数理解析研究所講究録
第 1557 巻 2007 年 179-186

179

Bayesian Communication Leading to a Nash
Equilibrium in Belief \starTakashi Matsuhisa1 and Pavel Strokan2
1

Department of Natural Sciences, Ibaraki National College of Technology
Nakane 866, Hitachinaka-shi, Ibaraki 312-8508, Japan.
E-mail: mathisa@ge.ibaraki-ct.ac.jp
2
Department of Applied Mathematics and Control Processes
Saint-Petersburg State University
Universitetskij pr., 35 Saint-Petersburg 198504 Russia
E-mail: stron@rambler.ru

Abstract. ABayaeian communication in the -belief system is presented which leads to aNash equilibrium of astrategic form game throughmessages as aBayesian updating process. In the communication processeach player predicts the other players’ actions under $his/her$ private information with probability at least $his/her$ belief. The players communicate privately theirthrough message according to the communication graph, where each player receiving the message learns $md$
revises $his/her$ conjecture. The emphasis is on that both any topologicalassumptions on the communication graph and any common-knowledge
assumptioo on the structure of communication are not required.
Keywords: -Belief system, Nash equilibrium, Bayesit communication, Protocol, Conjecture, Non-corporative game.
AMS 2000 Mathematics Subject lassification: Primary $91A35$ ,Secondary $03B45$ .
Journal of Economic Literature Classification: C62, C78.
$p$$co\iota\dot{\eta}ectures$

$p$

$C$

1

Introduction

This article relates equilibria and distributed knowledge. In game theoreticalsituations among a group of players, the concept of mixed strategy Nash equilibrium has become central. Yet little is known the process by which players
learn if they do. This article will give a protocol run by the mutual learning of
their beliefs of players’ actions, and it highlights an epistemic aspect of Bayesian
updating process leading to a mixed strategy Nash equilibrium for a strategic
form game.
As for as J.F. Nash [8] fundamental notion of strategic equilibrium is concerned, R.J. Aumann and A. Brandenburger [1] gives epistemic conditions formixed strategy Nash equilibrium: They show that the common-knowledge of thepredictions of the players having the partition information (that is, equivalently,$s$“This paper was presented in WINE 2005, 15-17 December 2005, Hong Kong, China
([6])

180
the S5-know1edge model) yields a Nash equilibrium of a game. However it is
not clear just what learning process leads to the equilibrium. The present article
aims to fill this gap from epistemic point of view.
Our real concern is with what Bayesian learning process leads to a mixed
strategy Nash equilibrium of a finite strategic form game with e\’ephasis on
the epistemic point of view. We focus on the Bayesian belief revision throughcommunication among group of players. We show that
Main theorem. Suppose that the players in a strategic form game have thep-belief system with a common prior distmbution. In a communication processof the game according to a protocol with revisions of their beliefs about the otherstrategy
players’ actions, the profile of their future predictions induces a
long
game
in
the
run.
equilibrium
the
Nash
of
Let us consider the following protocol: The players start with the same priordistribution on astate-space. In addition they have private information given
by apartition of the state space. Beliefs of players $axe$ posterior probabilities: A
player $prightarrow believes$ (simply, believes) an event with $0<p\leq 1$ if the posterior probability of the event given $his/her$ information is at least . Each player predicts
the other players’ actions as $his/her$ belief of the actions. $He/she$ communicatesprivately their beliefs about the other players’ actions through messages, andthe receivers update their belief according to the messages. Precisely, the players are assumed to be rational and maximizing their expected utility according
their beliefs at every stage. Each player communicates privately $his/her$ belief
about the others’ actions as messages according to aprotocol,3and the receivers
update their private information and revise their belief.
The main theorem says that the players’ predictions regarding the future
beliefs converge in the long run, which lead to amixed strategy Nash equilibriumof agame. The emphasis is on the two points: First that each player’s prediction
is not required to be common-knowledge among all players, and secondly thateach player send to the another player not the exact information about $his/her$belief about the actions for the other players but the approximate information
about the the other players’ actions with probability at lest $his/her$ belief of theothers’ actions.
This paper is organized as follows: In section 2we give the formal model ofthe Bayesian communication on agame. Section 3states explicitly our thmrem
and gives asketch of the proof. In final section 4we conclude some remarks. We
are planning to present asmall example to illustrate the theorem in our lecture
presentation in the Kyoto Symposium ‘Mathematical Economics.’$\gamma nixed$$p$2

The ModelLetbe a non-empty finite set called a state-space, $N$ a set of finitely manybe the family of all subsets
players $\{1, 2, \ldots n\}$ at least two $(n\geq 2)$ , and let$\Omega$$2^{\Omega}$

3

When a player communicates with another, the other players are not informed about
the contents of the message.

181
called aof . Each member ofis called an event and each element ofplayers.
forallwhich is common
state. Let
be a probability measure on
is a finite probability space with fullFor simplicity it is assumed that$\Omega$$2^{\Omega}$$\rfloor\Omega$$\Omega$$\mu$$(\Omega, \mu)$$\mu$

suppart.4

p-Belief System5

2.1

Let be a real number with $0<p\leq 1$ . The p-belief system associated with thepartition information structureis the tuple
is a finite probconsisting of the following structures and interpretations:such thatability space, and $is$ p-belief opemtor $B_{i}(*;p)$ is the operator on$E$$B_{i}(E,p)$ is the set of states ofin which p-believes that has occurred withprobability at least ; that is, $B_{i}(E;p)$ $:=\{\omega\in\Omega|\mu(E|\Pi_{i}(\omega))\geq p\}$ .
$p$$\langle N, \Omega, \mu, (\Pi_{i})_{i\in N}, (B_{i}(*,p))_{i\in N}\rangle$$(\Pi_{i})_{i\in N}$$(\Omega,\mu)$$2^{\Omega}$$i$$\Omega$$p$

Remark 1. When $p=1$ the l-belief operator $B_{i}(*;1)$ becomes knowledge operator.
Game on p-Belief System6

2.2

with theBy agame we mean afinite strategic form game $(N, (A_{i})_{i\in N},$$\{1,
2,
\ldots,i, \ldots n\}$$N$following structure and interpretations: is afinite set of playelS
pure strategies) and is anwith $n\geq 2,$
is afinite set ofactions (or,payoff function of into $R$ , where denotes the product. We denote by thethe product$(a_{1},\ldots,a_{i-1},a_{i+1}, \ldots, a_{n})$-tuplethe $(n-1)$ -tuple
and byfor each $I\subset N$ .
for of A. Furthermore we denoteoverall conjecture (orAprobability distribution
is saId to beon
conjecture). For each player other than , this induces the marginal
simplyindividual conjecture about (or simply
distribution on $j’ s$ actions; we call it
conjecture about ) Functions on
are viewed like random variables in the. If is asuch function and is avalue of it, we denoteprobability space
by $[x=x]$ (or simply by $[x]$ ) the set $\{\omega\in\Omega|x(\omega)=x\}$ .
The information structurewith acommon prior yields the distribution
overall conjecture
defined by $q_{i}(a,w)=\mu([a=a]|\Pi_{i}(\omega))$ ;and theon$q_{i}(a_{-i},\omega)=\mu([a_{-i}=a_{-i}]|\Pi_{i}(\omega))$which
defined by the marginal distribution
intersection
thebyis viewed as arandom variable of . We denote.
the intersection
and denote byarandom variable ofpayoff function andLet be arandom variable of$:=[a_{i}=a_{i}]$for allan action . Where we assume that;and for every ofaction is said to be actual at astate if$\omega\in[a_{I}=a_{I}]$and the profile
is said to be actually played at if$I\subset
N.$for
The pay off functions $9=(g_{1},g_{2}, \ldots,g_{n})$ is said to be actually$G$$(g_{i})_{i\in N}\rangle$$i’ s$$i’ s$$A_{i}$$A$$A$$A_{1}\cross A_{2}\cross\cdots\cross A_{i-1}\cross A_{i+1}\cross\cdots\cross A_{n}$$A_{-i}$$(g_{1},g_{2}, \ldots g_{n})$

$n$

$g$

$a_{-i}$$a_{-I}=(a_{i})_{i\in N\backslash I}$$a$$\phi_{i}$$i’ s$$A_{-i}$$i’ s$$i$$j$$i’ s$$i’ s$$j.$$(\Omega, \mu)$$j$$\Omega$$x$$x$$(\Pi_{i})$$\mu$$i’ s$$A\cross\Omega$$[q_{i}=\phi_{i}]$$\phi_{i}$$\bigcap_{a-:\in A-:}[q_{i}(a_{-i})=\phi_{i}(a_{-i})]$$A_{i}.i’ s$$a_{i}$$\omega\in[a_{i}]$$\omega$$a_{i}]$.
for every
That is;[7].
Monderer and Samet
6
Aumann and Brandenburger [1]$\mu(\omega)\neq 0$$\omega$$a_{i}$$a_{I}$

5$g_{i}$$\Pi_{i}(\omega)\subseteq[a_{i}]$$a_{i}$$a_{i}$

4$\bigcap_{i\in N}[q_{i}=\phi_{t}]$$[\phi]$$i’ s$$g_{i}$$i’ s$$i’ s$$g_{i}$$A_{1}\cross A_{2}\cross\cdots\cross A_{n}$$\omega\in\Omega$$\omega\in[a_{i}=a_{i}]$$:= \bigcap_{i\in I}[a_{i}=$182
if $\omega\in[g=g]$ $:= \bigcap_{i\in N}[g_{i}=g_{i}]$ . Let Exp denote theplayed at a state.
expectation defined by $Exp(g_{i}(b_{i}, a_{-i});\omega)$ $:=$$\omega$$\sum_{a-:\in A-i}g_{i}(b_{i}, a_{-i})q_{i}(a_{-i},\omega)$

maximizes
actual action
A player is said to be mtional at if eachat when the otherthe expectation of his actually played pay off function. ; . Formally,
conjecture
players actions are distributed according to his
and $a_{i}=\Re(\omega),$ $Exp(g_{i}(a_{i}, a_{-i});\omega)\geq Exp(g_{i}(b_{i}, a_{-i});\omega)$ for
letting
denote the set of all of the states at which is rational.
every in . Let$i$$i’ s$$\omega$$a_{i}$$\omega$$g_{i}$$q_{i}$$($$\omega)$$g_{i}=g_{i}(\omega)$$i$$R_{i}$$A_{i}$$b_{i}$

Protocol 7

2.3

We assume that the players communicate by sending messages. Let be the time
horizontal line $\{0,1,2, \cdots t, \cdots\}$ . A protocol is a mapping $Pr$ : $Tarrow N\cross N,trightarrow$$(s(t),r(t))$ such that $s(t)\neq r(t)$ . Here stands for time and $s(t)$ and $r(t)$ are,respectively, the sender and the receiver of the communication which takes placeat time . We consider the protocol as the directed graph whose vertices are theset of all players $N$ and such that there is an edge (or an arc) from to if andonly if there are infinitely many such that $s(t)=i$ and $r(t)=j$ .
A protocol is said to be fair if the graph is strongly-connected; in words,
every player in this protocol communicates directly or indirectly with every
other player infinitely often. It is said to contain a cycle if there are players
communicates directly with, with $k\geq 3$ such that for all $m<k,$$i_{m+1}$ , and such that
communicates directly with . The communications is
assumed to proceed in rounds 8
$T$

$t$

$t$

$i$

$j$

$t$

$i_{1},$$i_{2},$$\ldots$$i_{m}$$i_{k}$$i_{1}$$i_{k}$

2.4

Communication on p-Belief System

A Bayesian belief communication process $\pi(G)$ with revisions of players’ conjecaccording to a protocol for a game is a tuple
tures
$G$

$(\phi_{i}^{t})_{(i,t)\in N\cross T}$$\pi(G)=(Pr, (\Pi_{i}^{t})_{i\in N},$ $(B_{i}^{t})_{i\in N},$$(\phi_{i}^{t})_{(i,t)\in N\cross T}\rangle$

with the following structures: the players have a common prior on , theprotocol $Pr$ among $N,$ $Pr(t)=(s(t), r(t))$ , is fair and it satisfies the conditions
that $r(t)=s(t+1)$ for every and that the communications proceed in rounds.
forat time is the mapping of into
The revised information structure. An
player . If $i=s(t)$ is a sender at , the message sent by to $j=r(t)$ isis a revision process of individual conjectures. These structures
n-tupleare inductively defined as follows:$\Omega$

$\mu$

$t$

$i$

$t$

$i$

$\Omega$$t$$\Pi_{i}^{t}$$2^{\Omega}$$M_{i}^{t}$$(\phi_{i}^{t})_{i\in N}$.
-Set
is defined. It yields the distribution $q_{i}^{t}(a,\omega)=\mu([a=$-Assume that
. Whence$\Pi_{i}^{0}(w)=\Pi_{i}(\omega)$$\Pi_{i}^{t}$$a]|\Pi_{i}^{t}(w))$

7
8

C.f.: Parikh and Krasucki [9]There exists a time $m$ such that for all
protocol is the minimal number of all$m$$Pr(t)=Pr(t+m)$ . The period of thesuch that for every $Pr(t+m)=Pr(t)$ .
$t,$

$t,$

183
$\bullet$denotes the set of all the state at which is rational according to. ; ; that is, each $is$ actual action
his conjecture
maximizes the
expectation of his pay off function being actually played at when theother players actions are distributed according to his conjecture
at time
The message:
sent by the sender at time is defined by$R_{i}^{t}$$i$$\omega$$q_{i}^{t}$$\omega)$$($$a_{i}$$\omega$$g_{i}$$q_{i}^{t}(\cdot ; \omega)$$t^{9}$$\bullet$$\Omegaarrow 2^{\Omega}$$M_{i}^{t}$

$i$

$t$

$M_{i}^{t}( \omega)=\bigcap_{a-:\in A_{-i}}B_{i}^{t}([a_{-i}];q_{i}^{t}(a_{-i},w))$

where$B_{i}^{t}$:,is defined by$2^{\Omega}arrow 2^{\Omega}$$B_{i}^{t}(E;p)=\{\omega\in\Omega|\mu(E|\Pi_{i}^{t}(\omega))\geq p\}$

Then:
-The revised partition

at time $t+1$ is defined as follows:
if $i=r(t)$ ;otherwise,
The revision processof conjectures is inductively defined as$\bullet$$\bullet$-.

$\Pi_{i}^{t+1}$$\Pi_{i}^{t+1}(w)=\Pi_{i}^{t}(\omega)\cap M_{s(t)}^{t}(\omega)$$\Pi_{i}^{t+1}(\omega)=\Pi_{i}^{t}(\omega)$$(\phi_{i}^{t})_{(i,t)\in N\cross T}$

follows:$\bullet$$\bullet$

Let
Take

$\omega_{0}\in\Omega$, and set$\phi_{s(0)}^{0}(a_{-s(0)})$$:=q_{s(0)}^{0}(a_{-s(0)},\omega_{0})$$\omega_{1}\in M_{\epsilon(0)}^{0}(\omega_{0})\cap B_{r(0)}([g_{s(0)}]\cap R_{s(0)}^{0}; p)^{10}$and set$:=$$\phi_{s(1)}^{1}(a_{-s(1)})$$q_{s(1)}^{1}(a_{-s(1)},w_{1})$$\bullet$Take$\omega_{t+1}\in M_{s(t)}^{t}(\omega_{t})\cap B_{r(t)}([g_{s(t)}]\cap R_{s(t)}^{t};p)$$q_{i}^{t+1}(a_{-\epsilon(t+1)},\omega_{t+1})$, and set$\phi_{s(t+1)}^{t+1}(a_{-s(t+1)})$

$:=$

.

The specification is that a sender $s(t)$ at time informs the receiver $r(t)his/her$individual conjecture about the other players’ actions with a probability greaterthan $his/her$ belief. The receiver revises $her/his$ information structure under the
information. $She/he$ predicts the other players action at the state where theplayer p-believes that the sender $s(t)$ is rational, and $she/he$ informs $her/his$ thepredictions to the other player $r(t+1)$ .
. ; $\omega)=$
We denote bya sufficient large such that for all $w\in\Omega,$.;.
by. Hence we can write byand$t$$\infty$$q_{i}^{\tau+1}$$($$q_{i}^{\tau}($$\tau$$\omega)=q_{i}^{\tau+2}(\cdot ; \omega)=\cdots$$q_{i}^{\infty}$$q_{i}^{\tau}$$\phi_{i}^{\tau}$$\phi_{i}^{\infty}$

Remark 2. The Bayesian belief communication is a modification of the communication model introduced by Ishikawa [3].
910$Exp^{t}$ , isFormally, letting $g_{i}=g_{i}(w),$ $a_{i}=a_{i}(\omega)$ , the expectation at time$Exp^{t}(g_{*}(a_{i},a_{-i});\omega)$$:=$
fined by. An player$\sum_{a-:\in A-i}g_{i}(a_{i}, a_{-i})q_{i}^{t}(a_{-i},w)$$t,$said to be rational according to his conjecture
Exp $(g_{l}(a_{i},a_{-i});w)\geq Exp^{t}(g_{i}(b_{i},a_{-i});w)$

We denote $[g_{i}];=[g_{i}=9i]$.

.

$q^{t}$ $($.,$\omega)$at$w$

if for all$b_{i}$

in$d\triangleright$

$i$

is

$A_{:}$,

184
3

The Result

We can now state the main theorem :Theorem 1. Suppose that the players in a stmtegic form gamehave the pbelief system with a common prior. In the Bayesian belief communication
process $\pi(G)$ according to a protocol among all players in the game with revisions,
there evists a timesuch that for eachof their conjectures
game.
stmtegyequilibrium
a
mixed
Nash
the
n-tupleinduces
the
of
$G$

$\mu$

$(\phi_{i}^{t})_{(i,t)\in N\cross T}$$t\geq\infty$$\infty$$(\phi_{i}^{t})_{i\in N}$

The proof is based on the below proposition:
Proposition 1. Suppose that the players in a strategic form game have thep-belief system with a common prior. In the Bayesian belief communication
process $\pi(G)$ in a game $G$ with revisions of their conjectures, if th protocol
$\mu$

$e$

has no cycle then both the conjecturesandonmust coincide;that is,for $(i,j)=(s(\infty), s(\infty+t))$ and for any$q_{i}^{\infty}$$q_{j}^{\infty}$$A\cross\Omega$$q_{i}^{\infty}(a;\omega_{\infty})=q_{j}^{\infty}(a;\omega_{\infty+t})$$t=1,2,3,$$\cdots$.

Proof. Let us first consider the case that $(i,j)=(s(\infty), r(\infty))$ . We denote.
$W_{i}^{\infty}(w)=\{\xi\in\Omega|M_{i}^{\infty}(\xi)=M_{i}^{\infty}(w)\}$

In view of the construction of

we can observe that$\{\Pi_{i}^{t}\}_{t\in T}$

for all$\Pi_{j}^{\infty}(\xi)\subseteq W_{i}^{\infty}(\omega)$

It immediately follows thatponentsfor$W_{i}^{\infty}(\omega)$$\xi\in\Pi_{i}^{\infty}(w)$$\Pi_{j}^{\infty}(\xi)$$\xi\in W_{i}^{\infty}(\omega)$.

(1)

is decomposed into a disjoint union of com-;

$W_{i}^{\infty}( \omega)=\bigcup_{k=1,2,,m}\ldots\Pi_{j}^{\infty}(\xi_{k})$where$\xi_{k}\in W_{i}^{\infty}(\omega)$.

(2)

It can be observed that(3)$\mu([a=a]|W_{i}^{\infty}(\omega))=\sum_{k=1}^{m}\lambda_{k}\mu([a=a]|\Pi_{j}^{\infty}(\xi_{k}))$, wefor all
Since. On noting thatis decomposed, we can obtainfor$q_{i}^{\infty}(a;\omega)=\mu([a=a]|W_{i}^{\infty}(\omega))=\mu([a=a]|\Pi_{i}^{\infty}(\xi_{k}))$ for any. Itsuch that
there exists a state
follows by (3) that, for each
for $(i,j)=(s(\infty), t(\infty))$ .
On continuing this process according to the fair protocol, the below facts,
and for sufficient large
can be plainly verified: For eachwithfor some
can observe that
into a disjoint union of components$\omega\in[a_{i}]$$\Pi_{i}(\omega)\subseteq[a_{i}]$$\sum_{k=1}^{m}\lambda_{k}=1^{11}$$\lambda_{k}>0$$q_{i}^{\infty}(a_{-i};w)=q_{i}^{\infty}(a;\omega)$$\Pi_{i}^{\infty}(\xi)$$W_{i}^{\infty}(w)$$\xi\in\Pi_{i}^{\infty}(\omega)$$\xi_{k}\in W_{i}^{\infty}(\omega)$$\omega\in\Omega$$\xi_{\omega}\in\Pi_{i}^{\infty}(\omega)$$q_{i}^{\infty}(a;\omega)\leq q_{j}^{\infty}(a;\xi_{\omega})$$w\in J\Omega$$\tau\geq 1$

11 This property is called the convexity for the conditional probability $\mu(X|*)$ in Parikh

and Krasucki [9].

185
1. For any $t\geq 1,$2.

$q_{i}^{\infty}(a;\omega)\leq q_{i}^{\infty}(a;\xi)\leq q_{i}^{\infty}(a;\zeta)\leq,$

Since$\Omega$for some. . for some$q_{s(\infty)}^{\infty}(a;\omega)\leq q_{s(\infty+t)}^{\infty}(a;\xi_{t})$$\xi,$$\zeta,$$\xi_{t}\in\Omega$$\cdots\in\Omega$

; and
.

is finite it can be obtained thatfor every , in completing the proof.

$q_{i}^{\infty}(a;\omega_{\infty})=q_{j}^{\infty}(a;\omega_{\infty+t})$$(s(\infty), s(\infty+t))$

for $(i,j)=$$a$

Proof of Theorem 1: We denote by
the set of all the players who directly
receive the message ffom on $N$ ; i.e., $\Gamma(i)=\{j\in N|(i, j)=Pr(t)$ for some. Let
denote. It is noted that$\Gamma(i)$$t\in$$i$$T\}$$F_{i}$$[\phi_{i}^{\infty}]$$:= \bigcap_{a-:\in A_{i}}[q_{i}^{\infty}(a_{-i}; *)=\phi_{i}^{\infty}(a_{-i})]$for each $i\in N,$ $j\in\Gamma(i)$ .
We observe the first point that for each $i\in N,$ $j\in\Gamma(i)$ and for every $a\in A$ ,. Then summing over , we can observe that$\mu([g=a_{i}]|F_{i}\cap F_{j})=\phi_{j}^{\infty}(a_{i})$for any $a\in A$ . In view of Proposition 1 it canbe observed that
; I.e.,
for eachis independentof the choices of every $j\in N$ other than . We set the probability distribution
byon, and set the profile.
We observe the second point that for every: In fact, viewing the definition of
we shall show that. To verify this it suffices toshow that for every $k=1,2,$ ,: We
prove it by induction on . For $k=1$ the result is immediate. Suppose it is true
for $k\geq 1$ . On noting the protocol is fair, we can take the sequence of sets ofplayers
with the following properties:..
. . $\subset I_{m}=N$ :(a)(b) For every $k\in N$ there is a player.
with$F_{i}\cap F_{j}\neq\emptyset$$\mu([a_{-j}=a_{-j}]|F_{i}\cap F_{j})=\phi_{j}^{\infty}(a_{-j})$$a_{-i}$$j,$ $k,$ $\neq i$$\phi_{j}^{\infty}(a_{i})=\phi_{k}^{\infty}(a_{i})$$\phi_{j}^{\infty}(a_{i})$$i$$A_{i}$$\sigma_{i}$$\sigma_{i}(a_{i})$$\sigma=(\sigma_{i})$$:=\phi_{j}^{\infty}(a_{i})$$a \in\prod_{i\in N}Supp(\sigma_{i}),$$\phi_{i}^{\infty}(a_{-i})=$$\sigma_{1}(a_{1})\cdots\sigma_{i-1}(a_{i-1})\sigma_{i+1}(a_{t+1})\cdots\sigma_{n}(a_{n})$$\sigma_{i}$$\phi_{i}^{\infty}(a_{-i})=\prod_{k\in N\backslash \{i\}}\phi_{i}^{\infty}(a_{k})$$\cdots$$n,$$\phi_{i}^{\infty}(a_{-i})=\phi_{i}^{\infty}(a_{-I_{k}})\prod_{k\in I_{k}\backslash \{i\}}\phi_{i}^{\infty}(a_{k})$$k$$\{I_{k}\}_{1\leq k\leq n}$$I_{1}=\{i\}\subset I_{2}\subset.$$\subset I_{k}\subset I_{k+1}\subset.$$I_{k+1}\backslash I_{k}=\{i_{k+1}\}$$i_{k+1} \in\bigcup_{j\in I_{k}}\Gamma(j)$$:=[\Re_{k+1}=a_{i_{k+1}}]\cap F_{j}\cap$
We let take $j\in I_{k}$ such that $i_{k+1}\in\Gamma(j)$ . Set. It can be verified that $\mu([a_{-j-i_{k+1}}=a_{-j-i_{k+1}}]|H_{i_{k+1}})=\phi_{-j-i_{k+1}}^{\infty}(a_{-j})$ .
$H_{i_{k+1}}$$F_{i_{k+1}}$

Dividing$\mu(F_{j}\cap F_{i_{k+1}})$yields that.

$\mu([a_{-j}=a_{-j}]|F_{j}\cap F_{i_{k+1}})=\phi_{i_{k+1}}^{\infty}(a_{-j})\mu([a_{i_{k+1}}=a_{i_{k+1}}]|F_{j}\cap F_{i_{k+1}})$; then summing over
we obtain. It immediately follows from Proposition 1 that, as required.
Furthermore we can observe that all the other players than agree on thesame conjecture
about . We conclude that each action
appearing with positive probability inagainst the product of
maximizes
the distributions
with $l\neq i$ . This implies that the profile
is amixed strategy Nash equilibrium of , in completing the proof.

Thus$\phi_{j}^{\infty}(a_{-j})=\phi_{i_{k+1}}^{\infty}(a_{-j-i_{k+1}})\phi_{j}^{t}(a_{i_{k+1}})$$a_{I_{k}}$$\phi_{j}^{\infty}(a_{-I_{k}})=\phi_{i_{k+1}}^{\infty}(a_{-I_{k}-i_{k+1}})\phi_{j}^{\infty}(a_{i_{k+1}})$$\phi_{i}^{\infty}(a_{-I_{k}})=\phi_{i}^{\infty}(a_{-I_{k}-i_{k+1}})\phi_{i}^{\infty}(a_{i_{k+1}})$$i$$\sigma_{j}(a_{j})=\phi_{i}^{\infty}(a_{j})$$j$$\sigma_{i}$$a_{i}$$g_{i}$$\sigma=(\sigma_{i})_{i\in N}$$\sigma_{l}$

$G$

4

$j$

$\square$Concluding remarks

We have observed that in a communication process with revisions of players’
beliefs about the other actions, their predictions induces a mixed strategy Nash

186
equilibrium of the game in the long run. Matsuhisa [4] established the same assertion in the S4-know1edge model. Furthermore Matsuhisa [5] showed a similar
result for -mixed strategy Nash equilibrium of a strategic form game in the$S4$-knowledge model, which gives an epistemic aspect in Theorem of E. Kalai
and E. Lehrer [2]. This article highlights the Bayesian belief communication
with missing some information, and shows that the convergence to an exact
Nash equilibrium is guaranteed even in such the communication on approximate
information.
$\epsilon$

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$2003rightarrow 6$$\bm{t}d$$T$$n$