm}{12mm}$Y$&\Cell{$\bbb$}{$\ccc$}&\Cell{$\ddd$} 
%{$\ddd$}\\ 
%\cline{2-3} 
%\MC{3}{c}{\rule{0em}{1ex}}\\ 
%\MC{3}{c}{\ (b) Fitness Game ${\cal F}$}\\ 
%\end{tabular} 
%\quad 
%\rule{1ex}{0em}\\ 
%\begin{array}{c} 
%\alpha =\\ 
%\prob(A+ R<\Delta) 
%\end{array} 
%\] 
%\caption{The Reward and Fitness Games.} 
%\end{figure} 
 
 
\bibliography{larrybib,kenbib} 
 
 
\end{document} 
 


