\documentstyle{article}
\begin{document}
\title{Communication, Risk and Efficiency \\ in Games\thanks{I am
grateful to Eddie Dekel, Roger Lagunoff, Joel Sobel, Eric van
Damme and two referees for helpful comments and suggestions. I
have benefitted from comments by seminar participants at
Northwestern University, the Midwest Mathematical Economics
Meetings (University of Michigan) and the Econometric Society
Winter Meetings (Washington D.C.). Support by the National Science
Foundation under award number SBR-9410588 is gratefully
acknowledged.}}
\author{Andreas Blume \\
Department of Economics \\  
University of Iowa} 
\date{April, 1996}
\maketitle
\vglue .6in
\begin{abstract}
This paper studies the evolution of effective pre-play
communication in games where a single
communication
round precedes a simultaneous-move, complete-information game. The
paper identifies stable outcomes under population learning dynamics
in which individuals with some probability replace their current
strategy with a best reply against beliefs supported on a sample of
currently used strategies. It is shown that under these
conditions the effectiveness of one-sided pre-play communication is
inversely
related to risk in the underlying game, and to the size of the
message space. Multi-sided communication can be shown to be more
effective than
one-sided communication; i.e., risk and the size of the message
space play no role. This requires that all players communicate,
have the same
preferred equilibrium and messages have some small a priori
information content that identifies message profiles that signal
agreement on a strict equilibrium in the underlying game.
\end{abstract}
\vglue 1in
\pagebreak
\setcounter{page}{1}
\newenvironment{proof}{\hfil\break {\bf
Proof:}}{\hspace*{\fill} $\Box$} 
\newtheorem{dfn}{Definition}
\newtheorem{lem}{Lemma}
\newtheorem{thm}{Theorem}
\newtheorem{prp}{Proposition}
\newtheorem{cor}{Corollary}

\section{Introduction}

This paper studies pre-play communication in games where a single
communication round precedes a simultaneous-move,
complete-information game. In the communication round, a subset of
players send a message from a finite message space. Messages are
costless and have no (or limited) a priori meaning. The main
concern of the
paper is whether meaningful communication can evolve endogenously
under a large class of dynamic rules. We show that this is the
case,
even if the dynamics are not restricted to be gradual. In addition,
the departure from gradual dynamics reveals two issues that are
novel in the evolutionary literature on pre-play communication;
these are a role for risk in the underlying game, and for the size
of the message space. Both factors potentially hinder the evolution
of effective pre-play communication.  

The result, that Pareto efficiency of an equilibrium alone is no
guarantee for it being reached via pre-play communication has an
interesting precedent outside of an evolutionary formulation.
Aumann [1990] has argued that pre-play communication may not lead
to Nash equilibrium, even if the underlying game has a unique
strict and efficient equilibrium.\footnote{Farrell [1988] argues
that pre-play communication
need not lead to a Nash equilibrium even in a game with a unique
Nash equilibrium.} He illustrates his argument with a version of
the familiar {\em Stag Hunt} game, in which there is a tension
between Pareto dominance and Harsanyi and Selten's [1988] risk
dominance. It is interesting that the concern about the
effectiveness of pre-play communication in this game can be made
operational in a setting where messages have no a priori meaning.

The evolutionary literature on pre-play communication, as
exemplified by Bhaskar [1992], Fudenberg and Maskin [1991], Kim
and Sobel [1993], Matsui [1991], Sobel [1993] and W\"arneryd
[1991], has shown that meaningful communication can evolve
endogenously. Roughly, this approach envisions a large population
of players who are repeatedly and randomly matched to play a given
communication game. Players gradually adjust their strategies in
the direction of successful ones. If the players' interests are
sufficiently closely aligned, they will learn over time to
communicate successfully. 

The evolutionary approach to pre-play communication thus far does
not distinguish among equilibria according to their risk. In games
with
common interests, the common interest outcome cannot be
destabilized
through evolutionary forces. Intuitively this is so because the
evolutionary process moves the population via mutations that
affect only a small fraction of the entire population. Since the
mutant population is small, the strategic problem faced by the
mutants in the post-entry population reduces essentially to an
optimization problem. The mutants face no significant strategic
uncertainty. This point is expressed most clearly in Sobel [1993]
where one player at a time gets to adjust his strategy while the
strategies used by the rest of the population remain unchanged.

The present paper refers to the same scenario, a large population
of players who are repeatedly and randomly matched. The paper
departs from the above mentioned
evolutionary studies in not postulating that the
population dynamics are gradual. It does not rule
out the possibility that large fractions of the population adjust
their strategies simultaneously or that some players try to
anticipate the
population dynamics. 

We propose a class of population learning dynamics where each
individual with some probability replaces its current strategy with
a best reply to beliefs that are supported on a sample of currently
used strategies. The stable outcomes of such dynamics can be
conveniently characterized in terms of subsets of the strategy
space. {\em Curb (closed under rational behavior) retracts} (Basu
and
Weibull [1991]) are minimal sets of strategies closed under
inclusion of best replies; they are convex and spanned by pure
strategies. Curb retracts exist in every game and always contain a
{\em (curb) equilibrium.}\footnote{Hurkens [1995] examines an
alternative dynamic whose stable outcomes can also be characterized
via curb set. His dynamic is inspired by Young's [1993] learning
dynamic.} We will show that if the dynamics reflect multiple levels
of depth of
reasoning and cautious behavior among members of the population,
then in games
with one-sided communication, curb equilibria will be observed with
high frequency. With multi-sided communication curb retracts have
no
predictive power, unless we permit some prior differentiation of
messages. With such differentiation, curb retracts consist entirely
of
equilibria.

The paper contains two major results. The first deals with
two-player games in which only one of the players can send a
message. Assume that in the underlying game there is a unique
strategy combination that maximizes the
communicating player's payoff. Let this strategy combination be
a strict equilibrium, i.e. its own unique best reply. Fix the size
of the message space and the number of strategies in the underlying
game. Then the payoffs associated with the communicating player's
favorite equilibrium will be the only curb equilibrium
payoffs in the communication game, provided the communicating
player's {\em risk} at that equilibrium
is sufficiently low; with a message space of size two, the
appropriate risk measure is related to, and in the case of a
symmetric $2 \times 2$-game, coincides with Harsanyi and Selten's
[1988] definition of risk-dominance for $2 \times 2$-games. 

The second major result concerns multi-sided communication in
$N$-player games in which all players can talk and messages have
{\em limited information content (LIC).} {\em LIC} is modelled via
an (arbitrarily) small variation in payoffs in the communication
game that links message profiles to equilibria in the underlying
game.\footnote{Examples in the literature where
communication games are analyzed after adding such small payoff
variations include Blume, Kim and Sobel [1993] and Hurkens [1993].}
We suggest a way to amend our
dynamics that would generate such message space differentiation
endogenously. 

Without {\em LIC,} the curb condition has no predictive power in
games with multi-sided communication. With {\em LIC,} the curb
concept
distinguishes two-sided from one-sided communication. {\em LIC}
does not affect the results with one-sided
communication. However, with two-sided communication, and a unique
strict common interest equilibrium in the underlying game, only
the payoffs of that equilibrium are curb equilibrium payoffs in the
communication game.   

The paper is organized as follows. The next section discusses two
examples to motivate the results on one-sided communication.
Section 3 introduces the dynamics and characterizes their stable
outcomes. Sections 4 and 5 deal
with one- and multi-sided communication, respectively. Section 6
concludes with a discussion of the literature and some thoughts
about the appearance of risk dominance here and elsewhere in the
literature.  

\section{Examples: Dodo and Stag Hunt} 

This section examines the evolution of effective pre-play
communication for two underlying games. The examples show that risk
in the underlying game matters under a simple adaptive rule.  

For future reference it is useful
to recall Harsanyi and Selten's definition of risk dominance. We
will confine ourselves to symmetric games to simplify the
exposition. Consider the game ${\bf G_0}$ below and assume
that $a>c$ and
$b<d$ such that $(U,L)$ and $(D,R)$ are two strict Nash equilibria.
According to Harsanyi and Selten $(U,L)$ risk dominates $(D,R)$ if
$a-c > d-b$ and vice versa, where $a-c$ ($d-b$) is the deviation
loss associated with the risk dominant (risk dominated)
equilibrium.\footnote{It is also true that $(U,L)$
is risk
dominant exactly when the probability of the column player playing
$L$ which makes the row player indifferent between $U$ and $L$ is
less than $1/2.$} 

\begin{center}
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\put(50,40){\line(1,0){50}}
\put(75,15){\line(0,1){50}}
\put(35,50){$U$}
\put(35,25){$D$}
\put(55,70){$L$}
\put(80,70){$R$}
\put(55,50){a,a}
\put(80,50){b,c}
\put(55,25){c,b}
\put(80,25){d,d}
\put(70,5){${\bf G_0}$}
\end{picture}
\end{center}

An example of such a game is ${\bf G_1},$ which Binmore calls {\em
Dodo.} 
Assume for the moment that players can rely on a commonly
understood language. Then, it is commonly accepted (see for example
Farrell and Rabin [1996] and the references therein) that we would
expect the row player's announcement ``I will play $U$" to be
believed by the column player. After all, the message is {\em
self-signaling}; i.e., the row player wants it to be believed if
and
only if it is true. It is also {\em self-committing,} because
conditional on it being believed the row player prefers to conform
to his announcement.

\begin{center}
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\put(50,40){\line(1,0){50}}
\put(75,15){\line(0,1){50}}
\put(35,50){$U$}
\put(35,25){$D$}
\put(55,70){$L$}
\put(80,70){$R$}
\put(55,50){3,3}
\put(80,50){0,0}
\put(55,25){0,0}
\put(80,25){1,1}
\put(70,5){${\bf G_1}$}
\put(125,15){\framebox(50,50)}
\put(125,40){\line(1,0){50}}
\put(150,15){\line(0,1){50}}
\put(110,50){$U$}
\put(110,25){$D$}
\put(130,70){$L$}
\put(155,70){$R$}
\put(130,50){9,9}
\put(155,50){0,8}
\put(130,25){8,0}
\put(155,25){7,7}
\put(145,5){${\bf G_2 }$}
\end{picture}
\end{center}

Game ${\bf G_2}$ is a version of Rousseau's {\em Stag
Hunt.} Just like in game ${\bf G_1}$ players have common interests;
i.e., there
is a unique efficient payoff vector. Nevertheless Aumann [1990] has
used the {\em Stag Hunt} game to argue that pre-play communication
need
not lead to Nash equilibrium. Suppose that without pre-play
communication players are (say by convention) coordinated on
the inefficient $(D,R)$ equilibrium. This could be the result of
payoffs having changed over time; e.g., the Pareto dominant
equilibrium could correspond to switching to a technology that has
recently improved.
Would it help if a player were
able to make a pre-play announcement? Not necessarily. Presumably
the row player wants to convince the column player to play $L$ and
thus might announce to play $U$ herself. The problem is that he
wants to induce the column player to play $L$ no matter what he
intends to play herself. If he is not completely convinced that
his announcement is successful, he may well play $D.$ If that
possibility is given sufficient weight by the column player, he
will
play $R$ himself. According this argument pre-play communication is
not successful in the {\em Stag Hunt} game because a message does
not reveal anything about the intentions of the announcer; the
message ``I will play $U$" is not self-signaling. 

Aumann's analysis of his example suggests one reason for why
pre-play communication
need not be effective in games with multiple Pareto-ranked
equilibria. It is less clear what conclusions to draw from it about
the role of pre-play communication in general. There are at least
four reasons for reexamining the difficulty with pre-play
communication in Aumann's example. Two of these can
be illustrated by varying the stag hunt game, $G_3$ and $G_4$
below.

\begin{center}
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\put(50,40){\line(1,0){50}}
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\put(35,25){$D$}
\put(55,70){$L$}
\put(80,70){$R$}
\put(55,50){9,9}
\put(77,50){6.9,8}
\put(52,25){8,6.9}
\put(80,25){7,7}
\put(70,5){${\bf G_3}$}
\put(125,15){\framebox(50,50)}
\put(125,40){\line(1,0){50}}
\put(150,15){\line(0,1){50}}
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\put(110,25){$D$}
\put(130,70){$L$}
\put(155,70){$R$}
\put(130,50){9,9}
\put(155,50){0,6}
\put(130,25){6,0}
\put(155,25){7,7}
\put(145,5){${\bf G_4}$}
\end{picture}
\end{center}

First, Aumann's argument does apply to game $G_3$ and yet its force
seems to be diminished in this game. $U$ is optimal for player 1
for a wide range of beliefs, which lessens the burden
on communication to discredit the $(D,R)$
equilibrium.\footnote{According to Aumann [1990] David Kreps has a
raised a similar point. Aumann argues, the fact that the efficient
outcome appears likely in this game should not be  attributed to a
role for communication.} Second, there may be reason's for
communication to be ineffective, even if
Aumann's argument does not apply; this is illustrated by game
$G_4.$ Having made an utterance such as ``I intend to play $U,"$ in
game $G_4,$ player one must have a high degree of confidence in its
effectiveness before following his own recommendation because $D$
is a best reply against a large set of beliefs. Third, Farrell and
Rabin [1996] insist even in Aumann's original example that the {\em
self-committing} property of the message ``I will play U" is enough
to ensure its effectiveness. Fourth and related to the last point,
the distinction made between the roles of pre-play communication in
{\sl Dodo} vs {\sl Stag Hunt} is informal and relies on an
interpretation of the status of a commonly understood language. 

This paper attempts to formalize the concern about the
effectiveness of pre-play communication in Aumann's example without
assuming that messages have an {\em a priori} meaning; rather, like
the
evolutionary
literature, we ask under what conditions meaningful communication
can evolve endogenously, and show that these conditions are related
to risk in the underlying game. The above examples show that there
is no containment relationship between risk considerations and
Aumann's argument. And yet, the two are related; in $G_0,$ let
$a>d>b,$ and $a>c.$ By raising $c,$ we both increase the risk of
$U$ and make it more likely that Aumann's argument applies; loosely
speaking both Aumann's argument and the risk argument are monotonic
in $c.$

Consider two large finite equal-sized populations of communicating
and silent players. They repeatedly play the
following communication game. In each period each of the
communicating players is randomly matched with a silent player. The
communicating player sends one of two messages, $m_1$ or $m_2$, and
then both players play the underlying game, which is either ${\bf
G_1}$ or ${\bf G_2}$. The reduced normal
forms of the communication games corresponding to {\em Dodo} and
{\em Stag Hunt,} denoted by ${\bf
\Gamma_1}$ and ${\bf \Gamma_2}$ are shown below.  

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\put(10,110){$(m_1,D)$}
\put(10,85){$(m_2,U)$}
\put(10,60){$(m_2,D)$}
\put(55,155){$LL$}
\put(80,155){$LR$}
\put(105,155){$RL$}
\put(130,155){$RR$}
\put(55,135){3,3}
\put(80,135){3,3}
\put(105,135){0,0}
\put(130,135){0,0}
\put(55,110){0,0}
\put(80,110){0,0}
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\put(130,110){1,1}
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\put(80,85){0,0}
\put(105,85){3,3}
\put(130,85){0,0}
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\put(80,60){1,1}
\put(105,60){0,0}
\put(130,60){1,1}
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\put(160,85){$(m_2,U)$}
\put(160,60){$(m_2,D)$}
\put(205,155){$LL$}
\put(230,155){$LR$}
\put(255,155){$RL$}
\put(280,155){$RR$}
\put(205,135){9,9}
\put(230,135){9,9}
\put(255,135){0,8}
\put(280,135){0,8}
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\put(230,110){8,0}
\put(255,110){7,7}
\put(280,110){7,7}
\put(205,85){9,9}
\put(230,85){0,8}
\put(255,85){9,9}
\put(280,85){0,8}
\put(205,60){8,0}
\put(230,60){7,7}
\put(255,60){8,0}
\put(280,60){7,7}
\put(245,35){${\bf \Gamma_2}$}
\end{picture}

Suppose play evolves according to a simple adaptive rule. In each
period each player uses last period's strategy with probability
one-half or else moves to a best reply against last period's
population
play; if there are multiple pure best replies, he randomizes
uniformly. 

What can be said about the long run properties of this dynamic
process in either game? Consider first $\Gamma _1.$ 
One can show that irrespective of the initial distribution of
population play, the postulated dynamic process will almost surely
converge to the set $Q$ of strategies consisting of all mixtures of
$(m_1,U)$ 
and $(m_2,U)$ for the row player and of all mixtures of $LL,$
$LR$ and $RL$ for the column player. To see this note that from any
initial state, there is positive probability, bounded away from
zero, that in two steps the population
moves to a state where all players of a given kind use the same
strategy. From any such state, there exists a sequence of 
best-reply iterations which takes the population into $Q$ in no
more than two steps; these sequences all have probability bounded
away from zero. Thus, from any initial state, the probability that
after four periods $Q$ is not reached is bounded away from one.
This implies that from any initial state, $Q$ is reached almost
surely. Furthermore, once the process has entered $Q$ it can never
leave $Q$ because strategies outside of $Q$ are never best replies
against beliefs concentrated on $Q.$  

Next consider the set $Q$ in the game $\Gamma _ 2.$ The same
argument
as above shows that from any initial state, $Q$ is reached almost
surely. However, for $\Gamma _ 2,$ the set $Q$ is not invariant
under the dynamic process. To understand this observe that there is
a bounded
number of steps, such that from any initial state in $Q$ the
population moves to a state where half of the silent players use
the strategy $LR$ and the other half uses the strategy $RL.$ At
that point none of the strategies in $Q$ are a best reply for the
communicating players, and the process exits $Q$ with positive
probability.

In the next section we characterize the stable outcomes of a class 
of dynamics in the spirit of this example. We will use this
characterization in the subsequent sections to analyze the roles of
message space size, risk and message differentiation in
communication games 

\section{Dynamics}

This section proposes a class of population learning dynamics for
finite games and characterizes its stable outcomes in
terms of the curb retracts of those games. 

For a finite strategic form game $G$ with player set ${\cal P}$ and
$P= \#({\cal P})$ let $S_p$ be player $p$'s set
of pure
strategies, with typical element $s_p$ and let $\Sigma_p$ be his
set of mixed strategies, with typical element $\sigma_p.$ $S$ and
$\Sigma$ are the sets of pure and mixed strategy profiles. $\sigma$
is a strategy profile and $\sigma_{-p}$ a partial profile that
excludes player $p$'s mixed strategy. Denote the convex hull of any
set $Z$ by ${\rm co}(Z).$ ${\rm MBR}_p( \cdot)$ is player $p$'s
(mixed) best reply correspondence and ${\rm MBR}( \sigma ) =
\times_{p=1}^P {\rm MBR}_p (\sigma _{-p}) $. $Q \subseteq \Sigma$
is
a retract if $Q = \times_{p=1}^P Q_p$ where $Q_p \subseteq
\Sigma_p$
is nonempty, closed and convex. A retract $Q$ is a {\em curb
retract} if for all $\sigma \in Q,$ ${\rm MBR}( \sigma) \subseteq
Q$,
and if $Q$ is a minimal retract with that property. A Nash
equilibrium
contained in a {\em curb retract} is called a {\em curb
equilibrium.} 

In every finite game there is at least one {\em curb retract} and
a {\em curb equilibrium.} {\em Curb retracts} are spanned by pure
strategies; i.e., with any mixed strategy $\sigma_p \in Q_p,$ $Q_p$
contains all pure strategies in the carrier of $\sigma_p.$
Therefore we can (and will) identify any curb retract with the set
of
pure strategies that span the retract. The intersection of two
retracts closed under
inclusion of best replies is itself closed under inclusion of best
replies.\footnote{For more on curb sets see Basu and Weibull
[1991],
Balkenborg [1992] and Hurkens [1993].} 
     
Curb retracts are attractive because they conveniently characterize
the
stable outcomes for a large class of learning dynamics.
We propose a class of {\em population learning dynamics} for finite
games played
between multiple populations of players. We show that every dynamic
in this class converges almost surely to a set of states that is
supported on a
curb retract and that from states supported on
a curb retract every other state supported on the same curb retract
is
reached with positive probability in finite time. Those
dynamics in this class in which players are cautious and reason at
sufficient depth have an additional property: If a
player has
a dominant strategy in a curb retract, then conditional on reaching
that set individuals representing that player will use the dominant
strategy with high probability.

Consider any finite game $G$ with player set $\cal P$ whose
cardinality we denote by $P.$ $\cal P$
also denotes the set of populations from which individuals are
drawn at random in any given period to play the game $G.$ Let $S_p$
denote
the finite set of strategies available to a player from population
$p \in {\cal P},$ and $S := \times _ {p \in {\cal P}} S_p.$ Assume
that $\#(p) > \#(S_p)$ for every population $p.$ For every set of
strategies $X_p \subseteq S_p$ let
$\Delta(X_p)$ stand for the set of probability distributions with
support $X_p,$ and for $X \subseteq S,$ let $\Delta (X) :=
\times_{p \in {\cal P}} \Delta (X_p)$ indicate
the set of uncorrelated distributions supported on $X,$ and let
$\Delta^0(X)$ denote all those uncorrelated distributions on $X$
that have full support. Let $BR_p
(\Delta(X))$ denote the set of player $p$'s pure best replies
to uncorrelated beliefs supported on $X$ and define
$BR(\Delta(X)) := \times _{p \in {\cal P}} BR_p(\Delta (X)).$

At each time $t$ the state of population $p \in {\cal P}$ is the
vector
$\omega _{pt} = \{s_{it} \}_{i \in p}$  of strategies adopted by
each member $i$ of population $p.$ The state $\omega _t = \{\omega
_{pt} \}_{p \in {\cal P}}$ at time
$t$ is the concatenation of all time $t$ population states. Write
$\Omega$ for the set of all possible states.  

The dynamic process on $\Omega$ follows a transition rule $\phi (
\cdot \vert \cdot)$ where $\phi ( \omega_{t+1} \vert \omega_t) $ is
the probability that the state in period $t+1$ will be
$\omega_{t+1}$ given that it is $\omega_t$ in period $t.$ The
transition rule is
a function of individual agent behavior. Agents enter period $t$
with their adopted strategy $s_{i, t}$ and use that strategy to
play the game $G$ with various random selections of players from
other populations. Through these random matches agents gain
information about the strategies currently in use by members of the
different populations. With probability $\lambda,~1 > \lambda > 0,$
an
individual is active and uses his information to update his
strategy, with probability $1-\lambda$ he is inactive and adopts
last period's strategy, in which case $s_{i,t+1} = s_{i , t}.$ 

Denote by $X_{itp'}$ the set of pure strategies that an active
individual $i
\in p$ observes at time $t$ among individuals in population $p'.$
This will be a subset of the strategies currently in use in that
population. Call $X_{it} := \times_{p' \in {\cal P}} X_{itp'}$
individual $i$'s
sample at
time $t.$ Active individuals adopt strategies that are best
replies to some belief based on their sample.

Define the operator $B$ by $B(X) := X \cup BR(\Delta (X))$ and let
$B^K$
be its $K$-fold iteration. For each
active
individual $i \in p$ and for all $p \in {\cal P}$ define 
$$S_K(X_{it}) := \{s_p \in S_p \vert s_p \in
BR_p(\Delta(B^K(X_{it}))) \},$$ and $$S_K^0(X_{it}) := \{s_p \in
S_p
\vert s_p \in BR_p(\Delta^0(B^K(X_{it}))) \}.$$ 

We complete the description of individual agent behavior by
imposing the following restrictions on the rules by which active
agents adopt new strategies. For an active agent $i$ with period
$t$ sample $X_{it},$ who {\em reasons at depth $K$,} $K \geq 0,$
assume that
$\mbox{Prob}\{s_{i,t+1} = s_p \vert X_{it}\} > 0$ if and only if
$s_p
\in
S_K(X_{it}).$ If in addition, the agent is {\em
$\epsilon$-cautious} then $\mbox{Prob}\{s_{i,t+1} \in S_K^0(X_{it})
\} \geq
1-\epsilon,$ where $\epsilon \in (0,1).$ These transition
probabilities are time-invariant. Note that we allow that $K=0$ and
agents need not be $\epsilon$-cautious, unless explicitly assumed
otherwise.

For any $X \subseteq S,$ there
exists a $T$ such that for all $t>T,~~ B^{t+1}(X) =
B^t(X),$ since
$B(\cdot)$ is monotone and $S$ is finite. Let $\mbox{supp} (\omega)
:= \{ s \in S \vert \forall
p \in {\cal P,}~\exists i \in p, \omega _i = s_p \}$ and define
$t(\omega) := \min \{t \in {\cal N} \vert B^{t+1}( \mbox{supp}
(\omega)) = B^t( \mbox{supp} (\omega)) \}.$
Then $W(\omega) := B^{t(\omega)}(\mbox{supp}(\omega))$ is closed
under the inclusion of best replies and thus contains a curb
retract for all $\omega \in \Omega.$ For any curb retract $Q$
define $K(Q) := \max \{ t(\omega) \vert \mbox{supp}(\omega) \in Q
\}.$ 

For any curb retract $Q$ call the set of states $\omega \in
\Omega$ in which every individual $i \in p$ has adopted a strategy
$s_{it} \in Q_p$ the {\em curb-state set} supported on $Q.$ The
class of dynamic processes described above consists of Markov
chains
with
stationary transition probabilities $\phi( \cdot \vert \cdot)$ on
the state space $\Omega.$ The stable set of states for such a
process are the so called {\em recurrent communication classes} of
the process. The recurrent
communication classes are subsets of $\Omega$ such that (i)
from every state there is a finite length sequence of positive
probability transitions to at least one of these classes, (ii)
within each class every state can be reached from every other state
via a finite length sequence of positive probability transitions,
and (iii) no state outside one of the classes can be reached from
a state inside through a positive probability transition. The
following result identifies the curb-state sets as the
recurrent communication classes for any population learning dynamic
$\phi.$

\begin{prp}(1) The curb-state sets are the recurrent
communication classes of every population learning dynamic $\phi.$
(2)
From
any initial state, the population learning dynamic converges almost
surely to a curb-state set. (3) If $Q \subset S$ is a
curb retract in which a player $p \in {\cal P}$ has a dominant
strategy (with respect to $Q$) $s_p \in S_p$, and players reason at
least at depth $K(Q)$
and are
$\epsilon$-cautious, then for all active $i \in p$ and $t>0,$
$\mbox{Prob}(s_{i,t+1} = s_p \vert \mbox{supp} (\omega_t) \in Q)
\geq 1- \epsilon.$  
\end{prp}
\begin{proof}
For any $\omega \in \Omega$ and any $X \subseteq S,$
$\phi$ induces a probability $P^\tau (X \vert \omega)$ that
the support of the state $\tau$ periods from the present will be
$X.$ There exists a number of periods $T$ such that
$P^T(W(\omega) \vert \omega) > 0~~\forall \omega \in \Omega.$ To
see
this note that each $s \in BR(\Delta (\mbox{supp} (\omega_t)))$ has
positive probability of being in the support of $\omega_{t+1}$.
Since $1- \lambda > 0,$ any $s \in \mbox{supp} ( \omega_t)$ also
has
positive probability of being in the support of $\omega_{t+1}.$
Since
$\#(p) > \#(S_p),$ there is positive probability that all
strategies
in $B(\mbox{supp} (\omega_t))$ are present in the population state
$\omega _{t+1}$ in period $t+1.$ Let $T = \max \{t(\omega) \vert
\omega \in \Omega \}.$ 

Once  $\mbox{supp}(\omega_t) = W(\omega) $ for some
$\omega \in \Omega,$ there is positive probability of all agents
being active and drawing a sample whose support is contained in the
same curb retract. Thus, from all initial states the dynamic
ends up in a curb-state set with positive probability after
no more than $T'=T+1$ periods. Once the dynamic has entered a
curb-state set $\hat \Omega,$ it will not leave it and
since
$P^T(W(\omega) \vert \omega) > 0~~\forall \omega \in \hat \Omega
\subset
\Omega,$ all states in $\hat \Omega$ are reached with positive
probability from any other state in $\hat \Omega.$ 

Let $\tilde \Omega \subset \Omega$ be the subset of states that
belong to some curb-state set. Since there are only
finitely many states, $\exists ~ \pi > 0: ~ \forall \omega \in
\Omega, $
$$\mbox{Prob} (\omega_{t+T'} \in \tilde \Omega \vert \omega_t =
\omega) \geq \pi  \Rightarrow $$
$$\mbox{Prob} (\omega_{t+kT'} \not\in \tilde \Omega \vert \omega_t
= \omega) \leq (1-\pi)^k .$$
Therefore the probability that the dynamic process will never reach
$\tilde \Omega$ equals
$$\lim_{k \rightarrow \infty} (1- \pi)^k  = 0.$$ 

If $\mbox{supp}(\omega_t) \in Q,$ where $Q$ is a curb set and
players reason at least at depth $K(Q),$ then the support of their
sample $X_{it}$
will be contained in $Q$ for all $i \in p,~ p \in {\cal P}$ and
$S(X_{it})=Q.$ Thus, with probability of at least
$1-\epsilon,$ active players will use a best reply against a belief
in $\Delta ^ 0 (Q),$
which must be the dominant strategy with respect to $Q$ for players
with such a
strategy. 
\end{proof}

Thus, in the long-run we expect the dynamic to end up in a 
curb-state set; once a curb-state set is reached, the dynamic never
leaves it and returns to every one of its member states with
positive probability, regardless of the initial state. This much
requires neither multiple levels of depth of reasoning nor caution
on part of the individuals. With caution and depth of reasoning,
strategies that are dominated with respect to a curb set will be
used with low probability. The communication games considered in
the next section have curb sets in which every strategy profile
that does not use a strategy dominated with respect to that curb
set is an equilibrium. This characteristic, shared with many normal
form games derived from an extensive form, lends credence to the
focus on curb equilibria in our analysis.  

\section{One-Sided Communication}

In this section we generalize our observation about {\em
Dodo} versus {\em Stag Hunt.} First we derive a necessary condition
on message space size for one-sided communication to be effective.
Next we provide a complementary sufficient condition. Then
we specialize the sufficient condition for the case of only two
messages, and
finally we specialize it further for the case where the underlying
game
is a symmetric $2 \times 2$-game. In each of these cases the
result is that effective communication is a stable outcome for 
our population learning dynamics, if the {\em risk} of the
preferred equilibrium is low relative to a {\em standard of
comparison}
that depends on the size of the underlying game and the size of the
message space. 

In the general setting the standard of comparison
is inversely related to both the size of the message space and the
size of the underlying game. Both of these factors increase the
number of possible strategies, which can be interpreted as
increasing strategic uncertainty. With only two message the
standard of comparison
becomes independent of the size of the underlying game (and
trivially the message space). If the underlying game is in addition
a symmetric $2 \times 2$-game, then the appropriate condition
reduces to
Harsanyi and Selten's [1988] {\em risk dominance} criterion.

Denote the strategies of the row (column) player in the underlying
game $G$ by $i \in I$ ($j \in J$). Let the row player be player one
and the column player be player two with payoffs $u_k(i,j),$
$k=1,2.$ Assume that the underlying game has multiple strict Nash
equilibria; this is the interesting case. Assume also that there is
a unique strategy combination $(\hat i, \hat j)$ that maximizes
the row player's payoff, i.e.
$$ (\hat i , \hat j) = \arg\max_{(i,j)}u_1(i,j),$$ and that this
strategy combination is a strict Nash equilibrium.  Examples of
such games are {\em Dodo,} {\em Stag Hunt} and the {\em Battle of
the Sexes.} Let $M$ be the
message space available to player one in the communication game
$\Gamma_1(G,M).$ In this game player one first sends a message from
the set $M,$ then both players play the game $G.$ Player one's
strategies in the reduced normal form of the communication game are
of the form $(m,i)$ with $m \in M$ and $i \in I.$ Player two's
strategies are functions $f$ that map messages $m\in M$ into
actions $j \in J.$ Let $F(G,M)$ be the set of pure strategies of
player two in the communication game induced by $G$ and $M,$ and
denote the cardinality of any finite set $X$ by $\# (X).$ We will
require $\#(M) \geq 2$ to ensure that the communication game does
not become degenerate. 

Our first result demonstrates that a restriction on the size of the
message space is a necessary condition for one-sided communication 
to be effective. 
\begin{prp}
For every game $G,$ and every strict equilibrium $(\tilde  i ,
\tilde j) \not= (\hat i , \hat j) $ in $G,$ if the
message space $M$ is sufficiently large, i.e., $$\#(M)-1 > \max_{i
\not= \tilde i}
{u_1( i
, \hat j) - u_1 ( \tilde i , \hat j) \over u_1 (\tilde i , \tilde
j)
- u_1( i , \tilde j) }, $$ then there exists a curb equilibrium
in the communication game with payoff $(u_1(\tilde  i , \tilde j),
u_2(\tilde  i , \tilde j)).$
\end{prp}

Thus, for any game $G$ we can find a message space large enough to
ensure that communication does not discriminate among the game's
strict equilibria. Note that if the underlying game $G$ is a $2
\times 2$-game, then the bound on message space size is the ratio
of the deviation losses that enter Harsanyi and Selten's definition
of risk dominance.

The idea of the proof is simple. First, one shows that every curb
set of the communication game contains every strategy of the silent
player that responds to one message with $\hat j$ and to all other
messages with $\tilde j.$ This follows from iterating best replies
and the fact that any best reply against a strategy of the sender
can be altered arbitrarily after unused messages, and still be a
best reply. Second, one considers beliefs of the communicating
player that are concentrated on this set of strategies and assign
equal probability to all strategies in the set. Clearly, the more
messages, the more attractive is $\tilde i$ against such beliefs.
In contrast taking
action $\hat i$ becomes increasingly risky as the number of
messages increases. How attractive $\tilde i$ will be depends on
how well $\tilde i$ does against mixtures of $\hat j$ and $\tilde
j$ versus how well other actions $i$ do against such mixtures. This
creates the trade-off between message space size and payoffs that
appears in the statement of the proposition. Since $(\tilde i ,
\tilde j)$ is a strict equilibrium, a sufficiently large message 
space guarantees that $\tilde i$ will be the unique best reply
against the postulated beliefs. 

\begin{proof} First, we show that player one's preferred
equilibrium
$(\hat i, \hat j)$ appears as a continuation equilibrium in any
curb retract of the communication game, regardless of the size of
the
message space. Let $(m,i)$ be any strategy of the
sender that is part of a curb retract $Q.$ With at least two
messages,
there exists a best reply, $f,$ for player two such that $f(m')=
\hat j,~m' \not= m.$ Therefore, $(m', \hat i) \in Q_1,$ and the
profile $((m', \hat i), f)$ is a curb equilibrium. 

Next we show that if the size of $M$ satisfies the condition stated
in the proposition, then other strict equilibria will also appear
in the curb set.
Let $(\tilde i, \tilde j)$ be any other strict equilibrium of $G.$
Consider the set of strategies $\tilde F$ of the form
$$\tilde F := \left\{f \in F(G,M) \vert   f_l(m) = \cases { \hat j
& if  $ m=m_l $  \cr
\tilde j & otherwise \cr }, ~~l=1,2,... \right\} $$ Note that each
of the
strategies in $\tilde F$ is in $Q_2,$ where $Q$ contains the curb
equilibrium $((m', \hat i), f).$ To see this observe that the
strategy $f'$ that responds to all messages with $\hat j$ is a
best reply to $(m', \hat i),$ and in turn all strategies of the
form $(m,\hat i)$ are in $Q_1. $ Each of the strategies in $\tilde
F$ is a
best reply against one of the latter strategies.

The set $\tilde F$ is constructed such that if player one believes
that player two uses a strategy in $\tilde F,$ but is not certain
which of these strategies is used, then regardless of which message
he considers, action $\tilde i$ is very attractive.
Suppose player one has beliefs corresponding to uniform
randomization over the set of strategies $\tilde F$ of player two.
Against
such beliefs, any strategy $(m, \tilde i)$ gives player one
the payoff 
$${1 \over \#(M) } u_1(\tilde i , \hat j) + {\#(M) - 1 \over \#(M)}
u_1 (\tilde i , \tilde j) . $$
Any strategy of the form $(m,i),$ $i \not= \tilde i$, yields the
payoff
$${1 \over \#(M) } u_1( i , \hat j) + {\#(M) - 1 \over \#(M)}
u_1 ( i , \tilde j). $$ The former payoff is larger than the
latter exactly when
$$\#(M)-1 > {u_1(i
, \hat j) - u_1 ( \tilde i , \hat j) \over u_1 (\tilde i , \tilde
j)
- u_1(i , \tilde j) }. $$
\end{proof}

Having demonstrated that in our environment a restriction on
message space size is necessary for effective communication to
emerge, we now turn to developing a sufficient condition. In doing
so, we let Proposition 2 guide us. From that proposition we learn
that message space size matters because it increases the number of 
strategies in which player two responds to {\em some} message with
(player one's) preferred equilibrium action but responds with other
actions
after different messages. We learn also that deviation losses
matter, because they provide a measure of the payoff impact of
these other actions.

Define the set of strategies of player two which sometimes respond
with the preferred action $\hat j$ as
$$\hat F (G,M):= \lbrace f \in  F(G,M) \vert ( \exists m \in M :
f(m) =
\hat j)\rbrace. $$ We will develop a sufficient condition, with an
analog to the
ratio of deviation losses from Proposition 2. This will be a
measure of
risk. Harsanyi and Selten's deviation losses only involve
equilibrium actions. Here however, player one has
to consider {\em all} of his alternative actions if he
is uncertain about which messages induce the preferred response
{\em and} which responses are induced otherwise. If effective
communication is to be stable, it will be necessary that even under
unfavorable beliefs, player one does not abandon the efficient
action $\hat i.$ The least favorable condition for action $\hat i$
is one where player two is omniscient, minimizes player one's
payoff from action $\hat i$ and maximizes player one's payoff from
every other action. This concern with extreme beliefs is necessary
because player two's responses are mediated by messages and there
may be only one (unknown) message that induces the desired
response $\hat j.$ In that vein, define a risk measure $\rho ((\hat
i , \hat j),G)$ for the equilibrium $(\hat i, \hat j)$ that trades
off player one's payoff from that equilibrium against the
possibility of facing the kind of omniscient player two described
above. Let $$ \rho ((\hat i, \hat j), G) := \max_{i \not= \hat i}
{{\max_j u_1(i,j)-\min_j u_1 (\hat i , j)} \over {u_1(\hat i , \hat
j) - \max_j u_1(i,j)}} . $$ This is a measure of the risk of player
one at the
equilibrium $( \hat i, \hat j).$ The risk of the equilibrium $(
\hat i,
\hat j)$ to
player one decreases if his payoff at that equilibrium increases or
if the worst outcome from playing his equilibrium strategy $\hat i$
improves. Relative to any alternative strategy $i,$ the risk
increases if the maximum payoff from that strategy increases.
Finally, only the maximal risk relative to any alternative strategy
matters. Note that this measure
is invariant with respect to positive affine transformations of the
payoff function. 

We introduce other risk measures below that are appropriate for the
case of only two messages or can be used to
compare risks among multiple strict Nash equilibria. However, the
measure $\rho$ remains central because it permits statements about
the most general class of games. In that regard, it may appear as
a limitation that $\rho$ applies only to games in which there is a
unique maximizer for player one's payoff and this is a strict Nash
equilibrium. Thus, it is worth pointing out that outside this class
of games no general results on effective communication are
available. This follows because in every curb set of the
communication game the communicating player's maximum payoff will
be attained by some strategy combination. If in the underlying game
this maximum payoff is not attained at a strict equilibrium then
iterating best replies can lead to large curb sets which may
include multiple equilibria with different payoffs. For example,
take any version of {\em Dodo} and add a dominated strategy $F$
(far right) for the silent (column) player such that $(D,F)$
maximizes the communicating (row) player's payoff. In any
corresponding communication game with at least two messages, there
is a unique curb set, which contains of all strategies of the
communicating player and nearly all strategies of the silent player
(only the strategy which uses the dominated reply after every
message is ruled out). This explains our restriction on the
underlying class of games, which is reflected in our risk measure. 

Our next result states that, for a given size of the message
space, if risk is sufficiently small, then communication will be
effective. We will construct a curb retract $Q$ that for the
communicating player is spanned by all strategies of the form $(m,
\hat i),$ and for the silent player is spanned by all strategies
that respond with $\hat j$ to {\em some} message. A condition
relating the risk of $(\hat i , \hat j)$ to the size of the message
space ensures that strategies of the form $(m,i),$ $i \not= \hat
i,$ are never best replies against any beliefs over $Q.$ The max
and min operators in the risk measure come from constructing a
worst case scenario for beliefs concentrated on $Q,$ in which
player one believes that the message $m$ induces player one's
favorite reply given $i,$ and alternative messages $l$ induce
player one's worst reply given $\hat i$ with high probability. Even
in such a worst case scenario one can always find a message $l$
that induces the reply $\hat j$ with probability of at least
$1/\#( \hat F(G,M)).$ The cardinality of the set $\hat F (G,M)$ is
also the number of
strategies of the silent player in $Q$ and is an increasing
function of the number of messages; no belief concentrated on $Q$
can assign weight less than the reciprocal of that number to all
strategies in $\hat F(G, M).$  Thus, for any beliefs concentrated
on $Q,$ the communicating player can ensure that the weight of
$u_1( \hat i, \hat j)$ in his expected payoffs is at least $1/\#(
\hat F(G,M))$ for one of his strategies in $Q,$ say $(l, \hat i).$
Since $(\hat i , \hat j)$ is the unique profile that maximizes her
payoff, if the possible payoffs from other strategy profiles
$(i,j),$ $i \not= \hat i ,$ in the underlying game are not ``too
high," this suffices to make $(l, \hat i)$ a strictly better reply
than $(m,i).$ The risk measure makes the meaning of ``too high"
precise. The following proposition summarizes this discussion.    

\begin{prp} Let $\Gamma_1 (G,M)$ be a one-sided communication game
with underlying game $G,$ and message space $M,$ and risk
$\rho((\hat i , \hat j), G)$ at player one's preferred equilibrium.
Suppose the following relation holds between $\rho((\hat i , \hat
j), G)$  and the cardinality $\#(\hat F (G,M))$ of player two's
set of strategies that respond with $\hat j$ after some message.
$${1
\over \# ( \hat F(G,M))
-1} > \rho ((\hat i, \hat j), G).$$ Then (a) the retract $Q=Q_1
\times Q_2 := \lbrace
{\rm co} \lbrace (m, \hat i)\rbrace_{m \in M} \rbrace \times
\lbrace  {\rm co} \lbrace \hat F(G,M) \rbrace \rbrace $ is the
unique
curb retract in $\Gamma_1 (G,M),$ and (b) the payoffs in all
curb equilibria of the communication game $\Gamma_1 (G,M)$ are
$u_k(\hat i , \hat j),$ $k=1,2.$ \end{prp}  

\begin{proof} We begin showing that $Q$ is closed under inclusion
of best replies. Consider player two. By assumption, $\hat j$ is
player two's unique best reply to
$\hat i$ in the underlying game. Thus, against beliefs concentrated
on $Q_1,$ as defined in the statement of the proposition, any
strategy $f'$ with $f'(m) \not= \hat j \ \ \forall m
\in M$ has a strictly lower expected payoff than $\hat f$ where
$\hat f (m) = \hat j \ \ \forall m \in M.$ 

Turn to player one and suppose, to derive a contradiction, that
$(m,i),$ $i \not= \hat i$, is a best reply for player one against
beliefs $\lambda$
concentrated on $Q_2.$ Let $\lambda (f)$ be the probability
assigned to strategy $f$ by $\lambda.$ Let $\hat
F(m,j) := \lbrace f \in \hat F(G,M) \vert f(m)=j \rbrace;$ this is
the set of all strategies in $\hat F (G,M)$ that respond to message
$m$ with action $j.$ Then the payoffs from strategies $(m,i)$ and
$l, \hat j$ satisfy
$$\sum_{j \in J} \sum_{f \in \hat F(m, j)} u_1(i , j)
\lambda (f) \geq  \sum_{j \in J} \sum_{f \in \hat F(l, j)} u_1(
\hat i
, j) \lambda (f) \ \ \forall l \not= m.$$ Note that finiteness of
$\hat F(G,M)$ implies that at least one strategy in this set has
probability bounded away from zero, i.e.
$$\max_{f \in \hat F(G,M)} \lambda (f) \geq {1 \over \# (\hat
F(G,M))} > 0.$$ Let $\tilde f \in \arg\max_{f \in \hat F(G,M)}
\lambda (f)$ be a strategy with maximum probability in $\hat
F(G,M),$ and without loss of generality let $\tilde f \in
\hat F(l, \hat j).$ Then the above payoff relation between
strategies $(m,i)$ and $(l, \hat i)$ implies that
$$u_1(i, \tilde f(m)) \lambda(\tilde f) + (1 - \lambda
(\tilde f)) \max_j u_1 (i,j) \geq$$
$$u_1( \hat i, \hat j) \lambda(\tilde f) + (1 - \lambda
(\tilde f)) \min_j u_1 ( \hat i,j).$$ For this condition to be
satisfied it is necessary that $\lambda(\tilde f) < 1;$ thus we can
rewrite it as 
$${\lambda (\tilde f) \over 1 - \lambda (\tilde f)}
(u_1(\hat i , \hat j) - \max _j u_1 (i,j)) \leq \max_j u_1(i,j) -
\min_j u_1 (\hat i , j), $$ which implies
$${1 \over \# ( \hat F(G,M)) -1} \leq \rho
((\hat i, \hat j), G),$$ in violation of the condition in the
proposition. Thus, for a strategy $(m,i)$ outside of
$Q_1$ to
be a best reply to beliefs concentrated on $Q_2,$ the condition in
the proposition must be violated. If it holds $Q$ is closed under
inclusion of best replies.

To see that $Q$ is minimal among retracts closed under inclusion of
best replies
consider the following: The strategy $f(m)= \hat j \ \ \forall m
\in M$ is a best reply to any strategy in $Q_1.$ All strategies in
$Q_1$ are best replies to $f(m)= \hat j \ \ \forall m \in M.$ Any
strategy in $Q_2$ is a best reply to some strategy in $Q_1$ by
construction.

To establish uniqueness it suffices to show that for any set
$\tilde
Q \not= Q$ that is closed under best replies, $Q \cap \tilde Q
\not= \emptyset:$ If $(\tilde m , i) \in \tilde Q_1,$ $i \not= \hat
i,$ and $j$ is a best reply to $i,$ then there exists an $f \in
\tilde Q _2$ such that $f( \tilde m) =j ,$ and $f(m)= \hat j \ \
\forall m \not= \tilde m.$ Thus for all $m' \not= \tilde m,$
$\lbrace (m', \hat i), f\rbrace \in Q \cap \tilde Q.$ This shows
that $Q$ is the unique curb retract in the communication game.  

It remains to show that all equilibria in the game restricted to
$Q$ have payoffs $u_k(\hat i , \hat j).$ Since $(\hat i , \hat j)$
is a strict Nash equilibrium in the underlying game, $u_2(\hat i ,
\hat j)$ is player two's maximal payoff in the communication game
restricted to $Q.$ Player two can guarantee himself that payoff in
$Q.$ Whenever player two gets $u_2(\hat i, \hat j),$ player one
gets $u_1( \hat i , \hat j )$ in the game restricted to $Q.$
\end{proof}

The preceding two propositions concern the impact of message space
size on the dynamic stability of efficient communication outcomes.
And yet, it takes only two messages to destabilize inefficient
outcomes. Moreover, we will show that with only two messages we can
employ a less conservative risk measure. Define this alternative
risk measure as $$ \tilde
\rho ((\hat i, \hat j), G) := \max_{i \not= \hat i}
{{u_1(i, \hat j)-\min_{j \not= \hat j} u_1 (\hat i , j)} \over
{u_1(\hat i , \hat
j) - \max_{j \not= \hat j}  u_1(i,j)}}. $$ This measure turns out
to
be closely related Harsanyi and Selten's definition of risk
dominance in two-player games.
The motivation for the measure $\tilde \rho$ is similar as for
$\rho;$ the value $\tilde \rho ((\hat i , \hat j), G)$ is low
whenever $\hat i$ is an ``acceptable" reply not only to $\hat j$
but also to other strategies $j.$ It is easily verified
that $\rho (( \hat i , \hat j ), G) \geq \tilde \rho (( \hat i ,
\hat j ), G).$ In this sense the risk measure $\tilde \rho$ is less
conservative than $\rho.$ Like $\rho$ this measure is invariant
under positive affine transformations. The differences between the
two measures are threefold. $u_1(i, \hat j)$ replaces $\max _j
u_1(i,j)$ in the numerator, which means that less weight is given
to the best possible outcome under an alternative strategy $i.$ The
other two differences concern the ranges of $j$ over which the min
in the numerator and the max in the denominator are taken. Under
$\tilde \rho$ in both cases $\hat j$ is excluded, which means that
a
(weakly) higher payoff is considered under the status quo and a
weakly lower payoff under the alternative. In the symmetric $2
\times 2$ game discussed in Section 2 there is only one alternative
and therefore there is no need to maximize over alternatives.
Moreover in this case the limitations on the range of the max and
min
operators imply that these operators can be dropped. By doing so
one
arrives at what one might call the Harsanyi-Selten risk measure $$
\rho_{\rm HS}((\hat i , \hat j),G) := {{u_1(i, \hat j)- u_1 (\hat
i ,
j)} \over
{u_1(\hat i , \hat
j) - u_1(i,j)}}$$ for $2 \times 2$ games. Note that in such a game
the equilibrium $( \hat i , \hat j)$ is risk dominant in the
Harsanyi-Selten sense if $$
u_1(\hat i, \hat j)- u_1 ( i , \hat j) > u_1(i , j) - u_1( \hat
i,j),$$ which is equivalent to $$
\rho_{\rm HS}((\hat i, \hat j), G) < 1.$$ 

The following result resembles
Proposition 3, with three differences. They concern the cardinality
of
the message space (two messages instead of finitely many), the risk
measure ($\tilde \rho$ instead of $\rho$) and the standard of
comparison ($1$ instead of $1 \over \#(\hat F(G,M)) -1$).

\begin{prp} For any one-sided communication game $\Gamma_1 (G,M)$
with  underlying game $G$ and size of the message space $\#(M) =2,$
suppose that player one's risk, $\tilde \rho((\hat i,\hat j), G),$
at his preferred equilibrium satisfies $$1 > \tilde \rho
((\hat i, \hat j), G).$$ Then (a) the retract $Q=Q_1 \times Q_2 :=
$ $$\lbrace
{\rm co} \lbrace (m, \hat i)\rbrace_{m \in M} \rbrace \times
\lbrace  {\rm co} \lbrace \hat F(G,M) \rbrace \rbrace $$ is the
unique
curb retract in $\Gamma_1 (G,M),$ and (b) the payoffs in all
curb equilibria of $\Gamma _ 1 (G,M)$ are $u_k(\hat i , \hat j),$
$k=1,2.$ \end{prp}  

\begin{proof} The proof of Proposition 2 is identical to the proof
of Proposition 1 with one exception that concerns showing that $Q$
is closed under inclusion of best replies.

Let $M := \lbrace m_1 , m_2 \rbrace.$ Suppose, in order to derive
a contradiction, that $(m_1,i),$ $i
\not= \hat i$, is a best reply against
the belief $\lambda$
concentrated on $Q_2.$ With only two messages it is convenient to
represent a strategy of player two as a vector $(j^1,j^2)$ with the
first (second) element being the response to message $m_1,$
$(m_2).$ Let $\lambda (j^1,j^2)$ be the probability
assigned to strategy $(j^1,j^2)$ by $\lambda.$  Then, since
$(m_1,i),$ $i \not= \hat i,$ must be at least as good a reply as
$(m_1, \hat i)$ to the belief $\lambda ,$ we have
$$ \sum _{j \not= \hat j } u_1 (i , \hat j) \lambda ( \hat j , j)
+ \sum _{j \not= \hat j } u_1 (i , j) \lambda ( j , \hat j) \geq $$
$$ \sum _{j \not= \hat j } u_1 ( \hat i , \hat j) \lambda ( \hat j
, j) + \sum _{j \not= \hat j } u_1 ( \hat i , j) \lambda ( j , \hat
j), $$ where we have used the fact that $u_1(\hat i , \hat j) > u_1
(i, \hat j).$

Therefore
$$[u_1( \hat i , \hat j) - u_1 (i , \hat j )] \sum_{j \not= \hat j}
\lambda ( \hat j , j ) \leq $$
$$[\max_{j \not= \hat j } u_1( i , j) - \min_{ j \not= \hat j } u_1
( \hat i , j )] \sum_{j \not= \hat j} \lambda (  j , \hat j )  .$$
Together with the condition $1 > \tilde \rho ((\hat i , \hat j),
G),$ this implies
$$ \sum_{j \not= \hat j} \lambda (  j , \hat j ) > \sum_{j \not=
\hat j} \lambda ( \hat j , j ),$$ since at least one of these sums
must be positive for $(m_1,i),$ $i \not= \hat i ,$ to be a best
reply.

Since $(m_1,i),$ $i \not= \hat i,$ must be at least as good a reply
as $(m_2, \hat i)$ we also have the following condition

$$ \sum _{j \not= \hat j } u_1 (i , \hat j) \lambda ( \hat j , j)
+ \sum _{j \not= \hat j } u_1 (i , j) \lambda ( j , \hat j) \geq $$
$$ \sum _{j \not= \hat j } u_1 ( \hat i , j) \lambda ( \hat j , j)
+ \sum _{j \not= \hat j } u_1 ( \hat i , \hat j) \lambda ( j , \hat
j). $$

Therefore
$$[u_1( \hat i , \hat j) - \max _ {i \not= \hat j} u_1 (i , j )]
\sum_{j \not= \hat j} \lambda ( j , \hat j ) \leq $$
$$[u_1( i , \hat j) - \min_ { j \not= \hat j } u_1 ( \hat i , j )]
\sum_{j \not= \hat j} \lambda (  \hat j , j ).$$
Together with the condition $1 > \tilde \rho ((\hat i , \hat j),
G),$ this implies
$$ \sum_{j \not= \hat j} \lambda ( \hat  j , j ) > \sum_{j \not=
\hat j} \lambda ( j , \hat j ).$$
Therefore we have reached a contradiction. Hence, $(m_1,i),$ $i
\not= \hat i,$ cannot be a best reply to any beliefs $\lambda$
concentrated on $Q_2.$ An analogous argument works for $(m_2,i),$
$i
\not= \hat i .$
\end{proof}

Proposition 4 shows that with only two messages a less stringent
risk measure, $\tilde \rho,$ can be used to formulate a condition
guaranteeing effective communication. We argued before that this
measure is
a generalization of the
Harsanyi-Selten risk measure $\rho_{\rm HS}$ while for $\rho,$
which we used for general message space sizes, the link to risk
dominance is more tenuous.
For
that reason let me examine why $\rho$ and not $\tilde \rho$ is the
appropriate risk measure in
games with more than two messages. Consider the following version
of {\em Dodo}

\begin{center}
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\put(50,40){\line(1,0){50}}
\put(75,15){\line(0,1){50}}
\put(35,50){$U$}
\put(35,25){$D$}
\put(55,70){$L$}
\put(80,70){$R$}
\put(55,50){x,x}
\put(80,50){0,0}
\put(55,25){0,0}
\put(80,25){1,1}
\end{picture}
\end{center}
where $2>x>1.$ We will analyze the communication game where only
the row
player can talk and the cardinality of the message space equals
three. The reduced normal form of this game is given by

\begin{picture}(250,250)
\put(50,50){\framebox(200,150)}
\put(50,100){\line(1,0){200}}
\put(50,75){\line(1,0){200}}
\put(50,125){\line(1,0){200}}
\put(50,150){\line(1,0){200}}
\put(50,175){\line(1,0){200}}
\put(75,50){\line(0,1){150}}
\put(100,50){\line(0,1){150}}
\put(125,50){\line(0,1){150}}
\put(150,50){\line(0,1){150}}
\put(175,50){\line(0,1){150}}
\put(200,50){\line(0,1){150}}
\put(225,50){\line(0,1){150}}
\put(15,185){$m_1U$}
\put(15,160){$m_1D$}
\put(15,135){$m_2U$}
\put(15,110){$m_2D$}
\put(15,85){$m_3U$}
\put(15,60){$m_3D$}
\put(55,205){$\mbox{\tiny LLL}$}
\put(80,205){$\mbox{\tiny LLR}$}
\put(105,205){$\mbox{\tiny LRL}$}
\put(130,205){$\mbox{\tiny LRR}$}
\put(155,205){$\mbox{\tiny RLL}$}
\put(180,205){$\mbox{\tiny RLR}$}
\put(205,205){$\mbox{\tiny RRL}$}
\put(230,205){$\mbox{\tiny RRR}$}
\put(55,185){x,x}
\put(80,185){x,x}
\put(105,185){x,x}
\put(130,185){x,x}
\put(155,185){0,0}
\put(180,185){0,0}
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\put(155,160){1,1} 
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\put(155,60){0,0}
\put(180,60){1,1}
\put(205,60){0,0}
\put(230,60){1,1}
\end{picture}

It is evident that in this game any curb retract that supports the
efficient outcome in the underlying game must contain all
strategies that support the efficient outcome in the communication
game. This means that for example the strategies $RLR$ and $RRL$
for player two must be included in the curb retract. However, if
player
one believes that player two uses only these two strategies and
uses them with equal probability, then the strategy $m_1D$ is a
best reply, from which it follows easily that in fact all
strategies will be in any curb retract. Only if we made $x$ larger
than 2 would this argument not work. It is this distinction that
is captured by the risk measure $\rho.$ Note that $\tilde \rho (x)
= 0 \ \forall x > 1$ and $\rho (x) = {1 \over x-1} \ \forall x>1;$
the risk associated with the equilibrium $UL$ gets large as $x$
approaches one. 

This example does not show why in addition to changing the risk
measure we need to change our standard of comparison as we increase
the number of messages since $\rho (x) < 1$ is equivalent to the
condition that $x > 2$. The reader may check however that if we add
another message, $x$ needs to increase for it to be the unique curb
equilibrium payoff in the communication game.   

The risk measures introduced up to this point are all
structurally similar to $\rho.$ This measure relies on $( \hat i ,
\hat j)$ being the unique strategy
profile that maximizes player one's payoff in the underlying game.
Therefore it is worth pointing out that instead of $\tilde \rho$ we
could have considered an alternative measure in Proposition 4. For
an {\em arbitrary} finite game and {\em any} strict equilibrium
$(\hat i , \hat j),$ this equilibrium's risk can be measured by $$
\rho_{\rm GHS} := \max_{i \not= \hat i} { \max_{j \not= \hat j} u_1
(i,j) - \min_{j \not= \hat j} u_1 ( \hat i , j) \over u_1 (\hat i
, \hat j)- u_1 (i , \hat j) }.$$ The attraction of this measure is
that it can both replace $\tilde \rho$ in Proposition 4 and be used
to compare risk across different strict equilibria. We will refer
to this as the
Generalized Harsanyi-Selten measure of risk. Section 6 contains a
brief discussion on the relation of this measure to some examples
from the literature on stochastic evolutionary game dynamics. 

\section{Multi-Sided Communication}

In this section we introduce multi-sided communication and
compare it to one-sided communication. The result of this
comparison turns out to depend on whether or not we allow the
population learning dynamic to introduce permanent asymmetries into
the message space. 

So far we have examined a class of population learning dynamics in
which individuals make only limited use of the desymmetrizing
effects of history. They learn to distinguish messages according to
payoff differences in their current environment. They do
not however develop an affection for messages that have served
them well for a long time or impute meaning (beyond the current use
in the population) to messages that have had a long association
with an equilibrium in the underlying game. This is a useful
benchmark in the analysis of cheap-talk games but not very
realistic. One ought to at least acknowledge a secondary, 
tie-breaking role for such behaviors.

For the central result of this section we leave the population
learning dynamics unchanged and work with a message space that is
already differentiated. This allows us to capitalize on our
investment in establishing a
link between the stable outcomes of population learning dynamics
and curb retracts. Later, we propose a modification of the
population
learning dynamics with an endogenous process of message space
differentiation. 

To capture message space differentiation through minor alterations
of
the communication game, we assume that associations of
message profiles and equilibria induce minute differences in
payoffs. This link between message profiles and equilibria
introduces {\em and formalizes} a small measure of {\em a priori}
meaning of messages, which we refer to as {\em limited information
content (LIC).} To formalize {\em LIC,} we focus on the recipients
of messages. The small payoff changes that we introduce make
players in their roles as recipients of messages slightly more
``gullible;" if all messages sent agree on the same equilibrium,
then the recipients become more favorably inclined toward their
corresponding equilibrium actions.\footnote{Gullibility of
receivers is different from a ``disincentive to lie" for senders
and from the nominal message costs employed by Hurkens [1993]. The
latter are concerned with {\em sender} preferences and have
stronger implications than gullibility in games where only a subset
of the players send messages.} It is useful to emphasize that
the meaning appears via a
minimal differentiation of the message space, which itself can be
endogenized without much difficulty. 

Two observations emerge.
If costless messages have no a priori meaning whatsoever, 
multi-sided communication is {\em less} effective than one-sided
communication. Multi-sided communication is {\em more} effective if
all players communicate, have the same preferred equilibrium, and
there is some a priori information content of messages, however
small, that identifies message profiles that signal agreement on 
a strict equilibrium in the underlying game.  

Consider the first of these claims. With multi-sided communication,
and a priori meaningless messages, there is no result of the form:
``If the risk of an efficient equilibrium in the stage game is
sufficiently low, it will be the only curb equilibrium in the
communication game." To see this consider the following example of
a communication game derived from a version of {\em Dodo.}

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\put(50,100){\line(1,0){200}}
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\put(50,25){\line(1,0){200}}
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\put(50,150){\line(1,0){200}}
\put(50,175){\line(1,0){200}}
\put(75,0){\line(0,1){200}}
\put(100,0){\line(0,1){200}}
\put(125,0){\line(0,1){200}}
\put(150,0){\line(0,1){200}}
\put(175,0){\line(0,1){200}}
\put(200,0){\line(0,1){200}}
\put(225,0){\line(0,1){200}}
\put(15,185){\mbox{\tiny $m_1UU$}}
\put(15,160){\mbox{\tiny $m_1UD$}}
\put(15,135){\mbox{\tiny $m_1DU$}}
\put(15,110){\mbox{\tiny $m_1DD$}}
\put(15,85){\mbox{\tiny $m_2UU$}}
\put(15,60){\mbox{\tiny $m_2UD$}}
\put(15,35){\mbox{\tiny $m_2DU$}}
\put(15,10){\mbox{\tiny $m_2DD$}}
\put(55,205){\mbox{\tiny $m'_1LL$}}
\put(80,205){\mbox{\tiny $m'_1LR$}}
\put(105,205){\mbox{\tiny $m'_1RL$}}
\put(130,205){\mbox{\tiny $m'_1RR$}}
\put(155,205){\mbox{\tiny $m'_2LL$}}
\put(180,205){\mbox{\tiny $m'_2LR$}}
\put(205,205){\mbox{\tiny $m'_2RL$}}
\put(230,205){\mbox{\tiny $m'_2RR$}}
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\end{center}

In this game two players simultaneously exchange messages before
playing {\em Dodo;} with $x>1.$ The cardinality of each player's
message space equals two. It is easily checked that any curb
retract
that supports the efficient outcome must include all strategies
that are consistent with efficiency. However, the game induced by
the retract that is formed by all such strategies contains an
equilibrium in which the row player uses $m_1UD$ and $m_2DU$ with
equal probability and the column player uses $m'_1LR$ and $m'_2RL$
with equal probability. Thus, for any $x$ there is a curb
equilibrium with payoffs $x+1 \over 2$ for each player. Hence,
within the 
class of population learning dynamics considered above, we cannot
rule out inefficient outcomes.

Now assume instead that the message space is differentiated, that
message exchanges alter, if only marginally, the players'
perception of the underlying game. In the example assume that once
players have exchanged the message pair $(m_1 , m'_1),$
they are somewhat more inclined to use actions $U$ and $L$
respectively in the underlying game. Model this increased
inclination as an enlargement of the range of beliefs for which the
actions $U$ and $L$ are optimal in the underlying game. That is,
for
the row player,
for any given distribution over $L$ and $R,$ the action $U$ yields
a slightly higher payoff than it would without the prior exchange
of
messages $m_1$ and $m'_1.$ Let there be a similar link between the
message pair $(m_2, m'_2)$ and the action pair $(D,R)$ so that
there is no bias in favor of either equilibrium.\footnote{This
second link is present only to demonstrate that we can allow for a
whole range of meanings. It does not affect our conclusions.} In
the example
such a payoff boost affects the bold entries in the above
representation of the communication game. It is easily checked that
for any sufficiently small positive payoff boost, there is a
unique curb retract in the communication game and
it consists entirely of equilibria that support the efficient
outcome.

For a general definition of {\em limited information content (LIC)}
we want to introduce some minimal message space differentiation
that links messages with equilibria in the underlying game. To this
end, we
let messages have a
marginal effect on the players' perception of the game following
the message exchange. This introduces a measure of effective
communication when a message is sent that is independent of the
current use of the message. It appears essential for effective
communication
to occur that the sender's message affect the receiver's
perception. In contrast, while the act of sending a message may
also have a
self-committing effect on receivers, such an effect does not seem
to be
equally central to the notion of effective communication. The
effect on senders of messages having been exchanged appears to be
indirect. For a message to alter the receiver's
perception, it has to be ``believed." On the other
hand, for a message to alter the sender's perception, the sender 
must ``believe" that the message is ``believed" by the receiver,
and thus solve an {\em inference problem.}
Therefore we take the position that the primary effect
of the message exchange is on receivers' beliefs. 

Another issue that we have to address is that players may be both
senders and receivers of messages. Therefore, it can occur that a
player sends a message linked to one equilibrium and at the same
time receives a message linked to another equilibrium. In that case
it appears reasonable that perceptions remain unaltered. 

Accordingly, the formal definition of {\em LIC} reflects two
principles,
(1) that the primary effect of message exchange is on perceptions
of receivers, and (2)
that any effect on perceptions requires unanimity among senders.

These principles have strong implications for the impact of one- vs
two-sided communication in two player games. If only
one player communicates, only the receiving player will be swayed
by an
LIC message. If both players
communicate,
they will {\em both} be swayed by LIC messages, as long as there is
no
ambiguity. Two-sided communication avoids the above mentioned
inference
problem of senders. 

We express the altered perception of the underlying game, following
a message profile linked to an equilibrium in that game, by making 
the associated equilibrium actions more attractive.
An action in the underlying game becomes more attractive
if its payoffs increase. Therefore we will express the {\em limited
information content} of a message profile as an $\epsilon$-increase
in payoffs from corresponding equilibrium actions in the underlying
game. 

Since the main result refers to an $P$ player game, we adopt a
somewhat more general notation in this section than in the
previous one. Let each player's set of pure strategies in the
underlying game be denoted by $S_p,$ $p \in {\cal P},$ and let
$s_p$ be a
typical element of player $p$'s strategy set. Analogous to the
previous section assume that there is a strategy combination $\hat 
s =\lbrace \hat s_p\rbrace_{p=1}^P$ in the underlying game such
that $$\hat s = \arg \max_s u_p (s) \ \ \forall p \in {\cal P}; $$
i.e.,
for every player $p \in {\cal P},$ $\hat s$ is the unique strategy
combination that gives $p$'s maximum payoff in the game. Let there
be a set $C \subseteq {\cal P}$ of communicating players. 

Let ${\cal E}$ denote the
set of strict Nash equilibria in the underlying game, and for each
communicating player $p,$ let $\iota _ p$ be an injective function
$$\iota_p : {\cal E} \rightarrow M_p ,$$ which will be referred to
as player $p$'s {\em LIC} mapping. Thus, for each
strict equilibrium, player $p$ has exactly one message linked to
that equilibrium. This implicitly assumes
that a communicating player's message space is at
least as
large as the number of strict equilibria in the underlying game.
Let $\iota ( \cdot )$ be the vector of all {\em LIC} mappings for
all the communicating players.

Using the {\em LIC} mappings, we can define {\em LIC} preferences
with an $\epsilon$-payoff boost
in the communication game as follows. Denote payoffs in the
communication game by $U_p (m,f)$ and let

$$U_p(m,f) =  \cases{u_p(f(m)) +
\epsilon , & if $\exists e \in {\cal E}, ~ \iota (e) =m,~e_p = f_p
(m)$ and ${\cal P}
\backslash p \subseteq C$; \cr u_p(f(m)), & otherwise. \cr} $$

The {\em LIC} mapping establishes the meaning of messages but only
through {\em LIC} preferences do these meanings become operational.
Without {\em LIC} preferences, the game is completely unchanged,
and even with {\em LIC} preferences, the meanings play a limited
role because none of the equilibrium outcomes change and all
messages can acquire all meanings, in equilibrium. According to
this formulation of LIC preferences, 
communication makes an equilibrium more attractive
for a player if all other players
communicate, and all communicating players, ``agree" on the
equilibrium in question. 

It is easy to check that this way of introducing small a priori
information content has no effect with one-sided communication.
Even with only one-sided communication, an LIC message does alter
perceptions. However, it does so only for the receiver. As a
consequence, the sender's payoffs in the
communication game (including all ties) are exactly as before and
the
results of the previous section go through virtually unchanged. 

For the remainder of this section assume that ${\cal P}=C.$ Denote
by
$\Gamma_P (G,M)$ the communication game in which the play
of the underlying game $G$ is preceded by one round of simultaneous
communication in which all $P$ players announce a message from
their respective message spaces $M_p.$

The following proposition generalizes our example. If all players
communicate and have LIC preferences, every strategy in the unique
curb retract is an efficient equilibrium.

\begin{prp} Let $\Gamma_P(G,M)$ be a multi-sided communication game
with player set ${\cal P},$ underlying game $G,$ unique efficient
profile
$\hat s$ in $G,$ and message spaces $M_p$,
$p \in {\cal P},$ where all players have {\em LIC} preferences.
Then
there exists a bound $\bar \epsilon > 0$ on the LIC payoff boost
$\epsilon$ such that the retract $Q= \times_{p=1}^P Q_p,$
where $Q_p  := {\rm co} \lbrace (m_p, f_p)
\vert
m_p \in M_p, \  \iota (f(m))=m, ~ f(m) = \hat s \rbrace ,$ is
the unique curb retract in $\Gamma_P (G,M),$ for all $ \bar
\epsilon > \epsilon >
0.$ \end{prp}  

\begin{proof} Against any belief concentrated on $Q_{-p}$ player
$p$ can
achieve a payoff of $u_p(\hat s ) + \epsilon$ by using one of the
strategies in $Q_p.$ Any other strategy will at most yield a payoff
of $u_p( \hat s).$ This shows that $Q$ is closed under inclusion of
best replies.

Every strategy combination in $Q$ gives player $p$ his maximal
payoff in the communication game. Therefore no strict subset of $Q$
is closed under inclusion of best replies, which shows that $Q$ is
minimal.

It remains to show uniqueness. We will show that if $\tilde Q$ is
curb, then $Q \cap \tilde Q \not= \emptyset .$ Let $(m,f) \in
\tilde Q \backslash Q .$ Then there exists a strategy $(m',f') \in
\tilde Q$ with $f'_p(\tilde m ) = \hat s _ p \ \ \forall p, \
\forall \tilde m _{-p} \not= m_{-p}.$ Against $(m',f')$ player $p$
can guarantee that the other players will play $\hat s _ {-p} $ in
the underlying game by not sending message $m_p.$ Hence, there
exists $(m'',f'') \in  \tilde Q$ with $f '' (m '') = \hat s,$ which
implies that there exists $(m'', f''') \in \tilde Q$ with $f'''_p(
\tilde m ) = \hat s _ p$ $\forall p, ~ \forall \tilde m _ {-
p}.$ Hence, $(\hat m , f''') \in \tilde Q ,$ for $\hat m = \iota
(\hat s).$ But $(\hat m , f''') \in Q .$ \end{proof} 

Proposition 5 relies on message space differentiation already 
having been established. One could model this differentiation as
arising endogenously. Here is an example of one
message profile becoming distinct from all others: Consider a
variation on our population learning
dynamic applied to $P$-player games, in which (conforming to the
central result of this section) all players communicate and the
underlying game has a unique efficient strategy combination $\hat
s.$ Let each population member
behave exactly as in our basic population
learning dynamic, with one exception: For any strategy profile
$(m^*, f^*)$ with $f^*(m^*)=e^* \in {\cal E},$ define the set
$S(m^*, f^*) :=
\{ (m_p, f_ p (
\dot )) \vert \forall p \in {\cal P},~ m_p =
m_p^*,~ f_p (m^*) = f_p^* (m^*) \}$ of all strategies in which
player $p$ sends message $m_p^*$ and takes the equilibrium action
$e_p^*$ when the profile $m^*$ is sent. Suppose that individual
$p$'s sample consists entirely of strategies in $S(m^*,f^*).$ Then,
for any message $m'_p \not= m_p^*,$ let the probability of
individual
$i \in p$ adopting the strategy $(m'_i , f'_i ( \cdot ))$
equal zero, if $u_p (f(m'_p , m_{-p}^*)) \leq u_p (f ^*(m^*)),$ 
for all $f$ such that $(m,f) \in S(m^*, f^*).$ That is, if an
individual believes he can induce
strict equilibrium $e$ with certainty by sending the message he
believes everyone in his population to send, he will send the
same message unless sending an alternative message strictly
raises his payoff. Consider any set of states in which all
individuals use strategies in $S(m^*, f^*)$ for some $(m^*, f^*).$
It is straightforward (although tedious) to
check that (1) outside such sets of states, the dynamic
behaves just like the original population learning dynamic, (2) the
dynamic will leave any such set of states where $f^*
(m^*)$ is an inefficient equilibrium, and will never leave such a
set where $f^* (m^*)$ is the unique efficient strategy profile.
Adapting the proof of Proposition 5 to explicit dynamics, these
observation can be shown to imply that the altered
class of dynamics converge to a set of efficient equilibria almost
surely. In the limit players use a single endogenously determined
message even if they believe other messages to lead to the same
payoffs. 

Evidently, the dramatic effect of differentiating the message space
derives from its breaking of some of the numerous
payoff ties in the communication game. Not all plausible
tie-breaking rules will guarantee efficiency in the limit. Say,
players develop a
slight preference (in the above $\epsilon$-sense) for
strategies that ignore messages. It is easy to check that in that
case every strict equilibrium of the underlying game gives rise to
its own curb retract in the communication game. Other plausible
behaviors give rise to stronger results than Proposition 5. For
example, one can model a ``disincentive to lie" by linking a
player's actions to a subset his messages. Then the player gets
$\epsilon$-penalized for not following a given message with the
appropriate action. In that case both one- and two-sided
communication lead to efficiency. The same strong efficiency result
is derived by Hurkens [1993] who differentiates messages by nominal
message costs.

Whether message space differentiation occurs and in which form is
an empirical matter that should be addressed experimentally.
The point of this section was to show that (1) without message
space differentiation communication remains ineffective with 
multi-sided communication and (2) that message space
differentiation can lead to an interesting reversion in the
effectiveness of one- and multi-sided communication. 

\section{Related Literature}

This section discusses three strands of literature relevant to this
paper. It first looks at some experimental evidence on pre-play
communication. Then follows an overview of the evolutionary
approach to pre-play communication. The section concludes with
comments on the literature on stochastic evolutionary game dynamics
and its relation to generalized risk measures.

Cooper, DeJong, Forsythe and Ross (CDFR) [1992] report on
experiments they conducted on a version of the {\em Stag Hunt} game
($a=1000, \ b=0, \ c=800, d=800$ in game ${\bf G_0}$). They
repeatedly let players play one-shot communication games, where
messages had an exogenously given meaning. They find that without
communication the
risk dominant equilibrium will be played. With one-way
communication the frequency of the Pareto-efficient equilibrium
increases but there are also a significant number of coordination
failures. Two-way communication resolves the coordination problem;
almost exclusively the Pareto-efficient equilibrium is played. 

Analogous to LIC preferences, one can interpret CDFR's results as
indicating that the unanimous assertion
to play according to the preferred equilibrium is enough to reduce
strategic uncertainty; on the other hand, if only one player
communicates he maybe less confident of the response of the
listener. It makes a difference whether one is informed about
someone's intent, rather than having to make an inference about
this intent. In the words of
CDFR: ``This doubt about the action of a receiver is overcome by
the two-way communication design since both players receive
information about the likely play of their opponents." [1992,
p.757]        

While refinements in the spirit of strategic stability have little
power in games with pre-play communication the evolutionary
approach
yields sensible predictions in common interest games. Several
authors have used versions of Maynard Smith and Price's [1973]
notion of an
{\em evolutionarily stable strategy (ESS)} in these games. Roughly,
in a
symmetric game a strategy is evolutionarily stable if it is a
symmetric Nash equilibrium and, if it is played by all members of
a large population, it cannot be invaded by a small population of
mutants who use a different strategy. An {\em ESS} must be a best
reply to itself, and it must be a better reply to the post-entry
population than any potential entrant. 

W\"arneryd [1991] studies
{\em Dodo} preceded by one round of pre-play communication in which
each player sends a message from a common finite message space
that contains at least two messages. W\"arneryd shows that any
{\em neutrally stable strategy (NSS)} (a variant of {em ESS}) in
{\em Dodo}
preceded by one round of simultaneous pre-play communication leads
to the efficient equilibrium in the underlying game. W\"arneryd's
analysis does not extend to more general games because there the
use of unused messages may get penalized deterring players from
introducing them. Matsui [1991] arrives at similar conclusions for
a different solution concept, {\em cyclically stable sets (CSS)}
[Gilboa and Matsui, 1991]. Unlike {\em NSSs,} {\em CSSs} are set
valued
and exist in every game. Kim and Sobel [1993] obtain similar
results as W\"arneryd for more
general games than {\em Dodo} by using a set-valued solution
concept, Swinkels' [1992] {\em Equilibrium Evolutionarily Stable
(EES)
Set}. 

These evolutionary solutions assume that only a small fraction of
the
population moves
at any given point in time. This reduces the possibility for
coordination failures significantly. In the case of common interest
games, once the evolutionary process reaches the efficient point,
any potential invader must also play an efficient strategy in order
to maximize its payoff against the post-entry population. The
problem of a mutant in this situation
is reduced to a simple optimization problem  with no
strategic component. Sobel [1993] has a particularly transparent
version of the
evolutionary argument. In his model only one player at a time
adjusts his strategy. 

With simultaneous adjustments as in the present paper, either one
has to acknowledge the role of risk and strategic uncertainty, or
one must introduce some other source of inertia if one wants to
ensure that only efficient outcomes are stable. In Section 5 we
examined one such form of inertia. Hurkens [1993]
gets very strong result for the same solution concept, curb
retracts, and a different form of inertia, namely nominal message
costs. He shows that in two-player games the same result holds for
persistent
equilibria, which were defined by Kalai and Samet [1984]. Since
messages are costly in Hurkens' paper his work also provides a
link with the work on ``burning money" by Ben-Porath and Dekel
[1992] and van Damme [1989]. The players who are given the
opportunity
to burn money can achieve their preferred outcome, and no money is
actually burned in equilibrium. Ben-Porath and Dekel use iterative
deletion of weakly dominated
strategies as their
solution concept and require that message costs be nonnegligible. 

Risk becomes an issue in this paper because of the relative ease
with which one can travel from one strategy combination to another.
This is a consequence of the low entry requirements for new
strategies in our dynamic and the absence of other sources
of friction such as message costs, as in Hurkens'
work.\footnote{Hurkens' results holds also for the weaker
solution concept ``closed under inclusion of better replies." Such
a solution concept poses even less stringent entry requirements
than curb retracts. In Hurkens work this permissiveness is balanced
by the cost of messages. For a discussion and characterization of
this solution concept see Ritzberger and Weibull [1993].} One
reason it is so easy to travel from one profile to another is that
communication turns strict equilibria into weak ones. Thus
communication creates an escape route from strict equilibria. An
alternative escape route is analyzed in the literature on
stochastic evolutionary game dynamics, as for example in Kandori,
Mailath and Rob [1993], Young [1993] and Ellison [1993]. There too,
risk plays a role.\footnote{Risk dominance also plays a role in $2
\times 2$ games with small payoff uncertainty
and incomplete information. This has been demonstrated in a recent
paper by Carlsson and van Damme [1993].} Kandori, Mailath and Rob
for example prove that
in two-player two-strategy coordination games the limit of the
stationary distributions of their dynamics as the noise vanishes
puts all probability weight on the risk dominant equilibrium. Young
[1993] examines a similar kind of dynamics and comes to similar
conclusions. He also points that while in two-strategy games a
characterization of stochastically stable equilibria in terms of
risk dominance is possible no such
characterization may be available for games with three or more
strategies. He gives the following example.   

\begin{center}
\begin{picture}(150,100)
\put(15,15){\framebox(75,75)}
\put(40,15){\line(0,1){75}}
\put(65,15){\line(0,1){75}}
\put(15,40){\line(1,0){75}}
\put(15,65){\line(1,0){75}}
\put(0,75){$U$}
\put(0,50){$M$}
\put(0,25){$D$}
\put(22,95){$L$}
\put(47,95){$C$}
\put(72,95){$R$}
\put(20,75){6,6}
\put(45,75){0,5}
\put(70,75){0,0}
\put(20,50){5,0}
\put(45,50){7,7}
\put(70,50){5,5}
\put(20,25){0,0}
\put(45,25){5,5}
\put(70,25){8,8}
\end{picture}
\end{center}

Young points out that while the equilibrium $(D,R)$ pairwise risk
dominates the other two pure strategy equilibria in this game,
there are plausible dynamics under which $(M,C)$ is the unique
stochastically stable equilibrium. Against this background it is
perhaps interesting to calculate the Generalized Harsanyi-Selten
measures for this game. They are $\rho_{\rm GHS}(U,L)=7,$
$\rho_{\rm GHS}(M,C)=3/2,$ and $\rho_{\rm GHS}(D,R)=7/3.$ Therefore
the equilibrium picked by Young's dynamics is also the one with the
lowest GHS risk measure. This phenomenon does not generalize.
Moreover, Ellison [1993] points out that variations in matching
rules (e.g. more frequent matching with close neighbors vs. uniform
matching) alter the long-run predictions in games similar to the
example. Despite the weakness of the link to stochastic
evolutionary game dynamics, we believe that the GHS risk measure
identifies some of the dynamic forces acting on
multiple Pareto-ranked strict
Nash equilibria.

\section{Conclusion}

The chances for pre-play communication to allow players to
coordinate on efficient equilibria are tied to the risk associated
with these equilibria. This observation can be formally expressed
through examining stable outcomes of a class of best reply dynamics
that permit simultaneous strategy adjustments among a large
fraction of the population. This contrasts with and complements
results from
the evolutionary literature on pre-play communication in which risk
plays no role. One
benefit of this approach is that it permits one to make a
distinction between one- and two-sided communication, which
parallels some experimental results.

\vfill\eject

\hfill\break
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\end{document}
