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%\newtheorem{axiom}{Axiom}[chapter] \skip\footins=1.25\skip\footins %\addtolength{\arraycolsep}{-0.5\arraycolsep} %%%%%%%%%%%%%%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. 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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. 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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 $A' > A$, $C'> C$, $B' D$. 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 $A+Cs+t)/\prob(R>s)$ {\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}