\documentstyle{article}
\begin{document}
\title{Information Transmission and Preference Similarity\thanks{I
thank, without implicating, Doug DeJong and Joel
Sobel for comments. I have benefitted from comments by seminar
participants at the University of Indiana and the University of
Iowa. Support by the National Science Foundation
under award number SBR-9410588 is gratefully acknowledged.}} 
\author{Andreas Blume}
\date{May, 1996}
\maketitle
\vglue .3in
\begin{abstract}

This paper examines sets of Nash equilibria in sender-receiver
games that are stable against replacement by alternative Nash
equilibria. Such stable sets exist. In {\em partial common interest
games} they contain only informative equilibria. The stability
requirement sharpens currently available predictions for such games
by (1) weakening the partial common interest condition, (2) ruling
out strictly dominated actions, (3) reflecting the informativeness
of the sender's strategy in the receiver's reply, and (4) by ruling
out pooling actions. This approach is then used as a step toward
unifying the study
of partial common interest games and of Crawford and Sobel's
parametric model of preference similarity. 

\end{abstract}
\vglue 2in
\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}

The goal of this paper is twofold. One is two provide strong and
refutable predictions for sender-receiver games with imperfectly 
aligned interests, without having to rely on auxiliary assumptions
like exogenous meanings of messages, nominal signaling costs or
restrictions on the interpretation of zero-probability messages,
which are common in the literature. The second objective is to
begin a formal study of equilibrium selection in Crawford and
Sobel's [1982] model of strategic information transmission. This
dual purpose mirrors the two approaches in the literature to
expressing imperfect incentive alignment in sender-receiver games
formally, either through a {\em partial common interest} condition,
Blume, Kim and Sobel [1993], Rabin and Sobel [1996], Blume [1996],
or as in Crawford and Sobel's parametric model of preference
similarity. 

To this end this paper examines sets of Nash equilibria in
sender-receiver
games that are stable against replacement by alternative Nash
equilibria. Such stable sets exist. In {\em partial common interest
games} they contain only informative equilibria. The stability
requirement sharpens currently available predictions for such games
by (1) weakening the partial common interest condition, (2) ruling
out strictly dominated actions, (3) reflecting the informativeness
of the sender's strategy in the receiver's reply, and (4) by ruling
out pooling actions. This approach is then used as a step toward
unifying the study
of partial common interest games and of Crawford and Sobel's
parametric model.

In sender-receiver games a privately informed sender
sends a message to a receiver who takes an action. Payoffs are
determined jointly by the sender's private information and the
receiver's type; messages do not affect payoffs directly.
If the sender's and the receiver's
incentives are sufficiently closely aligned, the sender wants to
reveal at least some of his private information. However, there is
always an uninformative ``babbling" equilibrium, in which the
sender's messages contain no information and the receiver responds
to all messages with the same action. 

If the sender and the receiver can both gain from sharing some
information, we might expect them to do so if they are able to
focus on particular interpretation of messages. In the refinement
literature,
Farrell [1993], Matthews, Okuno-Fujiwara and Postlewaite [1991] and
Rabin
[1990], a focal mechanism is provided through an existing commonly
understood language. In this paper we take the view that focal
points, as considered by Schelling [1960], may emerge endogenously
in the process of repeated
play.\footnote{The endogenous assignment of meanings to messages is
not only relevant for a priori meaningless messages but also for
messages that are part of a commonly understood language and whose
meanings are subject to deterioration and recoding in specific
strategic environments.} 

In an environment where population play has settled on a particular
equilibrium, other
strategies can emerge through either mistakes or experimentation.
This paper takes the position that such strategies can become focal
points for the reorganization of the environment. They have
the best chance of doing so if (1) they are best replies to the
current environment and (2) they are themselves equilibria. Under
those circumstances no player incurs a loss by switching from the
status quo to the alternative equilibrium, even if he is matched
with a player who remains at the status quo. Furthermore, since the
alternative is an equilibrium, there is no ex post regret if other
players behave in the same manner, and switch. Finally, at least in
two-player games, if the two conditions are satisfied for a
candidate replacement equilibrium, they remain satisfied even if
initially only a fraction of the population adopts the replacement
strategy. Nonequilibrium
mistakes or experiments lack this coordinating potential.  

The notion that a solution should consist of equilibria and be
stable against the replacement by alternative equilibria is
formalized
in the
conditions for a set of strategies to be an {\em equilibrium entry
resistant (EER)} set. An {\em
EER} set is a minimal closed set of Nash equilibria that is closed
under
the inclusion of (replacement by) strategies that are best replies
both against an
element of the set, and against themselves. The solution concept
proposed here combines ideas from the
evolutionary literature and the refinement literature. From the
evolutionary literature it takes the idea of endogeneity, from the
refinement literature it borrows the idea of focal points. 

{\em EER} sets exist in every finite game and consist solely of
equilibria. This accords well with experimental results on the
emergence of meaningful communication in sender-receiver games.
Blume, DeJong, Kim and Sprinkle [1995] examine the endogenous
emergence of meaning for {\em a priori} meaningless messages
experimentally. They consider both {\em common interest games,} in
which there is a unique payoff pair that maximizes both the
sender's and the receiver's payoff, and {\em  divergent interest
games} in which the sender's and the receiver's preferences over
equilibrium outcomes are opposed. Equilibria play a central role in
classifying the data. This is even true in games of
divergent interests where many set-valued evolutionary solutions
either do not exist or contain nonequilibrium strategies. They find
that while both revealing and nonrevealing equilibria may emerge in
these games, equilibrium behavior itself is robust.  

{\em EER} sets are attractive for their conceptual simplicity. They
make powerful predictions in sender-receiver games without relying
on auxiliary assumptions like exogenously given meanings of
messages, nominal signaling costs, limitations on drift,
interpretation of zero-probability messages etc.\ that
are commonly made in this literature, both in static solution
concepts and in explicit dynamic models. 

The simplicity of {\em EER} sets comes at a cost. We do not have a
justification of {\em  EER} sets as the long-run limits of some
explicit dynamic
that relies only on simple behavioral rules governed by local
conditions. Equilibrium play is postulated rather than explained as
the result of dynamic adjustment. This does not detract from the
value of {\em EER} sets as a solution concept. In the laboratory
setting we deal with subjects who are capable of some degree of
strategic analysis of a game and who become more sophisticated
strategists during repeated play. Their behavior is likely to be
governed as much by a global analysis of the game as by myopic
payoff considerations based on the current state of play.
Undoubtedly, equilibria have a prominent place among the set of all
strategies in a game. In this paper it is taken as given that
players know an equilibrium when they see it. For such players an
emergent equilibrium can become a rallying point if playing
according to that equilibrium is to their advantage. Nonequilibrium
experiments or mistakes lack this property. They cannot serve as
organizing
precedents because they lack the ability to coordinate players'
expectations.      

We bring the EER condition and a weaker variant of it to bear
on the two approaches to imperfect
incentive
alignment in sender-receiver games that have been considered in the
literature. One is via a
{\em partial common interest condition,} the other is Crawford and
Sobel's [1982] parametric model. The former has played a role 
in evolutionary and refinement analyses of such games. This theme
is taken up in the first part of the paper. It is shown that {\em
EER} sets reject pooling equilibria under a considerably weaker
partial common interest condition than those proposed in the
literature. In addition, the set of strategies included in a
solution is in general much smaller and more tractable than if
nonequilibrium  strategies are admitted as part of the solution.
This is a great advantage if we want to test the theory
experimentally. 

One can enhance tractability further by strengthening the partial
common interest condition; the resulting condition is
still weaker than those in the literature. The stronger condition
permits a straightforward characterization of the equilibria that
form the solution without the need for characterizing the entire
set  of equilibria.

In two-player games (such as sender-receiver games) there is a
close relationship between {\em EER} sets and set-valued
evolutionary solution concepts. Essentially, if one of the
evolutionary solutions exists, it contains an {\em EER} set.
Therefore, if a certain partition of the type set is identified in
an evolutionary solution, there is also an {\em EER} set in which
it is identified. On the other hand, it turns out that
noninformative equilibria are just as easily destabilized under the
$\em EER$ condition as under the evolutionary solutions. As a
result, {\em EER} sets make more definite predictions in
sender-receiver games. 

The second approach to imperfect incentive alignment has attracted
less attention in the evolutionary and refinement literature.
Farrell [1993] pointed out that his {\em neologism proofness}
criterion tends to reject all equilibria in Crawford and Sobel's
quadratic model. The evolutionary literature on sender-receiver
games does not address the selection issue in Crawford and Sobel's
model at all. In the present paper it is shown that by weakening
the replacement condition in the definition of {\em EER} sets we
can reject pooling equilibria both when incentives are closely
aligned in Crawford and Sobel's model and in a class of partial
common
interest games.

The remainder of the paper is organized as follows. Section 2
introduces sender-receiver games and the solution concept. Section
3 establishes the important role played by unused messages. Section
4 characterizes stable outcomes in games satisfying a {\em partial
common interest} condition. In section 5 the partial common
interest
condition is strengthened to permit a characterization of stable
outcomes without having to identify the entire set of Nash
equilibria. Section 6 relates {\em EER} sets to evolutionary
solutions in two-player games. Section 7 explores ways of unifying
the analysis of partial common interest games and of 
Crawford-Sobel games. The final section discusses the literature. 

\section{Preliminaries}

This section describes sender-receiver games, defines and the
solution concept, and proves existence of a solution in finite
games. 

In a sender-receiver game player 1, the sender, has private
information and player 2, the receiver, takes an action. Before the
receiver takes his action, the sender sends a message. Payoffs to
both players depend solely on the sender's private information, his
type, and on the receiver's action. 

Let $T$ be the finite set of types and $\pi ( \cdot )$ the prior
distribution of types. The sender's set of pure strategies is the
set of mappings from the type set to the finite set of messages,
$M.$ The receiver's set of pure strategies is the set of mappings
from $M= \{m_i \}_{i=1}^k$ to the finite set of actions $A.$ Given
type $t \in T,$
message $m \in M, $ and action $a \in A,$ player $i$'s payoff is
$v_i(t,a),~i=1,2.$ Messages do not directly affect payoffs. For any
finite set $X,$ denote by $\Delta (X)$ the set of probability
distributions over $X.$ The
payoff from a mixed action $\alpha \in \Delta (A)$ is $v_i (t,
\alpha) = \sum _ {a \in A} v_i(t,a) \alpha (a), ~ i=1,2.$ 

$\sigma_1(m,t)$ is the probability of type $t$ sending message $m$
and $\sigma_2(a,m)$ stands for the probability that the receiver
will choose action $a$ in response to message $m.$ $\sigma =
(\sigma _1 , \sigma _2)$ denotes a mixed strategy profile. A
strategy profile $\sigma$ gives rise to a payoff $V(\sigma) = (V_1
(\sigma) , V_2 (\sigma)), $ where
$$V_i (\sigma) = \sum_{t \in T } \sum_{m \in M} \sum_{a \in A} v_i
(t ,a) \sigma _1 (m,t) \sigma _2 (a , m) \pi (t ), ~i=1,2.  $$
Also, let 
$$V_i (t, \sigma ) =  \sum_{m \in M} \sum_{a \in A} v_i
(t ,a) \sigma _1 (m,t) \sigma _2 (a , m)  ~i=1,2.  $$ 
$\sigma$ is a Nash equilibrium if $\sigma _1$ and $\sigma _2$ are
mutual best replies:
$$\mbox{if}~ \sigma_1 (m,t) > 0, ~ \mbox{then} ~m~ \mbox{solves}~
\max_{ m ' \in M} \sum_ {a \in A} v_1 (t,a) \sigma _ 2 (a , m'), $$
and $$\mbox{if}~ \sigma_2 (a,m) > 0, ~ \mbox{then} ~a~
\mbox{solves}~ \max_{ a' \in A} \sum_ {t \in T} v_2 (t,a) \sigma _
1 (m , t) \pi (t).$$ 

The message space is assumed to be large to ensure the availability
of sufficiently many unsent messages in any candidate for a
solution. If $\#(X)$ denotes the cardinality of the set $X,$ then
$$\#(M) > 2 ^ {[\#(A) + \# (T)]} + \# (T).$$ 

For any finite game, let $\cal N$ be the set of Nash equilibria of
the game.

We will define {\em EER} sets for a general $n$-player game
$(I,S,u)$
with player set $I,$ the set of strategy profiles $S=
\times_{i=1}^n S_i$ and
payoff
function $u= (u_i)_{i=1}^n.$ Denote by $\Delta (S_i)$ the set of
mixed strategies of player $i$ and by $C(s)$ the carrier of $s,$
the set of all pure strategies that have positive probability
under $s.$ Let $BR_i ( \cdot ) $ be the (pure strategy) best reply
correspondence of player $i ,~i=1,...,n.$ For $\sigma = (\sigma _1
,...,\sigma _n) \in \times_{i=1}^n \Delta(S_i)$ define $BR( \sigma)
:= (BR_1 (\sigma _ {-1}), ...,  BR_n (\sigma _ {-n}));$ define the
mixed best reply correspondence $MBR(\sigma)$ analogously.

\begin{dfn} A set $\Theta \subset \cal N$ is equilibrium entry
resistant (EER) if it is minimal with respect to
\begin{enumerate}
\item[{[R]}] $\forall \sigma \in \Theta,$ if $C(\sigma ') \subset
BR
(\sigma) \cap BR ( \sigma '),$ then $\sigma ' \in \Theta ,$ and
\item[{[C]}] $\Theta$ is closed and nonempty.
\end{enumerate}
\end{dfn}

Thus, an {\em EER} set is a closed and nonempty set of equilibrium
strategy profiles that satisfies a {\em replacement condition,} [R]
in the above definition. Only equilibrium strategies that are best
replies to
the status quo can replace the status quo. The focus on equilibrium
strategies is deliberate. Cheap-talk refinements in the tradition
of Farrell [1993] have for the most part considered equilibria as
candidate solutions. Rabin [1990] is an exception but unlike in the
present paper he is concerned with one-time play between rational
players who base their decisions of which strategy to play only on
common knowledge of rationality. The evolutionary approach to 
cheap-talk games, discussed in detail below, has for the most part
also considered conditions on equilibria or on sets of equilibria.
Even where there is no a priori restriction to equilibria, it is
sometimes the case that solutions either consist of
(self-confirming) equilibria, as in N\"oldeke and Samuelson's
[1992] discussion of cheap-talk games, or can be
guaranteed to contain an equilibrium, Blume [1996]. 

Restricting successful replacements to be themselves equilibria is
more novel.
As outlined in the introduction this is intended to capture the
intuition that because of their focal potential, equilibrium
experiments may be more important than nonequilibrium experiments.
Jointly these assumptions make it possible to insist on equilibria
and yet to guarantee existence. 


Existence of {\em EER} sets is proved next. First it is shown that
closed sets of Nash equilibria that satisfy the replacement
condition have a simple structure. Whenever one element of a closed
connected component of Nash equilibria is part of such a set, all
elements of the component must be included.

\begin{lem}
If $\sigma \in \Theta \cap K ,$ where $\Theta$ is a closed set of
Nash equilibria satisfying the replacement condition and $K$ a
connected component of Nash equilibria, then $K \subset
\Theta .$
\end{lem}

\begin{proof}
To derive a contradiction, suppose not. Then either (1) $\Theta
\cap K$ contains a limit point of $K \setminus \Theta$ or (2) $K
\setminus \Theta$ contains a limit point of $\Theta \cap K.$
[Munkres, p.147] (2) is ruled out because $\Theta$ is closed. If
(1) holds, then there exists a sequence of Nash equilibria $\sigma
^n \rightarrow \sigma,~\sigma^n \not\in \Theta,  ~ \sigma \in
\Theta
.$ For $\sigma^n$ not to be in $\Theta, ~\forall n,$ it is
necessary
that
$\sigma^n$ not be a best reply to $\sigma, ~\forall n.$ Thus for
each $n$ there exists a player $i$ such that $V_i(\sigma_i , \sigma
_{-i}) > V_i ( \sigma_i^n , \sigma _ {-i} ).$ Hence, there  exists
a subsequence indexed by $k$ such that for one of the players, $i,$
$V_i(\sigma_i , \sigma _{-i}) > V_i ( \sigma_i^k , \sigma _ {-i}
),~ \forall ~ k=1,2 ... .$ Since there are finitely many pure
strategies, there exists a pure strategy $s_i$ that is in the
support of $\sigma _ i^k$ infinitely often such that $V_i(\sigma_i
, \sigma _{-i}) > V_i ( s_i , \sigma _ {-i} ).$ Without loss of
generality reindex the
subsequence such that $s_i$ is in the support of $\sigma_i^k$ for
all $k.$ Convergence of $\sigma^k$ and continuity of $V( \cdot )$
imply that $V_i ( s_i , \sigma _ {-i}^k ) \rightarrow  V_i ( s_i ,
\sigma _ {-i} )$ and $V_i(\sigma_i , \sigma _{-i}^k) \rightarrow
V_i(\sigma_i , \sigma _{-i}).$ Hence there exists an $N$ such that
for $k > N$ we have $V_i(\sigma_i , \sigma _{-i}^k) > V_i ( s_i ,
\sigma _ {-i}^k ).$ This contradicts
$\sigma ^k $ being Nash equilibria.
\end{proof}

Lemma 1 establishes that in our search for a solution we can
concentrate on unions of closed connected components of Nash
equilibria. The lemma is used in the following proposition to
establish existence of EER sets.

\begin{prp} The strategy space of every finite game contains an EER
set.
\end{prp}

\begin{proof} The set of all Nash equilibria is closed and
satisfies the replacement condition. By Lemma 1, any other set with
these properties must be a union of connected components of Nash
equilibria. Following Kohlberg and Mertens [1986], the set of Nash
equilibria of a finite game consists of a finite union of closed
connected components. Therefore we need to consider only finitely
many possible such unions. Hence, a minimal set exists.  
\end{proof}

\section{Unused Messages}

We will show that in a class of sender-receiver games where the
sender's and the receiver's preferences are partially aligned, {\em
EER} sets contain only equilibria in which the sender reveals at
least some information. This involves showing that partially
informative equilibria form a set that is closed under the
replacement condition and that less informative equilibria can be
replaced by more informative ones. 

Unused messages play an important role in the replacement of
equilibria. They form natural entry points for candidate
replacement equilibria. The following result shows that for any
element of an {\em
EER} set there exists an essentially equivalent equilibrium that
is also an element of the EER set and does not use a certain number
of messages.\footnote{This and the following result are
equivalent to results in Blume, Kim and
Sobel [1993].}
\begin{lem} If $\Theta$ is an EER set of a 
sender-receiver game, then for any strategy $\sigma \in \Theta ,$
there exists a strategy $\sigma' \in \Theta$ such that (1)
$\sigma_2 ' = \sigma _2,$ (2) $\sigma_1 ' (m,t) > 0 \Rightarrow
\sigma_1  (m,t) > 0,$ and (3) and the sender assigns probability
zero to at least $\# (T)$ messages under $\sigma '$.
\end{lem}
\begin{proof}
Let $\Theta$ be an {\em EER} set and $\sigma \in \Theta.$ Call
messages $m_i$ and $m_j$ equivalent if
$$\{t : \sigma_1 (m_i , t) > 0 \}  = \{t : \sigma_1 (m_j , t) > 0
\},$$ and
$$\{a : \sigma_2 (a , m_i) > 0 \}  = \{a : \sigma_2 (a , m_j) > 0
\}. $$ Call a
message $m_i$ redundant if $\sigma_1(m_i ,  t) = 0 , ~ \forall t ,$
or if there exists an equivalent message $m_j$ with $j <i .$ The
cardinality assumption on $M$ implies that there are at least $\#
(T)$ redundant messages. From $\sigma$ construct $\sigma'$ by
having the sender move all weight from redundant messages to the
lowest index equivalent message; leave $\sigma_2' = \sigma_2 .$ It
is easily checked that $\sigma '$ is a Nash equilibrium and
$C(\sigma ') \subset BR( \sigma ). $ Thus $\sigma ' \in \Theta.$
\end{proof}

Lemma 2 says that we can replace any element of an {\em EER} set by
one that maintains the same separation of types, is payoff
equivalent for the sender and leaves a large number of messages
unused. The following result states that the resulting strategy can
in turn be replaced by one that alters the receiver's responses
after  unused messages. The new response after an unused message
can be any reply that supports the equilibrium.   

The definition of {\em EER} sets implies
\begin{lem} Let $\Theta$ be an {\em  EER} set in a sender-receiver
game, $\sigma \in \Theta ,$ and $\sigma_1 (\overline m , t)$ $ =0
~ \forall t.$ For any $m' \not= \overline m,$ the strategy
$\sigma'$
defined by $\sigma_1' := \sigma_1,$ $\sigma _2 ' ( \cdot , m) :=
\sigma _2 ( \cdot , m)$ $\forall m \not= \overline m , $ and
$\sigma
_2 ' ( \cdot , \overline m ) := \sigma_2 ( \cdot , m') $ is an
element of $\Theta .$  \end{lem}

If there is a sufficiently strong alignment of interests between
senders and receivers,
Lemma 2 and 3 can be used to construct an effective mechanism for
the replacement of equilibria that do not utilize this alignment.
First, using Lemma 2, the original equilibrium is replaced by one
in
which the receiver uses identical responses, the sender types who
are separated in the original equilibrium remain separated and at
least $T$ messages are unused. Second, using Lemma 3 the receiver's
responses after unused messages are changed to a reply to a message
that some pooled set of types use in equilibrium. Third, suppose
the alignment of interests is strong enough that additional
separation of pooled types is possible without the need to undo any
of the existing separation of types. Then, a replacement
equilibrium exists in which the newly separated types use the
``unused messages," the other types use the messages they used
before, the receiver responds to the old positive probability
messages as before and uses a best reply that reflects the
additional separation of types after the newly activated messages.

\section{Equilibrium Partial Common Interest}

This section defines {\em equilibrium partial common
interest (EPCI)} relative to an equilibrium and argues that if an
equilibrium is EPCI
dominated via a partition  $J=\{J_i \}, ~
i=1, ... , j,$ of the type set $T$, then (1) it is not part of any
{\em EER} set, and (2) there exists an {\em EER} set containing
only equilibria in which members of different elements of the
partition $J$ send different messages.

{\em EPCI} is one way of formalizing the notion of partial
alignment of interests between the sender and receiver. In keeping
with the postulate of this paper of the primacy of equilibrium
behavior, {\em EPCI} focuses on the benefits to the sender from
being revealed as the member of some type set in some equilibrium.
Roughly, other equilibria that undo the revelation either through
misrepresentation (in terms of the status quo equilibrium) or
through pooling formerly separated types must not be more
attractive. 
The idea of {\em EPCI} is that if the receiver uses best replies to
equilibrium strategies of the sender, then (1) types in $J_i$
prefer being
identified as members of $J_i$ rather than as members of $J_l,~i
\not= l$ and that (2) for any set $K$ containing types from
multiple elements of $J $ there is at least one type who prefers to
be identified as a member of his own element rather than as a
member of K.

In order to define equilibrium partial common interest, we need a
few
preliminary definitions. Define $\phi (J) ,$ the {\em set of
separating
strategies} relative to $J,$ as the set of Nash equilibrium
strategies $\sigma = (\sigma _1 , \sigma _2)$ such that, if
$\sigma
_ 1 (m , t_i ) > 0$ for $t_i  \in J_i , $ then $\sigma _1 (m,t) =
0,~ \forall ~ t \not\in J_i .$ For $L \subset T,$ let ${\cal N}
(L)$ be 
the set of Nash equilibria $\sigma$ such that there exists a
partition $J$ with $L \in J$ and $\sigma \in \phi (J).$ This is the
set of equilibrium strategies in which the set of types $L$ is
identified. Let $P(L)$
be the set of Nash equilibria $\sigma$ such
that $\exists m : \sum_{s \in T} \sigma_1(m,s) > 0$ and $ (
\sigma(m,t) > 0 \Rightarrow t \in L ).$ This is
the set of Nash equilibria in which there is a message that is
exclusively sent by types in $L.$ 

Whenever ${\cal N}(L)$ is nonempty, we can define
$$\underline v_1^{eb}(t;L) := \min_{\sigma_2}\max_{\alpha} \{v_1(t,
\alpha) \vert \theta \in {\cal N}(L),~\sigma_1 = \theta_1,~\sigma_2
\in MBR_2( \sigma_1), ~ \exists s \in L ,$$ 
$$
~m \in M , \sigma_1 (m,s)
> 0 , \alpha = \sigma_2 ( \cdot ,m) \}.$$ 
$\underline v_1^{eb}(t;L)$ is the payoff type $t$ can guarantee for
himself if the receiver best responds to an equilibrium strategy
$\sigma_1$
in which the set of types $L$ is separated and if the sender uses
messages that have positive probability for types in $L$ under
$\sigma_1.$ The value $\underline v_1^{eb}(t;L)$ is a lower bound
on type $t$'s payoff if he chooses to remain pooled with the type
set $L.$ The construction of this lower bound recognizes that in
any candidate replacement equilibrium, following a message that is
a positive probability messages under the status quo equilibrium,
the receiver has {\em equilibrium beliefs,} i.e. beliefs dictated
by the status quo equilibrium, and best responds to those beliefs.

With these preliminary definitions we can define equilibrium
partial common interest relative to a reference equilibrium.

\begin{dfn}
A sender-receiver game has $(J; \sigma ^0)$-equilibrium partial
common interest relative to an equilibrium $\sigma^0$ ($(J; \sigma
^ 0)$-EPCI) if there exists a
partition $J = \{ J_i\}_{i=1}^j$ of $T$ such that 
\begin{enumerate}
\item[{[0]}] $\phi(J) \not= \emptyset $
\item[{[1]}] $\forall  \sigma \in \phi (J),~ V_1(t_i, \sigma ) >
\sum_{a \in A} v_1(t_i , a) \sigma
_2 (a , m), ~ \forall ~t_i \in J_i,~ \forall J_i, ~\forall m$ \\ $
~\mbox{such that}~ \exists t_l \in J_l , i \not= l, ~ \mbox{with}~
\sigma _1 (m, t_l) > 0.$
\item[{[2]}] $\exists \sigma \in \phi (J)$ such that $\forall J_i
, \exists
m_i
: t_i
\in J_i \Rightarrow \sigma_1 (m_i , t_i ) = 1 ~\mbox{and}~
V_1(t_i,\sigma) \geq V_1 (t_i , \sigma ^ 0), $
\item[{[3]}] If $K \cap J_l \not= \emptyset$ for at least two $l,$
then
$\forall \sigma \in P(K), \exists i , t_i \in K \cap J_i :
\underline v_1^{eb}(t_i; J_i) > V_1(t_i, \sigma),$ and 
\item[{[4]}] $\forall i , \exists m_i $ such that $ J_i \subset \{t
:
\sigma_1^0(m_i, t) > 0 \}.$
\end{enumerate}
\end{dfn}

This definition of equilibrium partial common interest is with
respect to a reference equilibrium, $\sigma^0 .$ $(J; \sigma
^ 0)$-EPCI requires that $\sigma ^0$ be ``dominated" by a class of
equilibria that separate the partition $J.$ Domination requires
that types belonging to the same element of the partition send a
common message in the dominated equilibrium, [4], and that there
exists a dominating equilibrium that separates $J$ in which types
belonging to the same element
of the partition send a common message that yields payoffs at least
as large as $\sigma^0,$
[2]. Jointly, these conditions make it possible that types
belonging to a common element move from one common message that
does not identify them to an identifying message without being
penalized for doing so. Once $J$ is separated, it is an optimal
response for the receiver to accommodate the separation. And since
each sender type's payoffs do not decrease by moving to the 
$J$-separating equilibrium the sender has no incentive to continue
using the positive probability messages of the reference
equilibrium. This is how $J$-separation can become established,
starting with the reference equilibrium. It remains to ensure, that
$J$-separation, once achieved will not become undone. This is
accomplished by conditions [1] and [3]. The former guarantees
that in any $J$-separating equilibrium types in one element of the
partition strictly prefer not to mimic types in another element.
Thus, a $J$-separating equilibrium cannot be replaced by one in
which a type switches to a positive probability message of another
partition element. Finally, condition [3] ensures that in any
replacement equilibrium, types who start separated remain
separated. If members from across different partition elements were
to pool on an unused message in a candidate replacement
equilibrium, the replacement condition would
require the receiver to reply with a best response that is part of
an equilibrium. Because the replacement condition requires the
receiver to use a
best reply against the status quo, the sender of type $t_i \in J_i$
can guarantee himself $\underline v_1^{eb}(t_i; J_i)$ by using one
of the
status quo messages. According to condition [3] there is at least
one type in any set that pools types across partition elements, who
prefers $\underline v_1^{eb}(t_i; J_i)$ to the payoff he would
receive from following the candidate replacement equilibrium.

This discussion is summarized in the following proposition.
\begin{prp} If $G$ is a sender-receiver game with 
$(J; \sigma^0)$-EPCI, then
\begin{enumerate}
\item[{[5]}] $\phi (J)$ contains an EER set, and
\item[{[6]}] $\sigma ^ 0$ is not a member of any EER set.
\end{enumerate}  
\end{prp}
\begin{proof}
To prove [5], consider $\sigma \in \phi (J).$ Let $R(\sigma)$ be
the smallest closed set containing $\sigma$ that satisfies [R].
Clearly, $R (\sigma)$ contains an {\em EER} set. Also, $R(\sigma)
\subset \phi (J) .$ To see this, let $\sigma \in \phi(J)$ and
let $\sigma '$ be a replacement strategy. It will be the case that
types in different elements of {\em J} use different message under
$\sigma ' :$ If $m$ is a positive probability message under
$\sigma$
for types in $J_l,$ then under $\sigma ',$ $m$ will not be used by
types in $J_i,~l
\not= i,$ because of [1] and because $\sigma _1'$ must be a best
reply to $\sigma_2.$ Suppose then that $m$ is a message such that
$\sigma _ 1 (m,t) = 0 ~ \forall t$ and let $K \cap J_l \not=
\emptyset $ for at least two $l$ where $K= \{ t : \sigma _1 '(m,t)
> 0 \}.$ [R] implies first that $\sigma_2' ( \cdot , m)$ is an
equilibrium reply to a positive probability message that induces
beliefs concentrated on $K,$ i.e. $\sigma ' \in P(K),$ and second
that
$\sigma _ 2 '$ is a best reply to $\sigma _1,$ such that a type
$t_l \in J_l$
sender can guarantee himself at least $\underline v _1 ^{eb}
(t;J_i)$ by following $\sigma _ 1 ,$ and will receive at most $V_1
(t_l , \sigma ')$ from following $\sigma '.$ Hence (3) implies that
there is at least one type in $K$ who prefers following $\sigma _1$
to following $\sigma_1 '$ against $\sigma_2 '.$ This inconsistent
with $\sigma '$ being an equilibrium, which contradicts [R]. Thus,
$\sigma ' \in \phi (J).$ Also, if $\overline Q$ is the closure of
$Q \subset \phi(J),$ then $\overline Q \subset \phi (J).$ Claim (5)
follows.

To prove [6], suppose that $\sigma^0 $ is a member of an {\em EER
} set. We will derive a contradiction by showing that then
$R(\sigma
^ 0) \cap \phi (J) \not= \emptyset.$ Lemmata 2 and 3 imply that it
is without loss of generality to assume that there are $j$ unused
messages $\tilde m_i,~ i=1,...,j,$ under $\sigma^0$ such that
$\sigma _2 ^ 0 ( \cdot , \tilde m _i) = \sigma _2 ^0 ( \cdot , m_i
),~ i=1,...,j,$ where $m_i,~ i=1,...,j$ are the messages identified
in [4]. [2] implies that there exists $\sigma ' \in \phi (J)$ in
which the sender signals only with the unused messages of $\sigma
^ 0,$ such that if $t_i \in J_i , $ then $\sigma_1 ' (\tilde m _i
, t _ i) = 1 ,$ and the receiver responds to the positive
probability
messages of $\sigma ^ 0$ as he does under $\sigma ^ 0.$ These 
off-the-equilibrium path responses support the equilibrium $\sigma
'$ because of [2]. Thus $\sigma ' \in \phi(J) \cap MBR(\sigma ^
0),$
which implies that $\sigma ' \in R(\sigma ^ 0 ) \cap \phi (J).$ 
\end{proof}

The following game illustrates the equilibrium partial interest
condition and demonstrates how {\em EER} sets strengthen
predictions in games with partial alignment of interests. Let types
be equally likely. The first entry in each cell denotes the
sender's payoff, the second
entry is the receiver's payoff.

{{\footnotesize
\begin{center}
\begin{picture}(300,100)
\put(50,15){\framebox(200,75)}
\put(50,40){\line(1,0){200}}
\put(50,65){\line(1,0){200}}
\put(90,15){\line(0,1){75}}
\put(130,15){\line(0,1){75}}
\put(170,15){\line(0,1){75}}
\put(210,15){\line(0,1){75}}
\put(35,75){$t_1$}
\put(35,50){$t_2$}
\put(35,25){$t_3$}
\put(68,95){$a_1$}
\put(108,95){$a_2$}
\put(148,95){$a_3$}
\put(188,95){$a_4$}
\put(228,95){$a_5$}
\put(65,75){x,0}
\put(105,75){2,7}
\put(145,75){4,4}
\put(185,75){0,0}
\put(225,75){0,3}
\put(65,50){2,7}
\put(105,50){x,0}
\put(145,50){4,4}
\put(185,50){0,0}
\put(225,50){0,3}
\put(65,25){0,0}
\put(105,25){0,0}
\put(145,25){0,0}
\put(185,25){4,4}
\put(225,25){0,3}
\end{picture}
\end{center}}
\begin{center}
{Figure 1: Game 1}
\end{center}}
Independent of the value of $x,$ there exists an {\em EER} set in
which types $t_1$ and
$t_2$
separate from type $t_3$ in every strategy that is part of the set.
Also, the pooling outcome is not part of any {\em EER} set. Other
evolutionary solutions (discussed below) either do not exist or
predict communication only if $x$ is positive. Intuitively, if
nonequilibrium strategies are permitted as part of the solution,
then for $x < 0$ types $t_1$ and $t_2$ cannot be assured of 
positive payoffs, even if the set $\{t_1 , t_2 \}$ is separated
{\em and} the receiver responds accordingly.   



\section{Partial Common Interest}

The evaluation of the {\em EPCI} condition requires that the entire
set of Nash equilibria be determined. However, often it is possible
to make predictions on the basis of a simpler and more transparent
condition.\footnote{This is convenient also for the purpose of
parameterizing different incentive structures in game experiments.
} In this section we propose a slightly stronger partial
common interest condition {\em (PCI)} that avoids the need to
determine the
entire set of Nash equilibria. This concept is intermediate
between {\em EPCI} and partial common interest definitions
available
in the literature. 

{\em PCI} formalizes the requirement that if the sender uses a
strategy in which the elements of the partition $J$ are identified
and if the receiver uses a best reply to the sender's strategy,
then (1) $t_i \in J_i$ has no incentive to misrepresent himself as
a member of $J_l$, and (2) a set of types $K$ combining members
from different elements of the partition cannot all gain if the
receiver uses a best reply to beliefs concentrated on $K.$ 

For separation of the partition $J$ not to unravel, once it is
established, it is necessary that the minimum payoff from being
identified as the member of an element $J_i$ not be too low. In
the previous sections these bounds where determined by the fact
that players use equilibrium strategies. Here we will consider
bounds that only use the fact (given  the {\em EER} replacement
condition) that the receiver uses best replies to a strategy of the
sender.

If types separate according to
the partition $J$, then there {\em always} exists a message that
allows $t_i \in J_i$ not only to identify himself as a member of
$J_i$ but to rule out certain classes of beliefs over $J_i .$ The
following lemma verifies this claim. Let
$L_T^c$ denote the complement of $L$ in $T,$ and for $K \subset T
,$ let $\mu (K , m ; \sigma _ 1 )$ denote the posterior probability
of the type set $K$ given message $m$ for strategy $\sigma _ 1 .$

\begin{lem} Let $L \subset T $ and suppose that for $t_i \in L,$
$\sigma _ 1 (m,t_i) > 0  $ implies $\sigma_1 (m,t_j) = 0 ~\forall
~ t_j \in L_T^c$
and for $t_i \in L_T^c,$
$\sigma_1 (m,t_i) > 0  $ implies $\sigma_1 (m,t_j) =
0
~\forall ~ t_j
\in L.$ Then, $\forall ~ K \subset L,
~\exists~m ~:~ \mu (K,m; \sigma_1 ) \geq {\pi (K) \over \pi(L)}. $ 
\end{lem}
\begin{proof} Suppose not. Let $\tilde M := \{ m \in M : \sigma_1
(m
, t) > 0, ~ t \in L \}.$ Then $\forall m \in \tilde M$ 
\begin{center}
$\mu(K , m; \sigma_1 ) = {\sum_{t \in K} \sigma_1 (m,t) \pi(t)
\over
\sum_{t \in K} \sigma_1 (m , t) \pi (t) + \sum_{t \in K_L^c}
\sigma_1 (m , t) \pi(t)  } < { \pi (K) \over \pi (L)} \Rightarrow
$

$ \sum_{t \in K} \sigma_1 (m,t) \pi(t) < \Bigl( \sum_{t \in K}
\sigma_1 (m , t) \pi (t) + \sum_{t \in K_L^c} \sigma_1 (m , t)
\pi(t)
\Bigr)
{\pi (K) \over \pi (L)} ~~ $ \\ $\forall m \in \tilde M  ~
\Rightarrow
$

$ \sum_{m \in \tilde M } \sum_{t \in K} \sigma_1 (m,t) \pi(t) <
\sum_{m \in \tilde M}  \Bigl( \sum_{t \in K} \sigma_1 (m , t) \pi
(t)
+ \sum_{t \in K_L^c} \sigma_1 (m , t) \pi(t) \Bigr) {\pi (K) \over
\pi
(L)} ~ \Rightarrow   $

$ \sum_{t \in K} \sum_{m \in \tilde M } \sigma_1 (m,t) \pi(t) < 
\Bigl( \sum_{t \in K} \sum_{m \in \tilde M} \sigma_1 (m , t) \pi
(t)
+
\sum_{t \in K_L^c} \sum_{m \in \tilde M} \sigma_1 (m , t) \pi(t)
\Bigr)
{\pi (K) \over \pi (L)} ~ \Rightarrow   $

$\pi(K) < \pi(K).$
\end{center}
Thus we have established a contradiction.
\end{proof} 

In view of this result we can provide a lower bound on the payoff
of a type who is identified as a member of $J_i$ by a strategy
profile
$\sigma_1$ that separates $J$ and in which the receiver uses a
best
reply. 

For any $K \subset L \subset T,$ let
$$BR_2 (K,L) := \arg\max \Bigl\{ \sum_{a \in A} \sum_{t \in L } v_2
(t ,a
) \mu (t) \alpha (a): ~~~~~~~~~~~~~~~~~~~~~ $$
$$~~~~~~~~~~~~~~~~~~~~~~~~~~~
{\rm
supp}
(\alpha) = A , {\rm supp}
(\mu) = L, {\rm and}~ \mu (K ) \geq {\pi (K) \over \pi (L)}
\Bigr\}. $$ $BR_2 (K,L)$ is the set of best replies of the receiver
against beliefs concentrated on $L$ that put at least weight
$\pi(K) \over \pi(L)$ on the subset $K$ of $L.$
Let $${\underline v}_1 (t ; L) := \max_{K \subset L} \min_\alpha 
\{ v_1(t, \alpha): \alpha \in BR _2 (K,L) \}. $$

From the lemma, if types in $L$ are identified as such by their
messages
and the receiver uses a best reply, then for any $K \subset L ,$
the
sender can always find a message that
induces a reply in $BR _2 (K,L).$ The definition of ${\underline
v}_1 (t ; L)$ recognizes the fact that the sender can ``choose"
among $K.$ If types in $L$ are identified as such by their messages
and the
receiver uses a best reply, then
${\underline v}_1 (t ; L)$ is the payoff that a type $t$ sender
can guarantee himself by continuing to identify himself as
belonging to  $L.$

This motivates the following definition of partial common interest.
For $J_i \subset T,$ let $BR_2(J_i)$ be the set of best replies of
the receiver against beliefs concentrated on $J_i,$ and let
$BR_2(J_i;\pi)$ be the set of best replies against prior beliefs
restricted to $J_i.$ Denote the corresponding mixed best replies by
$MBR_2 ( \cdot ).$

\begin{dfn} A sender-receiver game G has $(J, \sigma^0)$-partial
common interest
($(J, \sigma^0)$-PCI) relative to an equilibrium $\sigma ^0$ if
there exists a partition $J= \{ J_i\},~ i=1, ... , j, $ of $T$ such
that 
\begin{enumerate} 
\item[{[7]}] ${\underline v}_1 (t_i ; J_i) > \max \{ v_1 (t_i ,
a_l)
: a_l
\in
BR_2 (J_l) \} $ for all $t_i \in J_i$ and $i \not= l .$ 
\item[{[8]}] for all $i=1,...,j,$ there exists $a_i \in BR_2 (J_i
; \pi)$
such that \\ $v_1
(t_i, a_i)
\geq V_1(t_i, \sigma^0)$ for all $t_i \in J_i,$ 
\item[{[9]}] if $K \cap J_l \not= \emptyset$ for at least two $l
,$
then
for each $\alpha \in MBR_2 (K)$ there exists $i$ and $t_i \in K
\cap
J_i$
such that ${\underline v}_1 (t_i ; J_i) >  v_1 (t_i , \alpha),$ and
\item[{[10]}] $\forall i,~ \exists m:~ J_i \subset \{ t: \sigma _
1
(m,t)
> 0 \}.$
\end{enumerate}
\end{dfn}

Comparing Definitions 2 and 3, it is easily verified that (1)
$\underline v_1 (t_i ; J_i) \leq
V_1(t_i , \sigma) \forall \sigma \in \phi (J) \forall t_i \in J_i$
(2) $\sum_{a \in A} v_1(t_i ,a) \sigma_2 (a,m) \leq \max \{
v_1 (t_i , a_l) : a_l \in BR_2 (J_l) \}, \forall t_i \in J_i ,
\forall J_i , \forall m$ $~\mbox{such that}~ \exists t_l \in
J_l, i \not= l, ~ \mbox{with} ~ \sigma_1 (m, t_l) > 0, ~\forall
\sigma \in
\phi(J),$ and thus [7] implies
[1], and [7] and [8] imply [2]. (3) [9] implies [3] because (1)
the set of best replies against beliefs concentrated on a set $K$
is
a superset of the set of equilibrium best replies against such
beliefs and (2) $\forall K, \sigma$ we have $\min_\alpha \{ v_1(t,
\alpha) \vert \alpha \in BR_2(K,L) \leq \max_\alpha \{v_1(t,
\alpha) \vert \theta \in {\cal N}(L) ,~\sigma_1 =
\theta_1,~\sigma_2
\in MBR_2( \sigma_1), ~ \exists s \in L , ~m \in M , \sigma_1 (m,s)
> 0 , \alpha = \sigma_2 ( \cdot ,m) \}  \}$. Thus we
have the following corollary to Proposition 2.

\begin{cor} If $G$ is a sender-receiver game with 
$(J; \sigma^0)$-PCI, then
\begin{enumerate}
\item[{[11]}] $\phi (J)$ contains an EER set, and
\item[{[12]}] $\sigma ^ 0$ is not a member of any EER set.
\end{enumerate}  
\end{cor}

\section{Entry as Replacement}

{\em EER} sets resemble static set-valued solution concepts from
evolutionary game theory. This section examines the relationship in
greater detail. It is shown that in two-player games, as for
example in 
sender-receiver games, the essential difference is in
turning the entry condition of the evolutionary solution concepts
into a replacement condition.  

Recall the following definition of {\em equilibrium evolutionarily
stable (EES)} sets, due to Swinkels [1992]. 

\begin{dfn}
A set $\Theta \subset \times_{i=1}^n \Delta (S_i)$ is {\em
equilibrium evolutionarily stable (EES),} if it is minimal with
respect to
\begin{enumerate}
\item [{[13]}] There exists $\epsilon ' \in (0,1)$ such that for
all
$\epsilon \in (0, \epsilon ')$ and for all $\sigma \in \Theta,$ \\
if $C(\sigma ') \subset BR ((1- \epsilon)\sigma + \epsilon \sigma
'), $  then $(1- \epsilon)\sigma + \epsilon \sigma ' \in \Theta.$
\item [{[14]}] $\Theta$ is closed and nonempty.
\item [{[15]}] $\Theta \subset {\cal N}$
\end{enumerate}
\end{dfn}   

An {\em EES} set is a closed set of Nash equilibria that meets an
entry condition, [13]. Blume, Kim and Sobel (BKS) [1993] showed
that {\em  EES} sets predict communication in sender-receiver games
of common interest, i.e. when there exists a unique 
Pareto-efficient point. {\em EES} sets need not exist in every game
and fail to exist in large classes of partial common interest
games. To address the existence issue BKS propose to drop condition
[15]. They call sets satisfying the remaining requirements {\em ER}
sets. {\em ER} sets are fairly permissive with regard to receiver's
actions; even if senders are
locked into separation according to a partition {\em J,} the
senders' actions after positive probability messages need not fully
reflect this separation. 

Consider Game 2 below, with types being equally likely. The
first entry in each cell denotes the sender's payoff, the second
entry is the receiver's payoff.

{\footnotesize
\begin{center}
\begin{picture}(250,80)
\put(50,15){\framebox(160,50)}
\put(50,40){\line(1,0){160}}
\put(90,15){\line(0,1){50}}
\put(130,15){\line(0,1){50}}
\put(170,15){\line(0,1){50}}
\put(35,50){$t_1$}
\put(35,25){$t_2$}
\put(68,70){$a_1$}
\put(108,70){$a_2$}
\put(148,70){$a_3$}
\put(188,70){$a_4$}
\put(65,50){2,4}
\put(105,50){4,4}
\put(145,50){0,0}
\put(185,50){0,3}
\put(65,25){0,0}
\put(105,25){0,0}
\put(145,25){3,4}
\put(185,25){0,3}
\end{picture}
\end{center}}
\begin{center}
Figure 2: Game 2
\end{center}

This game illustrates both the nonexistence problem of {\em EES}
sets and that in an {\em ER} set sender separation need
not imply that the receiver uses only separating replies.  

The pooling component of Nash equilibria can be invaded by a
separating equilibrium and thus cannot be part of an {\em EES} set.
The only other equilibria in this game are separating equilibria.
In such an equilibrium the receiver is indifferent among actions
$a_1$ and $a_2$ following any message that identifies type $t_1.$
A separating equilibrium $\sigma$ can invade where the receiver
uses only action $a_1$ after positive probability messages that
identify $t_1$ and action $a_4$ after zero probability messages.
This invasion can continue until the entire population plays
according to $\sigma .$ Our assumption on message space size
ensures that it is without loss of generality to assume that there
is at least one unused message, $m ,$ under $\sigma.$ $\sigma$
can  be invaded by a strategy $\sigma '$ that is identical to
$\sigma
,$ except that following $m ,$ the receiver responds with an equal
probability mixture over $a_2$ and $a_4.$ The invasion can continue
until $\sigma'$ represents population play. $\sigma '$ can be
invaded by a strategy $\sigma ''$ in which $t_2$ uses the same
messages as before, $t_1$ uses only $m,$ the receiver responds to
all messages other than $m$ as before, and responds to $m$ with
action $a_2.$ However, $(1-\epsilon ) \sigma' + \epsilon \sigma ''$
is not a Nash equilibrium for any $\epsilon \in (0,1).$ This
contradicts the requirement that {\em EES} sets consist only of
Nash
equilibria. 

One can show that there exists an {\em ER} set in this
game that contains a (separating) Nash equilibrium, $\sigma .$
From
there one can argue as above, except that now the strategy
$(1-\epsilon ) \sigma' + \epsilon \sigma ''$ becomes part of the
{\em ER} set for some $\epsilon \in [0,1].$ Because there is
positive weight on $\sigma '' ,$ $m$ is used with positive
probability, and because there is positive weight on $\sigma ' ,$
the message $m$ induces action $a_4$ with positive probability.  

The approach to preserving existence in this paper is different.
The requirement that solutions must consist of Nash equilibria is
maintained. What changes, is the entry condition [13]. It changes
in two ways; (1) entering strategies must be Nash equilibria and
(2) a strategy that passes the entry requirement replaces the
present population. This is best seen in the following version of
the definition of {\em EER} sets for two-player games. The
remainder of this section is concerned with two-player games.

\begin{dfn}
A set $\Theta \subset \Delta (S_1) \times \Delta (S_2)$ is {\em
equilibrium entry resistant (EER),} if it is minimal with
respect to
\begin{enumerate}
\item [{[16]}] There exists $\epsilon ' \in (0,1)$ such that for
all
$\epsilon \in (0, \epsilon ')$ and for all $\sigma \in \Theta,$ \\
if $\sigma' \in {\cal N}$ and $C(\sigma ') \subset BR ((1-
\epsilon)\sigma + \epsilon \sigma
'), $  then $\sigma ' \in \Theta.$
\item [{[17]}] $\Theta$ is closed and nonempty.
\item [{[18]}] $\Theta \subset {\cal N}$
\end{enumerate}
\end{dfn}   

In two-player games, Definitions 1 and 5 of {\em EER} sets are
equivalent. To show this
we have to establish the equivalence of [16] and [R]. To see that
[R] implies [16], let $\sigma \in  \Theta$ and let $\sigma '$
satisfy $\sigma ' \in {\cal N}$ and $C(\sigma ') \subset BR ((1-
\epsilon) \sigma + \epsilon \sigma ')$ for $\epsilon$ sufficiently
small. Then it must be that $C(\sigma ') \subset BR (\sigma) \cap
BR (\sigma ').$ Thus, if $\Theta$ satisfies [R] it satisfies [16]
as well. The converse follows from the following lemma.

\begin{lem}  Let  $C(\sigma ')
\subset BR(\sigma) \cap BR(\sigma ').$ Then 
$C( \sigma ' ) \in BR( \lambda \sigma
+ (1- \lambda) \sigma' ), \forall \lambda \in [0,1].$
\end{lem}

\begin{proof}
Suppose not. Then there exists
$\sigma '',~ i \in \{1,2\}$ such that $V_i (\sigma_i'', \lambda
\sigma_{-i} + (1- \lambda) \sigma_{-i}')$ $=\lambda V_i (\sigma_i''
, \sigma_{-i}) + (1- \lambda) V_i (\sigma _i'' , \sigma _{-i}')$
$>\lambda V_i (\sigma_i' , \sigma_{-i}) + (1- \lambda) V_i (\sigma
_i' , \sigma _{-i}'),$ which implies that we have either $V_i(
\sigma_i'' , \sigma _{-i}) > V_i( \sigma_i' , \sigma _{-i}) $ or we
have $V_i( \sigma_i'' ,$ $\sigma _{-i}') > V_i( \sigma_i' , \sigma
_{-i}') .$ Either inequality contradicts $C( \sigma ') \subset BR
( \sigma ) \cap BR (\sigma ').$
\end{proof}


According to Definition 5 an {\em  EER} set is a closed set of
Nash equilibria that cannot be replaced by entrants that (1) are
themselves equilibria and (2) are best replies to a post-entry
population consisting mainly of the original population and a small
fraction of entrants. This is close to the definition of {\em
weakly
equilibrium evolutionarily stable (WEES)} sets of Kim and Sobel
[1992]. WEES replaces condition [16] by: 
\begin{enumerate} 
\item [{[19]}] There exists $\epsilon '
\in (0,1)$ such that for all
$\epsilon \in (0, \epsilon ')$ and for all $\sigma \in \Theta,$ \\
if $\sigma' \in {\cal N}$ and $C(\sigma ') \subset BR ((1-
\epsilon)\sigma + \epsilon \sigma
'), $  then $(1- \epsilon)\sigma + \epsilon \sigma ' \in \Theta.$
\end{enumerate}
Thus {\em EER} sets differ from {\em WEES} sets only in that the
entry condition takes the form of a replacement condition. Both
concepts have the potential drawback that the sets need not contain
a proper equilibrium. {\em  WEES} sets need not exist in general
whereas {\em EER} sets do.

\begin{prp}
Every EES set and every WEES set contain an EER set. If an ER set
contains a Nash equilibrium, the ER set contains an EER
set.\footnote{Lemma 5 and Proposition 3 are only valid in 
two-player games. In the appendix an example is given in which an
{\em EES} set does not contain an {\em EER} set as defined in
Definition 1.} \end{prp}

The proof cannibalizes a result in Kim and Sobel [1992].

\begin{proof} Let $\sigma \in {\cal N} \cap \Theta $ where $\Theta
$ is either an
EES, a WEES or an ER set. If $C(\sigma ') \in BR ( \sigma)
\cap
BR (\sigma ') ,$ then $\sigma ' \in \Theta.$ This can be seen as
follows. By Lemma 5, $C( \sigma
' ) \subset BR ((1- \lambda) \sigma + \lambda \sigma ' ) ~ \forall
\lambda.$ Let $\epsilon '' = \sup \{ \epsilon ' : (1- \epsilon )
\sigma + \epsilon \sigma ' \in \Theta ~ \forall \epsilon \in (0,
\epsilon ') \}.$ $\epsilon '' $ could depend on whether $\Theta$ is
an EES, a WEES or an ER set. By [13], for EES and ER sets, or [19]
for WEES sets, $\epsilon '' >0 .$ Closedness of $\Theta$ implies
that $\sigma '' := (1- \epsilon '') \sigma + \epsilon '' \sigma'
\in
\Theta .$ By Lemma 5, $C (\sigma ') \subset BR( (1- \lambda)
\sigma '' + \lambda \sigma ') ~\forall \lambda \in [0,1].$
Therefore [13], for EES and ER sets, and [19] for WEES sets,
implies that $\epsilon '' = 1. $ Hence $\sigma ' \in \Theta.$ Now
consider $R( \sigma),$ the smallest closed set containing $\sigma$
that satisfies [R]. The foregoing showed that $R( \sigma ) \subset
\Theta.$ Combining this with the fact that $R(\sigma)$ contains an
EER set establishes the result.
\end{proof}

\section{Selection in Crawford-Sobel Games}

Crawford and Sobel (CS) [1982] launched the literature on cheap
talk
games with a parametric model of incentive alignment. Variations 
of their model have played an important role in applied work on
cheap talk in accounting, Newman and Sansing [1993] and Gigler
[1994], political science, Austen-Smith [1990], and economics,
Stein [1989]. Equilibria in CS's model have a
simple structure and an intuitive comparative statics result holds.
In every equilibrium, the sender's strategy partitions the
private information, the unit interval, into finitely many
intervals. The more closely incentives are aligned, the finer is
the partition under the most influential equilibrium\footnote{This
terminology is adapted from Austen-Smith and Banks [1995]}, the
equilibrium that induces the maximum number of distinct responses,
and in that sense the more information is revealed in equilibrium.

The applied literature has primarily focussed on the most
influential equilibria. This is sometimes justified by appealing to
the ex ante efficiency of these equilibria. No other argument, of
either the refinement sort, or an evolutionary one has ever been
given for why the most influential equilibria ought to be
selected or even for why the ever present pooling equilibria ought
to be rejected.

The CS game has been notoriously impervious to any
refinement or evolutionary analysis. Farrell [1993], for example
pointed
out that in CS's quadratic example (discussed below), whenever a
nontrivial partition equilibrium exists, none of the equilibria in
the game passes the neologism proofness test. EES sets typically
also do not exist in versions of the game with a plausible
discretization of the type space. ER sets are plainly nontractable
and, as we saw above, do not lead to plausible restrictions of
receiver behavior. This makes it interesting to ask
whether it is possible to say something about the CS-game within
our framework. A second reason for bringing up CS-games is that the
notion of partial alignment of interests in those games is quite
different from the various {\em PCI} conditions used here and
elsewhere in the literature. The alignment conditions in the
evolutionary/refinement literature are stronger than in the
CS-game. This ought to make it harder to reject uninformative
outcomes in CS-games using the approach taken in this paper.

It is desirable to have a unified approach to both {\em PCI}
and CS-games. This section shows that a modification of our
solution concept indeed rejects pooling equilibria in {\em both}
CS-games
and under a strengthened {\em EPCI} requirement. The modified
solution concept, {\em asymmetric entry resistant
(AER)} set, relaxes the replacement condition in the definition of
EER sets. It permits the
transition to a new equilibrium to be initiated also by a subset of
the
player set rather than only by the entire set.

For any strategy profile $\sigma,$ and player set $K \subset I,$
denote by $\sigma_K$ the partial profile which contains only the
strategies of players in the set $K.$ The complementary partial
profile is indicated by $\sigma _ {-K}.$ Similarly, we write $BR_K$
for partial best response correspondences.

\begin{dfn} A set $\Theta \subset \cal N$ is $m$-asymmetric entry
resistant ($m$-AER) for $m \in \{ 0,...,n \}$, if it is minimal
with
respect to
\begin{enumerate}
\item [{[20]}] $\forall \sigma \in \Theta,$ if $\exists ~K \subset
I,~\#(K) \geq m $ such that $C(\sigma_K ') \subset
BR_K(\sigma) \cap BR_K(\sigma'),$ and $C(\sigma_{-K} ')
\subset BR_{-K} ( \sigma'),$ then $\sigma ' \in \Theta ,$ and
\item [{[21]}] $\Theta$ is closed and nonempty.
\end{enumerate}
\end{dfn}

For $m=n,$ $m${\em -AER} sets are the $EER$ sets defined earlier,
for $m=0$ they coincide with the entire set of Nash equilibria, and
with increasing $m$ the replacement condition becomes increasingly
restrictive. For the consideration of sender-receiver games we will
let $m=1$ and simply refer to {\em AER} sets. This leads to a
simple distinction between {\sl sender-led replacements} of
$\sigma$ by
$\sigma ',$ where $C(\sigma_1') \subset BR_1(\sigma_2)$ (S-
replacements), and
{\sl receiver-led replacements,} where $C(\sigma_2') \subset
BR_2(\sigma_1)$ (R-replacements).

Existence of {\em AER} sets for finite sender-receiver game is
established in the same manner as for {\em EER} sets. It is also
straightforward to show that the results that guarantee
availability of unused messages continue to hold. This can be used
to prove a result on {\em EPCI}-games if one strengthens the {\em
EPCI}-requirement. Modify the {\em EPCI}-definition by replacing
condition [3] by 
\begin{enumerate}
\item[{[3']}] if $K \cap J_l \not= \emptyset$ for at least two $l,$
then for all $\alpha \in \Delta (A),~\exists i,~t_i \in K \cap J_i
: \underline v_1^{eb}(t_i,J_i) > v_1 (t_i , \alpha).$ 
\end{enumerate}

While this is a stringent condition, it is satisfies in many
interesting classes of games. Game 1 is one example.

\begin{prp} If $G$ is a sender-receiver game with 
$(J; \sigma^0)$-EPCI, then
\begin{enumerate}
\item[{[22]}] $\phi (J)$ contains an AER set, and
\item[{[23]}] $\sigma ^ 0$ is not a member of any AER set.
\end{enumerate}  
\end{prp}

\begin{proof} The proof of [23] is identical to the proof of [6].
To verify [22] it is necessary to deal with both
sender-led replacements and receiver-led replacements. To derive a
contradiction, let $\sigma'$ be a candidate replacement for $\sigma
\in \phi(J)$ under which types from different elements of $J$ send
a common message.

Under sender-led replacements, $C(\sigma_1') \subset BR_1
(\sigma_2).$ Against $\sigma_2,$ each type $t_i$ can guarantee
$V_1(t_i, \sigma) \geq \underline v_1^{eb}(t_i,J_i)$ by following
$\sigma_1.$ Condition [3'] ensures that for no strategy of the
receiver, including $\sigma_2,$ does there exists a message $m$
that would at least achieve a payoff of $\underline
v_1^{eb}(t_i,J_i)$ for all types in a set $K = \{t: \sigma_1'(m,t)
>
0 \}$ such that $K \cap J_l \not= \emptyset$ for at least two $l.$ 

Under receiver-led replacements $C(\sigma_2') \subset BR_2
(\sigma_1).$ Thus, type $t_i \in J_i$ can guarantee a payoff of at
least $\underline v_1^{eb}(t_i,J_i)$
against $\sigma_2'.$ According to condition [3'], there does not
exist a message $\tilde m$ and set $K = \{t: \sigma'( \tilde m,t)
> 0 \}$ such that $K \cap J_l \not= \emptyset$ for at least two $l$
and all types $t_i$ in $K$ would at least achieve a payoff of
$\underline v_1^{eb}(t_i,J_i).$ This contradicts $\sigma'$ being an
equilibrium. \end{proof}

In the remainder of this section we will show that AER sets also
have predictive power in CS games and thus get us closer to the
desired
unified treatment of {\em PCI} and CS games. In a sender-receiver
game of the Crawford-Sobel type, the sender's
type $t$ is drawn from a cumulative distribution $F( \cdot ),$ with
density $f( \cdot )$ and support $[0,1].$ The receiver's action is
a real number $a \in \Re.$ The sender's payoff function
$v^1(a,t,b)$ depends on a parameter
$b \in \Re$ that measures the degree of alignment of interests
between senders and receivers. Both the sender's and the receiver's
payoff function, $v^2(a,t),$ are twice continuously differentiable.
Indicating partial derivatives by subscripts, assume that for all
$t$ and $i=1,2,$ $v_1^i (a,t) = 0$ for some action $a,$ $v_{11}^i
< 0$ to guarantee a unique maximum in $a$ for any $(t,b),$ and
$v_{12}^i > 0,$ a sorting condition that ensures that for either
player under full information the optimal action is a strictly
increasing function of the sender's type. Assume that the message
space $M$ is a superset of $[0,1].$

In a {\em partition equilibrium,} the type set can be divided into
intervals (which we represent via their interiors) $(t_j,t_{j+1}),~
j=0,1,..., N-1,$ such that
$t_0=0,
~t_N=1,$ types in
$(t_j,t_{j+1})$ send a common message $m_j$ unique to that type set
and the receiver responds to each $m_j$ with the best reply to
prior beliefs restricted to $(t_j,t_{j+1}).$ Since there always
exists a
pooling equilibrium in which all types send the same message, the
existence of a partition equilibrium is guaranteed.

Let $a^1(t,b) := \arg \max v^1(a,t,b)$ denote the maximizer of the
sender's utility if his type is $t$ and the value of the alignment
parameter equals $b.$ Similarly, for the receiver, $a^2(t) := \arg
\max v^2 (a,t).$ Crawford and Sobel show that if $a^1(t,b) \not=
a^2(t)$ for all $t,$ then (1) in every equilibrium the relationship
between type and action is as in a partition equilibrium, (2) there
exists a partition equilibrium that maximizes the number of
elements of the partition, and (3) if the maximal partition
equilibrium has $N$ elements, then there exist a partition
equilibrium with $n$ elements for all $n \in \{1, ... ,N \}.$

To address the equilibrium selection issue, Crawford and Sobel
impose additional structure on their model. They examine a
quadratic example (that has become the basis for most of the
applications of cheap talk games). In that example $F( \cdot )$ is
uniform on $[0,1],$ $v^1(a,t,b) := -(a-(t+b))^2,$ $v^2(a,t) := -(a-
t)^2,$ and $b>0.$ This example satisfies all of the above
assumptions implying that all equilibria are partition equilibria.
In addition (1) the equilibrium with the maximal number of
partition
elements (the ``finest" equilibrium) is ex ante efficient among the
set of equilibria, and (2) as $b \rightarrow 0$ the finest
equilibrium becomes finer approaching full revelation in the limit.
Crawford and Sobel argue in favor of the finest partition
equilibrium. It is one of the two extreme and therefore distinct 
equilibria, and they find the coarsest equilibrium not to be
plausible. Secondly, the finest equilibrium is ex ante efficient. 

While this rule of selecting the finest partition equilibria has
been adopted widely in the applied literature, no argument besides
distinctness and efficiency has been advanced. We will show below
that with quasi-evolutionary arguments we can at least reject the
completely uninformative equilibria in a version of Crawford and
Sobel's model that generalizes their quadratic model.

To generalize the lessons learned from the quadratic model,
Crawford and Sobel add an additional assumption on the payoff
functions. Let $v^1(a,t,0) \equiv v^2(a,t)$ and $v_{13}^1 > 0$ to
guarantee
that for all $b>0$ the sender's preferred response strictly exceeds
the receiver's full information action. Our next proposition will
be proved under this assumption, and for $b > 0.$ 

In addition, we assume that receivers use pure replies off the
equilibrium path and we make a technical modification in the
definition of AER sets. We drop the requirement that AER sets be
closed to avoid having to introduce topologies on strategy spaces.
The restriction on receiver strategies is for tractability. The
modification of AER sets is purely cosmetic. In the case of finite
type, message and action sets closedness helps in characterizing
and proving existence of EER and AER sets. Fortunately, for CS
games, given our assumptions, characterizing AER sets and proving
existence is immediate without appealing to closedness of AER sets.
We know from Crawford and Sobel that there are only finitely many
partition equilibria and that every equilibrium induces a 
type-action association from a corresponding partition equilibrium.
The set of all equilibria is invariant under S- and R-replacement.
Clearly, if some equilibrium corresponding to a partition
equilibrium belongs to an AER set, so does every other equilibrium
corresponding to that AER set. Since there are only finitely many
partition equilibria, there is a minimal set that is closed under
S- and R-replacement.

Following Austen-Smith and Banks [1995], we will refer to any
equilibrium that induces multiple actions as {\sl influential.}
Define a pooling equilibrium as any equilibrium that induces only
the pooling action.

We will show that pooling equilibria can be replaced by influential
equilibria and that the reverse replacement is impossible. In
addition there are severe restrictions on replacements among
influential equilibria. Essentially an influential equilibrium
$\sigma^0$ can be replaced by another influential equilibrium
$\sigma^1$ only if the partition of the latter refines the
partition of the former. This is established in the following three
lemmata.

To state and prove these results it is convenient to adopt the
following conventions. Represent any partition by the collection of
interiors of its elements. For an equilibrium $\sigma,$ denote the
partition corresponding to that equilibrium by $J(\sigma).$ A
partition $J(\sigma^1)$ refines $J(\sigma^0)$ if the interiors of
elements of $J(\sigma^1)$ are subsets of interiors of elements of
$J(\sigma^0).$ If $(s,t)$ is a nonempty interval of types, let
$a(s,t)$ stand for the receiver's best reply to beliefs restricted
to $(s,t.)$ Also, given an equilibrium $\sigma,$ let $a_h (\sigma)$
denote the highest action taken after a positive probability
message of $\sigma .$ 

The first of our three preliminary results establishes limits on
sender-led replacement. 

\begin{lem}If the equilibrium $\sigma^1$ S-replaces the equilibrium
$\sigma^0,$ then $J(\sigma^1)$ refines $J(\sigma^0).$ \end{lem}

\begin{proof} In order to derive a contradiction, suppose not. Then
there are two adjacent elements of $J(\sigma ^ 0),$ say $(t_{j-1},
t_j)$ and $(t_j, t_{j+1}), $ and types $t' \in (t_{j-1}, t_j),$
$t'' \in (t_j, t_{j+1})$ who send a common message under
$\sigma^1.$
Under sender-led replacement this means that there exists a
message, $m,$ that is unused in $\sigma^0$ and in that equilibrium
induces a reply $a$ that makes both types $t'$ and $t''$
indifferent between their $\sigma ^ 0 $ messages and $m.$ Note that
for $t''$ to be willing to send $m$ we need $a > a(t_{j-1}, t_j).$
Similarly, we need $a < a (t_j, t_{j+1}).$ Since the optimal action
from the sender's perspective is an increasing function of the
sender's
type, there exists a type $t^* \in  (t' , t'')$ who strictly
prefers the action $a$ to any of the receiver's equilibrium replies
under $\sigma ^0.$ This is incompatible with $\sigma ^ 0$ being an
equilibrium. \end{proof}

The next two preliminary results establish analogous limits on
receiver-led replacement. 

\begin{lem} If the equilibrium $\sigma ^1$ R-replaces the
equilibrium $\sigma ^ 0,$ then the maximal element of $J(\sigma^0)$
contains the maximal element of $J(\sigma ^1).$ \end{lem}

\begin{proof} Suppose not. Then the maximal actions under the two
equilibria satisfy $a_h(\sigma^1) < a_h(\sigma^0).$ From $a^2(t) =
a^1(t,0),$ and $v_{13}^1 >0$ we have $a^1(t,b) \geq a^2(t).$
Therefore there is a positive measure of types who prefer
$a_h(\sigma^0)$ to $a_h(\sigma^ 1).$ Since under receiver-led
replacement the receiver's replacement strategy $\sigma_2 ^ 1$ is
a best reply to the sender's status quo $\sigma_1^0,$ the status
quo action $a_h (\sigma ^ 0)$ must be attainable for the
sender under $\sigma ^1.$ This contradicts $\sigma ^ 1$ being an
equilibrium. \end{proof}

This result shows that there is a sense in which an R-replacement
must induce a finer partition. A complementary result demonstrates
that just having a greater number of elements in the candidate
replacement partition is not enough. 

\begin{lem} If the equilibrium $\sigma^1$ R-replaces the
equilibrium $\sigma^0,$ then there do not exist actions $a_1^1 <
a^0 < a_2^1$ such that $a_1^1$ and $a_2^1$ are adjacent equilibrium
actions under $\sigma ^ 1,$ and $a ^ 0$ is an equilibrium action
under $\sigma ^ 0.$ \end{lem}

\begin{proof} Strict concavity in actions of payoff functions
implies that the marginal type who is indifferent between $a_1^1$ 
and $a_2^1$ strictly prefers $a^0.$ Under receiver-led
replacements, the action $a^0$ is available to senders in the
candidate replacement equilibrium $\sigma ^1.$ This breaks the
equilibrium $\sigma ^ 1.$ \end{proof}

In summary, influential equilibria are hard to replace. Sender-led
replacement requires nestedness and receiver-led replacement
requires analogous strong conditions. Fortunately this does not
prevent pooling equilibria from being replaced. Therefore we have
the following proposition.

\begin{prp} If G is a CS game in which the set of influential
equilibria is nonempty, then
\begin{enumerate}
\item[{[24]}] the set of influential equilibria contains an AER
set, and
\item[{[25]}] no pooling equilibrium is a member of an AER set.
\end{enumerate}
\end{prp}

\begin{proof} The first claim of the proposition, [24], follows
directly from Lemma 6 and Lemma 7.

For the second statement in the proposition, [25], let $\sigma^p$
be a
pooling equilibrium. We will show that $\sigma^p$ can be replaced
by
an influential equilibrium. First, $\sigma^p$ can be replaced by
$\sigma^1$ where $\sigma_1^1 = \sigma_1^p$ and under $\sigma_2^1$
the
receiver responds to all message with the pooling response $a_p.$
Second, note that, by assumption, there exists an influential
equilibrium $\sigma^2$ and without loss of generality types
belonging to the same partition element use the same message under
$\sigma^2$. Observe that $\sigma_1^2$ is a best reply to
$\sigma_2^1.$ Thus the pooling equilibrium $\sigma^1$ can  be
replaced by the influential equilibrium $\sigma^2.$  
\end{proof}

This result cannot be substantially strengthened because in general
partitions will not be nested. As an example consider Crawford and
Sobel's quadratic example with an alignment parameter $b=1/20.$
Crawford and Sobel show that in that case there are three partition
equilibria with partitions $\{(0,1)]\},$ $\{(0, {2 \over 5}), ({2
\over 5} ,1) \},$ and $\{(0, {2 \over 15}), ({2 \over 15}, {7 \over
15}), ({7 \over 15} , 1) \}.$ The two influential partitions are
not
nested; therefore by Lemma 6 they cannot be attained one from the
other via sender-led replacement. The finer partition among the
influential ones cannot be attained from the coarser one via
receiver-led replacement because of Lemma 8. The reverse
replacement is impossible because of Lemma 7. Thus, in this game
there are two AER sets, each corresponding to one of the
influential partitions.  

The type of incentive alignment considered by Crawford and Sobel
does play an important role in the rejection of pooling equilibria
by {\em AER} sets. The following example shows that outside the
class of Crawford-Sobel games pooling equilibria need not be
rejected even in the presence of partition equilibria. Let $F(
\cdot )$ be uniform on $[0,1],$
$v^1(a,t,b) := -(a-t)^2 - b(a-{1 \over 2})^2,$ $v^2(a,t) := -(a-
t)^2,$ and $b>0.$ Then, regardless of the value of $b$ there always
exists a partition equilibrium with a two-element partition in
which types $t \in [0, {1 \over 2})$ send a common message distinct
from the message sent by types in $[{1 \over 2},1].$ For
sufficiently large $b$ these are the only influential partition
equilibria. It is easily checked that in that case any 
{\em AER}-set contains pooling equilibria. This is so because for
large $b$ all types strictly prefer the pooling action $1 \over 2$
to either of the two separating actions $1 \over 4$ and $3 \over
4$. 

\section{Relation to the Literature}

Thus far equilibrium selection in sender-receiver games satisfying
a partial common interest condition has been approached with a
number of different quasi-dynamic solution concepts. Here, we
pursue a similar strategy. It yields the sharpest predictions to
date, guarantees existence, predicts tractable sets of equilibria
and does not rely on commonly used auxiliary assumptions like
nominal message costs, exogenously  given meanings of messages,
limitations on drift, limitations on the interpretation of
zero-probability messages, etc.. Of course, our paper owes a
considerable debt to these other works. We will conclude with a
brief discussion of how predictions differ across the various
approaches.

In two-player games, {\em EER} sets resemble static
evolutionary solutions. In both instances a strategy or a set of
strategies is a solution if it
satisfies an entry (or a replacement) condition. However, there are
substantial differences. Unlike {\em evolutionarily stable
strategies,} Maynard Smith and Price [1973], {\em EER} sets always
exist. Unlike {\em neutrally stable strategies}, Maynard Smith and
Price [1973], {\em EER} sets predict efficient outcomes in all
common 
interest games. As for set-valued solution concepts, again {\em ES}
sets, Thomas [1985 a,b], and {\em EES} sets, Swinkels [1992],
Blume, Kim and
Sobel [1993], do not exist in general. However, whenever an {\em
ES}
set or an {\em EES} set does exists, it contains an {\em EER} set. 

Blume, Kim and Sobel [1993] propose to consider {\em ER} sets to
address the existence problem of {\em EES} sets. {\em ER} sets do
exist and do not include pooling equilibria in partial common
interest games. However, in these games, they do not rule out
pooling actions, in general. {\em ER} sets are closely related to
the {\em cyclically stable sets (CSS)} of Gilboa and Matsui [1991]
and
Matsui [1992]. {\em CSSs}  can be given an explicit dynamic
interpretation. Like {\em ER} sets, {\em CSS} sets can be shown not
to rule out pooling actions in partial common interest games, and
not to rule out dominated actions in general. The latter point is
illustrated in the following game, Game 3.

{\footnotesize
\begin{center}
\begin{picture}(250,80)
\put(50,15){\framebox(120,50)}
\put(50,40){\line(1,0){120}}
\put(90,15){\line(0,1){50}}
\put(130,15){\line(0,1){50}}
\put(35,50){$t_1$}
\put(35,25){$t_2$}
\put(68,70){$a_1$}
\put(108,70){$a_2$}
\put(148,70){$a_3$}
\put(65,50){7,7}
\put(105,50){2,2}
\put(145,50){9,0}
\put(65,25){2,2}
\put(105,25){7,7}
\put(145,25){0,0}
\end{picture}
\end{center}}
\begin{center}
{Figure 3: Game 3}
\end{center} 

In this game no {\em EES} set exists. There exists an {\em ER} set
and a {\em CSS} in which the two types of the sender separate.
Either set contains a strategy in which the strictly dominated
action $a_3$ is used with
positive probability.

Blume's [1994] {\em perturbed message persistence (PMP)} does
eliminate
dominated actions. It also eliminates pooling actions in a class of
partial common interest games that includes Game 1 with $x>0.$ PMP
ensures
that the receiver's responses
reflect the separation of types in partial common interest games.
The partial common interest condition in Blume [1994] is more
restrictive, in part, because the retracts that form the solution
will in general include nonequilibrium strategies. {\em PMP}
retracts exist in every game.

Rabin and Sobel [1994] present yet another approach to partial
common interest games. Their solutions, {\em recurrent MOPs}, are
sets of equilibria and exist in every game. Unlike {\em EER} sets,
neither recurrent MOPs
nor PMP sets predict communication in Game 4; again, 
assume that types are equally likely. 

{\footnotesize
\begin{center}
\begin{picture}(300,100)
\put(50,15){\framebox(200,75)}
\put(50,40){\line(1,0){200}}
\put(50,65){\line(1,0){200}}
\put(90,15){\line(0,1){75}}
\put(130,15){\line(0,1){75}}
\put(170,15){\line(0,1){75}}
\put(210,15){\line(0,1){75}}
\put(35,75){$t_1$}
\put(35,50){$t_2$}
\put(35,25){$t_3$}
\put(68,95){$a_1$}
\put(108,95){$a_2$}
\put(148,95){$a_3$}
\put(188,95){$a_4$}
\put(228,95){$a_5$}
\put(65,75){5,5}
\put(105,75){6,0}
\put(145,75){4,4}
\put(185,75){0,0}
\put(225,75){3,3}
\put(60,50){.5,-1}
\put(145,50){4,4}
\put(100,50){-1,5}
\put(185,50){0,0}
\put(225,50){3,3}
\put(65,25){0,0}
\put(105,25){0,0}
\put(145,25){0,0}
\put(185,25){4,4}
\put(225,25){3,3}
\end{picture}
\end{center}}
\begin{center}
{Figure 4: Game 4}
\end{center}

In the present paper it is postulated that solutions consist of
sets of
equilibria, and that these sets must be stable only against other
equilibria. Both ideas are familiar. Farrell [1993, p.523]
discusses the
possibility of requiring a neologism to be itself part of an
equilibrium. Myerson's [1988] core mechanism examines this idea in
an
environment that permits correlation. As for the requirement that
solutions be sets of equilibria, Swinkels [1992] argues that his
requirement on entrants to use best replies against the post-entry
population is more plausible if this behavior does not lead outside
the set of Nash equilibria. There are also explicit dynamic models
that derive equilibrium behavior from simple behavioral adjustment
rules. For example, in the class of sender-receiver games they
examine, the limiting outcomes of N\"oldeke and
Samuelson's [1992] dynamic contain only (self-confirming)
equilibria. 

None of these solution concepts has been used to distinguish
influential from pooling equilibria in Crawford-Sobel games. In
CS's quadratic example typically no neologism-proof equilibrium
exists, for natural discretizations of the game no EES set exists,
and ER sets are not tractable. The present paper is the first one
to integrate the analysis of Crawford-Sobel games with that of
partial common interest games.

\vfill\eject
\appendix
\section{Appendix}

This appendix demonstrates that the equivalence of definitions 1
and 5 of EER sets
does not extend beyond the class
of two-player games. Consider the following three-player game in
which the row player chooses row $a_1$ or $a_2,$ the column player
chooses column $b_1$ or $b_2$ and the matrix player chooses matrix
$c_1$ or $c_2.$     

{\footnotesize
\begin{center}
\begin{picture}(300,100)
\put(50,15){\framebox(100,50)}
\put(200,15){\framebox(100,50)}
\put(50,40){\line(1,0){100}}
\put(200,40){\line(1,0){100}}
\put(100,15){\line(0,1){50}}
\put(250,15){\line(0,1){50}}
\put(35,50){$a_1$}
\put(35,25){$a_2$}
\put(68,75){$b_1$}
\put(118,75){$b_2$}
\put(185,50){$a_1$}
\put(185,25){$a_2$}
\put(218,75){$b_1$}
\put(268,75){$b_2$}
\put(98,0){$c_1$}
\put(248,0){$c_2$}
\put(60,50){3,3,3}
\put(110,50){2,2,0}
\put(60,25){2,2,0}
\put(110,25){2,2,0}
\put(210,50){0,0,0}
\put(255,50){1,1,100}
\put(205,25){1,1,100}
\put(260,25){2,2,0}
\end{picture}
\end{center}}

Both the equilibrium $(a_1,b_1,c_1)$ and the component
$(a_2,b_2,(\gamma, 1-\gamma)),~\gamma \in [0,1]$ are EER sets
according to Definition 5 (naturally extended to $n$-player games).
Both of these sets are also {\em
EES} sets. However, only the former is an EER set according to
Definition 1.

\vfill\eject

\hfill\break
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\end{description}
\end{document}