数理解析研究所講究録 1222 巻 2001 年 160-169

160

AWARENESS, BELIEF AND COMMUNICATION REACHINGCONSENSUS

福田恵美子 (EMIKO FUKUDA) * , 松久隆 (TAKASHI MATSUHISA) \dag er , AND 笹沼寿人

(HISATO SASANUMA)

ABSTRACT. We present communication process according to aprotocol whichis associated with an awareness and belief model. In the model we impose noneof the requirements for player’s knowledge as those in the standard model withpartitional information structure. We show that consensus on the posteriors
for an event among all players can still be guaranteed in the communication if
the protocol contains no cycle.

1. INTRODUCTION
Geanakoplos and Polemarchakis [5] investigated acommunication process inwhich two players announce their posteriors to each other. In the process players
learn and revise their posteriors and they reached consensus.
Krasudci [6] investigated the communication process according to aprotocol inwhich value of function are communicated privately through messages among atleast three players. He showed that in the process that in the process, consensus
on the values of adecision function can be guaranteed if the protocol contains nocycle. They both extend the agreement theorem of Aumann [1]. The information
structures of players for their model are given by partitions.
Bacharach [2] showed that the model with the partitional information structureis equivalent to his knowledge operator model with the three axioms for operators:
Kl axiom of knowledge, K2 axiom of transparency and K3 axiom of wisdom.
Matsuhisa and Kamiyama [7] introduced the lattice structure of knowledge forwhich non requirements is not imposed such as the three axioms. In lattice
structure Aumann’s notion of common-knowledge is the same as their notion ofcommon-belief. Matsuhisa and Usami [9] presented an awareness structure inwhich the players are not required to have logicaly omniscient. Matsuhisa andUsami obtained an extension of the ’Agreeing to disagree’ that Aumann showedin the awareness model.
The purpose of this paper is to introduce the revision process of the valuesof acommon decision function through the communication associated with theawareness structure. The main result of ours is an extension of Krasucki [6] asfollows:
Key words and phrases. Awareness, belief, communication process, consensus, protocol,
agreeing to disagree, no trade theorem.
This article is apreliminary version and the final form will be published elsewhere.
’Corresponding author.
tSupported by the Grand-in-Aid for Scientific Research(C)(2)(N0.12640145) in the Japan
Society for the Promotion of Sciences

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Theorem 1. In the communication process associated with awareness stmcture,consensus can be guaranteed if the communication protocol is an acyclic graph.
This paper organizes as follows: Section 2defines an awareness belief model,
associated information structure with awareness structure and communication
process. Further we present the notion of consensus of the values of adecisionfunction. Section 3ilustrates Theorem 1by using asimple example. In section 4
we show that consensus of the value of adecision function that satisfies the surething principle and is preserved under difference can still be guaranteed in thiscommunication process if the protocol contains no cycle.

2. THE Model
Let be anon-empty finite set called astate space, $N$ aset of finitely many
that is the family of subsets
players at least two, and let denote the field
Q.
Each member of is called an event and each element of called astate.
of$\Omega$au$2^{\Omega}$$\mathcal{F}$$\Omega$$\mathcal{F}$2.1. Belief and Common-Belief. A belief structure is atuple
is aclass ofwhich is astate pace andbelief operators on$B_{E}F$byisdefinedon
operator
mutual belief

in
. The

$\langle\Omega, (B:)_{i\in N}\rangle$$(B_{i})$$\Omega$$\mathrm{z}’ \mathrm{s}$$2^{\Omega}$$2^{\Omega}$.

$B_{E}F= \dot{.}\bigcap_{\in N}B.\cdot F$

Acommon-belief operator is an operator$B_{C}$

on$\Omega$ertysatisfying the fied point prop-$B_{C}F\subseteq B_{E}(F\cap B_{C}F)$
for every $F$ of .
FP
The interpretation of $B\{F$ is the event that believes $F,$ ’whereas $B_{E}F$ isinterpreted as the event that ‘everybody believes F.’ We say that an event $E$ iscommon-belief at if belongs to $B_{C}E$ .
$\mathcal{F}$$‘ i$$\omega$$\omega$

2.2. Awareness and Belief. We present amodel of awareness according toMatsuhisa and Usami [9]. This model follows from E. Dekel, B. L. Lipman and
A. Rustichini [3]. Adifferent approach of awareness models is discussed in R.
. Vardi [4].
. Halpern, Y. Moses andFagin,
isin which
An awareness stmcture is atuple
that
such
operatorsonarnarenessofaclassisabelief structure andAxiom PL (axiom of plausibility) is valid:$\mathrm{M}.\mathrm{Y}$$\mathrm{J}.\mathrm{Y}$$\langle\Omega, (B:)_{i\in N}\rangle$$\langle\Omega, (\mathrm{A}\mathrm{i})\mathrm{i}\mathrm{e}\mathrm{N}, (B:)_{i\in N}\rangle$$(A_{i}):\in N$

PL$B_{:}F\cup B_{i}(\Omega\backslash B_{:}F)\subseteq A\{F$

The mutual awareness operator$\mathcal{F}$$\mathrm{z}’ \mathrm{s}$$A_{E}$

on$\mathcal{F}$for every F in$\mathcal{F}$.

is defined by$A_{E}F=\cap A_{i}Fi\in N^{\cdot}$

An awareness structure is called finite if the state space is afinite set.
is the event that is aware of $F,$ ’whereas $A_{E}F$ is
The interpretation ofinterpreted as the event ‘everybody is aware of F.’ The axiom PL says that isaware of if he believes it or if he believes that he dose not believe it.l$A_{:}F$

$‘ i$

$i$

$F$

lThe axiom PL dues to E. Dekel, B. L. Lipman and A. Rustichini [3]162
and it is said to be publicly
We say that an event $F$ is self-aware of if, and it. An event is said to be evident belief if
aware if
. An event is public belief (oris said to be public belief at state ifaware)
if whenever it occurs all players believe it (orrespectively, it is publicly
they are ffi aware of it.) We can think of public belief as embodying the essenceof what is involved in all players making their direct observations.
$F\subseteq A_{:}F$$i$$i’ s$$T$$F\subseteq A_{E}F$$\omega$$\omega$$T\subseteq B_{:}T$$\in T\subseteq B_{E}T$

2.3. Associated Information Structure. M. Bacharach [2] introduces the strong
epistemic model that is just the Kripke semantics corresponding to the modal
logic . Further he defines the information partition of the state space induced
from the knowledge operator.’ Following his line we generalze the notion of
information partition to the associated information structure as below.
with an awareness structure
The associated information structure$\mathrm{S}5^{2}$$(P\dot{.}):\in N$$\langle$$\Omega$, (Ai),$(B:)\rangle$

is aclass of the mappings$P\dot{.}$of$\Omega$

intodefined by$\mathcal{F}$$P_{\dot{1}}(\omega)=\cap\{T|\omega T\in 2^{\Omega} \in T\subseteq B:T\}$.

to be non(If there is no event $T$ for which $\in T\subseteq B_{:}T$ then we take P.
.
defined.) We callthe ’s evidence set at
evident beliefs since eachAn evidence set is interpreted as the basis for all$T$evident belief is decomposed into aunion of all evidence sets contained in$\cdot(\omega)$$\omega$$i$$P.\cdot(\omega)$$\omega$$i’ \mathrm{s}$$i’ \mathrm{s}$$T$.

Remark 1. The strong epistemic model in M. Bacharach [2] can be interpreted
satisfies , , , 4 and 5such thatas an awareness structure , (Ai),.
operator;
for every
i.e.
Ai(E)
and so Ais the trivial awarenessThis says that an awareness structure is an extension of the strong epistemic
model.
$\langle$$(B:)\rangle$$\Omega$$\mathrm{N}$$B_{:}$$\mathrm{K}$$\mathrm{T}$$=\Omega$$E\in \mathcal{F}$2.4. Posterior Revised. We improve on the notion of posterior as follows: Let, (Ai),
be an awareness structure with acommon-prior . For every
real number , we denote$\langle$$(B:),\mu\rangle$$\Omega$$\mu$$q_{\dot{1}}$$[q\dot{.}]=\{\omega\in\Omega|\mu(X\cap \mathrm{A}.(X)|P.\cdot(\omega))=q:\}$.

We say to be the ’s posterior of $X$ at if belongs to $[q:]$ . We denote by
for aU of $N$ ;. An event is the intersection of the sets
the profilethat is,$i$$q_{}$$\omega$$\omega$

$q$

$[q]$

$(q_{\dot{1}}):\in N$$i$$[q_{\dot{1}}]$$[q]= \bigcap_{\in N}[q.\cdot]$.

We say that the players commonly believe their posteriors
common-belief at ;that is, $\in B_{C}([q])$ .

$q_{}$

of $X$ at$\omega$

if

is

$[q]$

$\omega$$\omega$, in which 0is astate space and K. isstrong epistemic model is atuple ( , (K. )an ’s knowledge operator satisfying the following axioms: For every $E,F$ of ,$K.\cdot(E\cap F)=K_{\dot{1}}E\cap K_{}F$ ;
;
;
.
;
5
4
$2\mathrm{T}\mathrm{h}\mathrm{e}$$\Omega$$\cdot$$:\in N\rangle$$\cdot$$2^{\Omega}$$i$$\mathrm{N}$$K\dot{.}\Omega=\Omega$$\mathrm{T}$$\mathrm{K}$$K.\cdot F\subseteq K.\cdot K_{\dot{1}}F$$3\mathrm{T}\mathrm{h}\mathrm{e}$$i$$\Omega\backslash K.\cdot F$’s information partition$\mathrm{n}_{:}(\omega)=\bigcap_{T}\{T\in 2^{\Omega}|\omega \in K\dot{.}T\}$.

$\mathrm{n}_{:}$$K.\cdot F\subseteq F$$\subseteq K\dot{.}(\Omega\backslash K.\cdot F)$

induced from the knowledge operator K. is defined by$\cdot$163
is the conditional probabilty of the
An interpretation ofawareness section of $X$ under his evidence set at . For the above example,. Therefore
letting $A_{:}(E)=B:(E)\cup B:(\Omega\backslash B.\cdot(E))$ we obtain that A2{{a}at state is
it follows that the player 2’s improved posterior of, as desired.
$\mu(X\cap \mathrm{A}.(X)|P.\cdot(\omega))$$i’ \mathrm{s}$$\omega$$)$$=\{\beta\}$$\{\alpha\}$$\mu(\{\alpha\}\cap$$\alpha$$A_{:}(\{\alpha\})|P_{\dot{1}}(\alpha))=0$

2.5. Decision Function 4. Let be aset of decisions for all players in $N$ . Andecision function is amapping of into . It is said to satisfy the surething principle if it is preserved under disjoint union; that is, for every pair ofdisjoint events and such that if $f\dot{.}(S)=f(T)=d$ then $f_{}(S\cup T)=d$ .
Adecision function is said to be preserved under difference if for all events, $f\dot{.}(S)=f.\cdot(T)=d$ then we have $f_{}(T\backslash S)=d$ .
and such that
If is intended to be aposterior probability, we assume given aprobabilty
measure which is common for all players and some event $X$ . Then is the
$Z$

$Z$

$\mathcal{F}$$f_{}$$\mathrm{z}’ \mathrm{s}$

$T$

$S$

$f_{}$$T$$S$$S\subseteqq T$$f\dot{.}$$f.\cdot$$\mu$mapping of the domain of$\mu$

into the closed interval $[0, 1]$ such that$f\dot{.}(E)=\mu(X\cap A_{i}(X)|E)$

where $\mu(E)\neq 0$ . We plainly observe that thisand is preserved under difference.

$f$,satisfies the sure thing principle

2.6. Communication with Awareness Structure. We assume that playersof the set of noncommunicate by sending messages. Aprotocol is amapping$N\crossN$that assigns to each apair of
into the product set
negative integers
players $(s(t), r(t))$ . Here stands for time and $s(t)$ and $r(t)$ are, respectively, thesender and the recipient of the communication which takes place at time . We
players
consider aprotocol as the directed graph whose vertices are the set of$N$ and such that there is an edge (or an arc) fromto if and only if there areinfinitely many such that $s(t)=i$ and $r(t)=j$ .
Aprotocol 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 the graph contains acyclic path;, with $k\geqq 3$ such that for all $m<k$ ,that is, there are players , ,, and such that communicates directly with
communicates directly with
Definition 1. Acommunication process of revisions of the values of decision, in which
is triple
functionsis afair protocol such that for every , $r(t)=s(t+1)$ , communications proceedis the mapping of into $T$ for at time that is definedin $rounds^{5}$ , andinductively as follows:.
We assume given amapping
Supposeis defined.
:
denotes the value
is the mapping of into that assigns to each state the event:
consisting of all the states 4such thatdefined byis the belief operator induced by:$\mathrm{P}\mathrm{r}$$t$$\mathbb{Z}_{+}$

$t$

$t$

$\mathrm{a}\mathbb{I}$

$i$

$j$

$t$

$i_{1}$$i_{2}$$\ldots$$i_{m}$$i_{k}$$i_{m+1}$$i_{k}$$i_{1}$$\pi$$(f\dot{.}).\cdot\in N$$(\mathrm{P}\mathrm{r}, (Q^{t}\dot{.})(:,t)\in N\mathrm{x}T,$$\mathrm{P}\mathrm{r}(t)=(s(t),r(t))$$(f\dot{.}):\in N\rangle$

$t$

$i$

$\Omega$$Q^{t}.\cdot$$t$$Q^{0}\dot{.}:=P_{}$$\bullet$$Q^{t}.\cdot$$\bullet$$f_{}(Q^{t}.\cdot(\omega))$$-d_{i}^{t}(\omega)$$-\mathrm{t}V_{\dot{1}}$$\Omega$$\mathcal{F}$$\omega$$d^{t}\dot{.}(\xi)=d^{t}\dot{.}(\omega)$$W\dot{.}(\omega)$$-B_{\dot{1}}^{t}$$Q^{t}.\cdot$$B^{t}.\cdot E=\{\omega\in$$\Omega|Q^{t}.\cdot(\omega)\subseteqq E\}$. Bacharach [2]$4\mathrm{C}.\mathrm{f}$$5\mathrm{T}\mathrm{h}\mathrm{a}\mathrm{t}$is, there exists anatural number $m$ such that for all , $s(t)=s(t+m)$
$t$

164
-The recipient $j=r(t)$ at send the message
at $t+1$, and he does not send the message otherwise.
when
is defined as follows:Then-If is not recipient of message at time $t+1$ , then.
$t+1$
, then-If is arecipient of amessage at time$t$

$B_{j}^{t}(W_{j}(\omega))$$\omega\in B_{j}^{t}(W_{j}(\omega))$$\bullet$$Q_{\dot{1}}^{t+1}(\omega)$$i$$Q^{t+1}.\cdot(\omega)=Q^{t}.\cdot(\omega)$$i$$Q^{t+1}.\cdot(\omega)=Q_{i}^{t}(\omega)\cap$$B_{s(t)}^{t}(W_{s(t)}^{t}(\omega))$.

Specificaly the sender sends to the message that he believes that his decision. The communication is said to be cut off in a state ifisfor some and the recipient $j=r(t)$ at .
$i$

$j$

$d_{\mathrm{j}}^{t}(\omega)$$\omega\not\in B_{j}^{t}(W_{j}^{t}(\omega))$$\omega$$t$$t$

2.7. Consensus. The family $\{Q_{}^{t}(\omega)|t=0,1,2, \ldots\}$ is adescending chain inand so the limit
exists in each state where the communication is never cut
off: In fact, there exists sufficiently large
such that for aU $t\geq T$ ,because is finite. Since the protocol is fair, there exist infinitely many such.
that $r(t)=i$ , it follows that
Remark 2. For each the sequence of the domains ofmakes adescending chain, and thus it may be occurred that forin some .
It can be easily observed that
Lemma 1. For every player , the sequence
converges tothe limiting valueof at each state where the communication is nevercut off andis defined; i.e., there exists.
$\mathcal{F}$$Q_{\dot{1}}^{\infty}$$Q^{t}.\cdot=Q_{i}^{T}$$T\in \mathbb{Z}_{+}$$\Omega$$t$$Q_{\dot{1}}^{\infty}$$( \omega)=\lim_{tarrow\infty}Q^{t}.\cdot(\omega)$$Q^{t}\dot{.}(t=0,1, \cdots)$$i$$Q_{\dot{1}}^{\infty}$$Q_{\dot{1}}^{\infty}$$\omega$$\{d_{\dot{1}}^{t}(\omega)|t=0,1,2, \ldots\}$$i$$d^{\infty}.\cdot(\omega)$$(\omega)=\emptyset$$f$$\omega$$(\omega)$$d_{\dot{1}}^{\infty}$$( \omega)=\lim_{tarrow\infty}d^{t}.\cdot(\omega)$

the limiting value of at for . We say thatconsensus on the lmiting values of decision functions
can be guaranteed
processin acommunicationifandare equal for each player $i,j$ andin all the states that the communication is never cut off and thatis defined
$i\in
N$
.
for all
Remark 3. In the communication process, the limiting value of can be reachedin finitely many rounds; i.e., there exists anon-negative integer independent
on such that for every player and for all $t\geqq T$ ,by Lemma 1.

We call$f$$d^{\infty}.\cdot(\omega)=f(Q^{\infty}\dot{.}(\omega))$$\omega$$i$$(f_{\dot{1}}):\in N$$d^{\infty}\dot{.}(\omega)$$d_{\mathrm{j}}^{\infty}(\omega)$$\omega$$Q^{\infty}\dot{.}$

$f$

$T$

$i$

$\omega$$d_{\dot{1}}^{\infty}$$(\omega)=d_{\dot{1}}^{t}(\omega)$

3. EXAMPLE
We show asimple example below in order to explain Theorem 1. We consider
three players , and $C$ , and give their associated information structures at time
$t=0$ defined by Figure 1.
In this case, we assume the event $X=\{1,3,5,7\}$ and the protocolthat contains no cycle. Let us give the common priorfor each state, and their decision functions are given by the common function definedby $f(E)=\mu(X|E)$ . It is noted that the common decision function satisfies thesure thing principle and is preserved under difference.
Then the calculation result with player at
be given in Table 1.
to the recipient B.
Sender sends the message
mixes the message
and his associated information structure, and he makes his revised information
structure then
$A$

$B$

$A_{arrow}^{arrow}Barrowarrow C$$\mu(\omega)=\frac{1}{8}$$\omega\in\Omega$

$f$

$A$

$A$

$B_{A}^{0}(\omega)$$t=\mathrm{O}\mathrm{w}\mathrm{i}\mathrm{u}$

$B$

165

$X$

$Q_{C}^{0}$$Q_{B}^{0}$$Q_{A}^{0}$

FIGURE 1. Associated information structure at the early time t $=0$

TABLE 1. The calculation result with

$Q_{B}^{1}(\omega_{1})$$=$$Q_{B}^{1}(\omega_{2})=\{\omega_{1},\omega_{2},\omega_{3},\omega_{4}\}$$Q_{B}^{1}(\omega_{3})$$=$$Q_{B}^{1}(\omega_{4})=\{\omega_{3},\omega_{4}\}$,$Q_{B}^{1}(\omega_{6})=\{\omega_{6}\}$,$A$$Q_{B}^{1}(\omega_{8})=\{\omega_{8}\}$are not defined here. Next the results in Table
Note that the state , of
2is also obtained by the same way.
$\omega_{5}$$Q_{B}^{1}$$\omega_{7}$TABLE 2. The calculation result with$B$

Then player $C$ ’s revised information structure at $t=2$ is as follows:,$Q_{C}^{2}(\omega_{1})=Q_{C}^{2}(\omega_{4})=\{\omega_{1},\omega_{4}\}Q_{C}^{2}(\omega_{2})=\{\omega_{2},\omega_{3}\}$$Q_{C}^{2}(\omega_{3})=\{\omega_{3}\}$,$Q_{C}^{2}(\omega_{6})=\{\omega_{6}\}$,

Similarly, the communication goes on as follows$Q_{C}^{2}(\omega_{8})=\{\omega_{6},\omega_{8}\}$

166

TABLE 3. The calculation result with $C$
Then Player$B$’s revised information structure at $t=3$ is the next result.

$Q_{B}^{3}(\omega_{1})$$=$$Q_{B}^{3}(\omega_{4})=\{\omega_{1},\omega_{4}\}$$Q_{B}^{3}(\omega_{3})$$=$$\{\omega_{3}\}$,$Q_{B}^{3}(\omega_{6})=\{\omega_{6}\}$,$Q_{B}^{3}(\omega_{8})=\{\omega_{8}\}$TABLE 4. The calculation result with $B$
Player$A$’s revised information structure at $t=4$ is given by as follows:$Q_{A}^{4}(\omega_{1})$$=$$Q_{A}^{4}(\omega_{4})=\{\omega_{1},\omega_{4}\}$$Q_{A}^{4}(\omega_{3})$$=$$\{\omega_{3}\}$,$Q_{A}^{2}(\omega_{6})=Q_{A}^{2}(\omega_{8})=\{\omega_{6},\omega_{8}\}$TABLE 5. The calculation result with$A$

Player $B$ ’s revised information structure at $t=5$ is given by as follows:$Q_{B}^{5}(\omega_{1})$$=$$Q_{B}^{5}(\omega_{4})=\{\omega_{1},\omega_{4}\}$$Q_{B}^{5}(\omega_{3})$$=$$\{\omega_{3}\}$,$Q_{B}^{5}(\omega_{6})=\{\omega_{6}\}$,$Q_{B}^{5}(\omega_{8})=\{\omega_{8}\}$TABLE 6. The calculation result with $B$

167$\omega_{2}$$\omega_{2}$$\omega_{5}$$\omega_{7}$$\omega_{2}$$\omega_{5}$$\omega_{7}$$\mathrm{O}\omega_{6}$($$\omega_{5}$$\mathrm{O}\omega_{8}$$Q_{A}^{6}$$Q_{B}^{6}$$Q_{C}^{6}$

FIGURE 2. Each player’s revised information structure at t $=6$
For $t\geqq 5$ eachis stationary given by Figure 2. AU values of decisionfunctions are equal. Therefore consensus can be guaranteed in this example.
(Note that the states ,andare not defined.)$Q^{t}.\cdot$$\square$$\omega_{2}$$\omega_{5}$$\omega_{7}$4. THE RESULT
4.1. We prove the generalized version of Theorem 1in Matsuhisa et al [8] asfollows:
Theorem 2. In a communication process associated with awareness structure,
suppose that the all decision functions are common for all players, and that thecommon decision function satisfies the sure thing principle and is preserved under difference. Consensus on the limiting values of the decision function can be
guaranteed if the protocol contains no cycle.
In the case that the decision function is given by posterior (Section 2.5), it
immediately follows from Theorem 2that
Corollary 1. Suppose that all players have a common-prior. Conse nsus on thelimiting values of the posteriors for a publicly aware event among all players canstill be guaranteed in the communication if the protocol contains no cycle.

4.2. $RT$-Information Structure. Before proceeding to the proof, we need thenotion of $RT$-iffiormation structure and the fundamental lemma:
Amapping :is said to be an $RT$ information $stmcture^{6}$ on if thefolowing two conditions are true: For each and for every state such thatis defined,$Q$$\Omegaarrow \mathcal{F}$$\Omega$

$i$

(Ref)

$\omega$(Rn)$\in Q(\omega)$$\xi\in Q(\omega)$$\omega$$Q(\omega)$;

implies$Q(\xi)\subseteqq Q(\omega)$.

The following lemma is akey to proving the theorem:-information structure stands for refieive and transitive information structure$6\mathrm{T}\mathrm{h}\mathrm{e}RT$168
Fundamental Lemma. Let be a state in , and an $RT$-infomation structure on O. Suppose that be a decision function that satisfies the sure thing
principle and is preserved under difference.
If there exists an event $H$ of such that the two conditions are $tme$ :For eachof $H$ ,(a) $Q(\xi)$ is defined and it always contained in $H$ , and$Q$$\Omega$$\omega$$f$$\mathcal{F}$$\xi$(b) it is always satisfied that $f(Q(\xi))=f(Q(\omega))$ ,then we obtain that$f(H)=f(Q(\omega))$ .

Asimilar lemma may be found in Matsuhisa and Usami [9].

4.3. Proof of Theorem 2. It is sufficient to prove the following theorem because
Theorem 2immediately follows from it.
Theorem 3. In a communication process associated with an awareness structure,
suppose that the the decision functions are common for all players, and that thecommon decision function satisfies the sure thing principle and is preserved under
difference, if player communicates $his/her$ message directly to player then theirlimiting values of the decision function must be equal.
$j$

$i$

be acommunication process associated with awareness structures where is the common decision function. The protocol has theproperty that, if $s(t)=i$ and $r(t)=j$ then $r(t+m)=i$ and $s(t+m)=j$ for

Prvof. Let$\pi=\langle \mathrm{P}\mathrm{r}, (Q_{\dot{1}}^{t}), f\rangle$

$f$

some

$m$

.

for
is an $RT$-infomation structure, and so is
It should be noted thatevery . Therefore in view of (Ref) and (Rn), it can be plainly observed that
$Q^{\infty}.\cdot$$P_{\dot{1}}$$i$$\mathrm{T}$4$B^{\infty}\dot{.}(F)\subseteqq F$;.

$B^{\infty}.\cdot(F)\subseteqq B^{\infty}.\cdot(B_{\dot{1}}^{\infty}(F))$

is defined, set
For each state at which
completethe proof, it suffices to verify the two.
To
for $l=i,j$andproperties:, and(a) If 4is amember of $H$ then(b) $H$ is contained in the set consisting of all the states 4such that

$H=B_{\dot{1}}^{\infty}$ $(W_{\dot{1}}^{\infty}(\omega))\cap B_{\mathrm{j}}^{\infty}(W_{j}^{\infty}(\omega))$$Q_{\dot{1}}^{\infty}$$\omega$$Q=Q_{l}^{\infty}$$Q(\xi)\subseteqq H$$d_{l}^{\infty}(\xi)=$$d_{l}^{\infty}(\omega)$.

Indeed, if it is the case then, viewing of Fundamental Lemma we obtain thatas required.
for each $l\in\{i,j\}$ , and thus$f(H)=f(Q_{l}^{\infty}(\omega))$$d_{\dot{1}}^{\infty}$, and it follows
,
,
for
that
$\in
H$
,
every
by T. We observe that for

Proof of (a): It foUows from 4thatfrom the definition of,,and thus$Q_{l}^{t+1}$$W_{l}^{\infty}$$B_{\dot{1}}^{\infty}$$(\omega)=d_{j}^{\infty}(\omega)$$Q_{l}^{\infty}(\xi)\subseteqq B_{l}^{\infty}(W_{l}^{\infty}(\omega))$$\xi\in Q_{l}^{\infty}(\xi)\subseteq B_{l}^{\infty}$ $(W_{l}^{\infty}, (\xi))$$(\xi)=W_{l}^{\infty}$ $(\omega)$$(W^{\infty}\dot{.}(\omega))\cap B_{j}^{\infty}(W_{j}^{\infty}(\omega))$$\xi$$Q(\xi)\subseteq$.

Proof of (b): In view of T it is easily observed that H$\Omega|d_{l}^{\infty}(\xi)=d_{l}^{\infty}(\omega)\}$$\mathit{1}’\in\{i,j\}\backslash \{l\}$, in completing the proof.

$\subseteqq W_{l}^{\infty}(\omega)=\{\xi\in$$\square$169
4.4. As aconsequence of Theorem 2we can obtain ageneralized version of theAgreement theorem of Geanakoplos and Polemarchakis [5] and that of Matsuhisa
and Kamiyama [7].
ucture,Corollary 2. In a communication process associated with awarenesssuppose that the all decision functions are common for all players, and that thecommon decision function satisfies the sure thing principle and is preserved under
difference, if the players in the process are only two then their limiting values ofthe decision function must be equal.
$st$

Proof. Immediately follows from Theorem 2.

口

4.5. Conclusion. In this paper we have shown that the nature that consensus
can be guaranteed in acommunication process is not the restrictions on theplayers’ knowledge operators but their awareness and belief.
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(Emiko Fukuda) 東京工業大学大学院社会理工学研究科, DEPARTMENT OF VALUE AND DECISION SCIENCE, GRADUATE School OF DECISION SCIENCE AND TECHNOLOGY, Tokyo
INSTITUTE OF TECHNOLOGY,-OKAYAMA 2-CHOUME12-1, MEGURO-KU, Tokyo 152$\mathrm{O}\mathrm{H}$8552. JAPAN.
-rnail address: emiko(Ovaldes.titech. ac.jp$E$(Takashi Matsuhisa) 茨木工業高等専門学校, DEPARTMENT OF LIBERAL ARTS AND Scl-

IBARAKI NATIONAL COLLEGE OF TECHNOLOGY, NAKANE 866, HITACHINAKA-SHI,
IBARAKI 312-8508, JAPAN
$E$-mail address: mathisa$ge. ibaraki-ct.ac.jp
ENCES,, UNDERGRADUATE Course OF ELECTRIC ENGI(Hisato Sasanuma)
NEERING, IBARAKI NATIONAL COLLEGE OF TECHNOLOGY. Current Address: NHK (Nippon
Houso KYOKAI), SHIBUYA-KU, Tokyo 150-8081, JApAN.
$ae*]\mathrm{i}\ovalbox{\tt\small REJECT}\overline{\ovalbox{\tt\small REJECT}}\Leftrightarrow\ovalbox{\tt\small REJECT} \mathrm{P}\ovalbox{\tt\small REJECT}\propto\mp \mathrm{R}$