数理解析研究所講究録 1222 巻 2001 年 152-159

152

No trade theorem in an S-4 logic model慶応義塾大学大学院経済学研究科(Graduate School of Economics, Keio University) *平瀬 和基 (Kazuki Hirase) \dag er

Abstract

This paper introduces an application of the S-4 logic. There are two aims in this paper. Aim
1is to check the relation between our model and the S-4 logic. We 11 see the soundness andcompleteness of the S-4 logic with respect to the model by using the concept of structure. Aim
2is to prove avariation of no trade theorem in the model.

1Introduction
The word “knowledge” and especialy “common knowledge” plays avery important rolein game theory. Intuitively, an event is common knowledge if everyone knows it, everyone
knows that everyone knows it, everyone knows that everyone knows that everyone knows
it and so on. Then how can we treat (common) knowledge fomffiy?
Aumann (1976) tried to solve this problem. He introduced the formal notion of common knowledge using set based and partitioned information structure and showed, so
caUd, agreeing to disagree
After Aumann, many papers have studied knowledge. Milgrom (1981), and Monderer
and Samet (1989) treated knowledge by different approaches. Milgrom (1981) applied axiomatic approach 2 to modelng knowledge. Monderer and Samet (1989) used probabilty
appr0ach3. They managed to approximate knowledge with belief. We note that theseapproaches also use partiotional information structure.
Samet (1990) have studied non-partitional information structure. He showed agreeing
to disagree theorem based on non-partitional information structure.
This paper also studies non-partitional information structure like Samet. We would
h.ke to prove kind of no trade theorem which is introduced by Milgrom and Stokey(1982)$\mathrm{t}\mathrm{h}\infty \mathrm{r}\mathrm{e}\mathrm{m}^{1}$*Mita2-15-45, Mnatoh,),$\mathrm{T}\mathrm{o}\infty$$108^{\mathrm{q}}-\vee \mathfrak{N}5$, Japanhighmy ne jp
insists that if players’ posteriors for agiven event are common knowledge, then thesemust be equal, even though they are based on different information.
is the approach which defines the set of all states in which aplayer knows agiven event. After
M 釉。 paper mny papers have boen 一一吋 this approach.
’Prob 市 ih.ty approach defines the event in which player believes $E$ with prolx 山 ilty at least .
$\uparrow \mathrm{B}\mathrm{m}\mathrm{a}\mathrm{i}\mathrm{l}$$\mathrm{d}\mathrm{d}\mathrm{r}\infty:\mathrm{h}\mathrm{h}\subset \mathrm{M}$$\mathrm{p}\mathrm{c}$lIhk$\mathrm{t}\mathrm{h}\infty \mathrm{m}$$2\mathrm{T}\mathrm{h}\mathrm{i}\mathrm{s}$$\mathrm{m}’\epsilon$

$n$

$p$

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after showing the relation between the model and the logic. In section 5, we will see thelogic. avariation of no trade theorem is proved in section 8.
model is one of the$\mathrm{S}\ovalbox{\tt\small REJECT}$

2S-4 logic
S-4 logic is denoted as $Sy$ , , $AR$, Prove.
$\langle$$S$means symbols. Symbols consists of $N$, $PV$ , logical connectives, and players’ modal
operators. $N$ means aset of players and . Now, we restrict the number of the players
to 2persons for simplicity. But we can extend the results to persons case easily. $PV$ isaset of propositional variables, or atomic sentences. Logical connectives are, ,and . Players’ modal operators areand .
The second element of S-4 logic is S. means aset of sentences, or aset of formulae.
$Sy$

$i$

$j$

$n$

$\wedge,$$\neg$$\coprod_{\dot{1}}$$\vee,$$arrow$$(, )$$\coprod_{j}$

$S$

$S$

is inductively constructed from$L$

inductively.

(S1) : $PV\subseteq S$(SI) :$\phi,\psi\in S\Rightarrow\neg\emptyset$,$\phi$$arrow\psi$,$\phi$$\wedge\psi$,$\phi$$\vee\psi,\square _{n}\phi\in S(n=i,j)$(53) :Every sentence is constructed by affiite number ofapplications of (SI) and (S2).

The third element of the S-4 logic is $AR$ . All means axioms and rules. $AR$ consists, modal axioms and rules.
of$PL$ is propositional logic, or aset of aU tautologies, that is, for all.
$PL$$\phi,\psi,\chi\in S$(PLl) :$\phiarrow(\psiarrow\phi)$(PL2) : $(\phiarrow(\psiarrow\chi))arrow((\phiarrow\psi)arrow(\phiarrow\chi)$(PL3) :$(\neg\emptysetarrow\neg\psi)arrow((\neg\emptysetarrow\psi)arrow\phi)$(PL4) $:wedge\psi$$arrow\phi$(PL5) :$\phi$(MP) :$\frac{\phiarrow\psi\phi}{\psi}$$arrow\phi\vee\psi$$(\wedge-rule)$:$\frac{\phiarrow\psi\phiarrow\chi}{\phiarrow\psi\wedge\chi}$$(\vee-rule)$:$\frac{\psi\phi\chiarrow\phi}{\psi\vee\chiarrow\phi}$

where$\phi,\psi,\chi\in S$.

For modal part, we assume axioms $K,T,4$ and N. $K$ is, in other words, the Axiom
of Distribution. is the Axiom of Knowledge. 4is the Positive Introspection. And $N$ is
$T$

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the Necessitation rule.
(K) :
(T) :

(4) :

$(\square _{1},(\phiarrow\psi)arrow(\square _{\hslash}\phiarrow\square _{n}\psi))$$\square _{\mathfrak{n}}\phiarrow\phi$$\square _{n}\phiarrow\square _{n}\coprod_{n}\emptyset$(N) :$\frac{\phi}{\square _{n}\phi}$

where$\phi,\psi\in S$

and $n=i,j$With these axioms and rules, we can deffie the provability Prov. of asentence in the
logic.
Definition 1. Aproof is afinite tree satisfying (PR1) and (PR2).
(PRl) :Asentence is associated with each node, and the sentence
associated with every leaf node is an instance of $(PL1)-(PL5)$ , $K$,
, or 4.
$T$(Pm) :Each adjoining node forms an instance of (MP), ( -rule),$(\vee-mle)$ , or (N).
$\wedge$

We say thatis provable in the S-4 logic if and only if there exist aproof whichroot is associated with .
$\phi(\in S)$$\phi$

3Structure.

is anonempty finite state space. So is called aset of events.
Players’ information functions and is afunction fiom the state space to the event
set . The setmeans the event which player recognize when the real state is .
The setis called player n’s information set or possibilty set at .
We assume that each players’ information function satisfy the following.

Structure is$\langle\Omega, P_{}, P_{j}\rangle$$2^{\Omega}$$\Omega$$P_{}$$2^{\Omega}$$\Omega$$P_{j}$$P_{n}(\omega)$$n$$\omega$$P_{n}(\omega)$$\omega$$(P-1)$ :$\omega$$(P-2)$ : $d$for$\in P_{n}(\omega)$$\in P_{n}(\omega)\Rightarrow P_{n}(d)\subseteq P_{n}(\omega)$$\forall\omega,d$$\in\Omega$

and $n=:,j$ .

P-l means the condition that aplayer never excludes the real state. When the realstate is , the playern thinks that may have occurred. Prom P-2, we have that if thereis astate , so that $\xi\in P_{n}(d)$ and. So, P-2 says that playerthen $d$at can make consideration as follows: uThe state is excluded. If it were the state $d$ ,Iwould not exclude . Thus it must be that the state is not .P-l and P-2 play very
important roles in the relation to the S-4 logic. We call these three tuples an information$\omega$$\omega$$\xi\not\in P_{||}(\omega)$$\xi$$\not\in P_{n}(\omega)$

$n$

$\xi$

$\omega$$\xi$$\omega^{\prime n}$155
structure.

Partiotional information structure need to assume P-3: $d$$\in P_{n}(\omega)\Rightarrow P_{n}(d)\supseteq P_{n}(\omega)$.
forw,
and there is astatethat
To interpret P-3, consider the case that $d$$P(d)$. Then, P-3 says that aplayer at can conclude, from the fact that heis not in(she) can not exclude 4, that the state is not $d$, astate at which he (she) would be able
to exclude 4.
$\omega’\in\Omega$$\xi\in P(\omega)$$\in P(\omega)$$\omega$

Note the following proposition holds.
information function
Proposition 1. Player
if there is apartition of such that for any
partition that contains .
$n’ \mathrm{s}$$P_{n}$$\omega\in\Omega$$\Omega$satisfies P-l, 2, and 3if and onlyis the element of thethe set$P_{n}(\omega)$$\omega$intersect andandsatisfies P-l, 2, and 3. If$P_{l}.(\omega)=P_{n}(d)$$=P_{n}(\xi)$. By P-l we havethen by P-2 and 3, we haveCl
. The other direction is obvious.

Proof. Suppose that
$P_{n}(\omega)\cap P_{\iota}.(\omega’)$$P_{*}.(\omega)$$P_{n}$$\xi\in$$P_{r\iota}(d)$$\bigcup_{\omega\in\Omega}P_{n}(\omega)=\Omega$

We don’t assume P-3. Thus we treat anon-partitional information structure.

4ModelThe model
consists of $Sy$ , , an information structure, atruth assignment , and a. Atruth assignment is afunction fromvalidity , i.e.,the valdity of the. From this truth assignment,
to the set
sentences. For any sentence and , we define the validity as follows.
$S$$\mathrm{M}$$\models$$PV\cross\Omega$$\mathrm{M}$$\pi$$=\langle Sy,S,\Omega, P_{\dot{\iota}}, P_{j},\pi, \models\rangle$$\pi$$\{\mathrm{T}, [perp]\}$$\models \mathrm{d}\mathrm{e}\mathrm{c}\mathrm{i}\mathrm{d}\mathrm{e}\mathrm{s}$$\psi$$\phi$(VI): For any $v\in PV,$$\models_{\omega}v\Leftrightarrow\pi(v,\omega)=\mathrm{T}$if and only ifdoes not hold.

(V2):$\models_{\omega}\neg\emptyset$(V3):$\models_{\omega}\emptysetarrow\psi\Leftrightarrow\models_{\omega}\neg\emptyset$(V4):$\models_{\omega}\emptyset\wedge\psi\Leftrightarrow\models_{\mathrm{I}\theta}\phi \mathrm{a}\mathrm{n}\mathrm{d}\models_{\omega}\neg\emptyset$(V5):$\models_{d}‘\emptyset\vee\psi\Leftrightarrow\models_{\mathrm{I}\theta}\phi \mathrm{o}\mathrm{r}\models_{\omega}\psi$(V6):$\models_{\omega}\square _{n}\phi\Leftrightarrow P_{n}(\omega)\subseteq\{\xi\in\Omega:\models_{\zeta}\emptyset\}$$\models_{\omega}\emptyset$or$\models_{\omega}\psi$

for $n=i,j$

5Soundness and CompletenessWith these preparations of logic, structure, and model, we can prove the following theorem, This theorem is well known by logicians as soundness and completeness (of the
S-4 logic). The theorem insists that asentence is provable in the logic if and only if themodelsentence is vald at every state in any$\mathrm{S}$

$4$

156
in theTheorem 1. A sentence is $p.m$vable in the S-4 logic
formodel M.
, we can verify that each sentence 4 in $AR$ is valid
sketch of the proof For soundness, first,in the model using the properties P-l and P-2. For completenessat. Second, we can see that asentence is valid in
we can constmct the canonicalthe canonical model if and only if the sentence is provable in the S-4 logic. Third, we
can easily show that if asentence is valid in any S-4 model then the sentence is valid
in the canonical model, because the canonical model is also amodel. These arguments0show the completeness, that is, asentence is provable if it is vald.
$\forall\omega\in\Omega$$\Leftrightarrow\models_{\omega}\phi$$\phi$$(\Rightarrow)$$\forall\omega\in\Omega$$(\Leftarrow)$$\mathrm{m}\mathrm{o}\mathrm{d}\mathrm{e}1^{5}$$\phi$$\phi$$\phi$$\phi$

6Knowledge
Hrom this section, we analyze the model, S-4 logic model. Since no trade Theorem treats
an epistemic condition for no trade, we have to define the concept of knowledge formally.
This section defines the knowledge, common knowledge and mutual knowledge.
. $(n=i,j)$ .
at if and only if
Definition 2. Player knows an event
From the meaning of the infomation function, the player knows that some state in, (of course) the player know the state in $E$ has
has occurred. Hence if
occurred. With this interpretation, we have defined the player’s knowledge.
Before defining common knowledge, we define the self-evident event.
is aself evident between:and if and only if
Definition 3. An event
for $n=:,j$ .
$F$ is aself-evident event among, if whenever it occurs players and know that it
knowledge.
occurs. Now we define common
Definition 4. An event $E$ is common knowledge at between and if and only if there.
exist aself evident event $F$ between :and such that
$E$ is common knowledge between:and, if there is aself-evident event between and$E$containing whose occurrence implies .
Here, we define mutual knowledge, concept between knowledge and common knowledge. Intuitively, an event is mutual knowledge when all players know the event. The
formal definition is as follows.
. An event $E$ is rnutual knowledge at between
Definition 5. Let.
and if and only if
It is obvious that the following proposition holds.
Proposition 2. If an event $E$ is mutual knowledge at between and , player knows$E$ at. And if an event $E$ is common knowledge at between :and , $E$ is mutual
knowledge at between :and .
$E(\in 2^{\Omega})$$n$$P_{n}(\omega)\subseteq E$$\omega$$n$$n$$P_{n}(\omega)\subseteq e$$P_{1l}(\omega)$$F(\in 2^{\Omega})$$j$$\in F\Rightarrow$$\omega$$P_{n}(\omega)\subseteq F$

$S$

$i$

$i$

$\omega$

$j$

$j$

$j$

$\omega\in F\subseteq E$

$i$

$j$

$j$

$\omega$$R:=P.\cdot(\omega)\cup P_{j}(\omega)$$R\subseteq E$$j$$\omega$$\omega$$\omega$$\omega$$j$.
must be modified by asentence\epsilon &Chellas (1980), Hughes and Cresswell (1996)$4\mathrm{A}$$i$$\omega$rule$\not\in$$\phi$$arrow\psi$

$i$

$i$

$j$

$j$

157
7Economy
In this section, we consider apure exchange economy where the players interact. Pureexchange economy is awell known concept in economics literature. It is denoted as$C,e,$,(,. $N$ is aset of players. (is an information structure.
Here, there are kind of commodities, and $C:=\Re_{+}^{l}$ is acommodity space,:$\Omegaarrow C\cross C$
is an initial endowment of commodities for players. \^U $Uj$ :is aplayer’s utility function. We assume that each player’s utilty function is strictlyincreasing with respect to . We suppose the existence of aprior and it is common forboth players. (So, aprior is called common prior.) Let the cornrnon prior be aprobabilty
measure on , . We denote the common prior to $E$ as $\mu(E)$ . And we assume $\mu(E)>0$
for any event $E$ .
In this economy, given : $\Omegaarrow C\cross C$ , players compute expected utility $ex$ ante,interim, and $ex$ post$\mathrm{N}$$P_{},$$U_{\dot{1}},$$P_{\mathrm{j}},$$\Omega,P_{\dot{1}}$$U_{j},\mu\rangle$$P_{j}\rangle$$e=(\mathrm{q}., e_{j})$$l$$\Omega \mathrm{x}Carrow\Re$$C$$\Omega$

$\mu$

$x$

$\bullet$$\bullet$$\bullet$player$n’ \mathrm{s}ex$

player$n’ \mathrm{s}$player$n’ \mathrm{s}ex$ante expected utilty is$\sum_{\omega\in\Omega}\mu(\{\omega\})U_{n}(\omega, x_{n}(\Omega))$interim expected utility at$\omega$post expected utilty at$\omega$

is
is

.

.

$\sum_{d\in P_{\mathfrak{n}}(\omega)}\frac{\mu(\{d\})U_{*}(\omega’,x_{n}(d))}{\mu(\{P_{n}(\omega)\})}$$U_{n}(\omega,x_{*},(\omega))$

8No Trade Theorem
To show no trade theorem, we have to define trade, feasible trade, and Pareto optimalty.
Definition 6(trade and feasible trade). Atrade $t=(t:, t_{j})$ is afunction from,. Atrade is feasible if and only if for all$\Re^{l}\cross\Re^{l}$$\Omega$to$\omega\in\Omega$$t$$e_{n}(\omega)+t_{n}(\omega)\geq 0$$\forall n\in N$

and$t_{:}(\omega)+t_{\mathrm{j}}(\omega)\leq 0$.

Feasibilty of atrade means that each player’s trade have to be within the budget ofhis (her) own initial endowment and that trade does not make any commodity.

Definition 7(Pareto optimality).
only if there does not exist :
$y$

$x$

:

$\Omegaarrow C\mathrm{x}C$$\Omegaarrow C\mathrm{x}C$

is ($ex$ ante) Pareto optimal if andsuch that$\forall n\in N,E_{n}[U_{n}(y_{n})]\geq E_{n}[U_{n}(x_{*}.)]$

and$\exists n\in N,E_{n}[U_{n}(y_{n})]>E_{n}[U_{n}(x_{n})]$.

158
Pareto optimalty is one of the most famous normative concept in economics.
With these preparations, we can prove the following theorem.
Theorem 2(No Trade Theorem). Suppose that an initial endowment is $ex$ ante
Pareto optimal and that is a feasible trade. Then, if it is mutual knowledge at thateach player $ex$ post weakly prefers to , every player at is interim indifferent berween
and the zero trade, that is,
$e$

$t$

$\omega$$t$$e$$\omega$$t$$\sum_{d\in P_{\hslash}(\omega)}\frac{\mu(\{d\})U_{n}(d,\mathrm{e}_{n}(d)+t_{n}^{*}(\omega’))}{\mu(\{P_{1*}(\omega)\})}=\sum_{d\in P_{n}(\cdot)}\frac{\mu(\{d\})U_{ll}(\omega,e_{n}(d))}{\mu(\{P_{n}(\omega)\})}$, $n=i,j$ .

The result says that the trade is meaningless for every player at with his (her)infomation. So Iwould like to think this as some stabilty, or an equilbrium. Hence, notrade theorem shows an epistemic condition for an equilbrium.
$t$

Proof Denote$R,\forall n\in N$$R:=P_{\dot{1}}(\omega)\cup P_{\mathrm{j}}(\omega)$$\omega$. Rrom the condition of mutual knowledge,,$U_{n}(\omega’,e_{n}(\omega’)+t_{n}(\omega’))\geq U_{*}.(\omega’,e_{n}(\omega’))$Now, define $t_{n}^{*}:=tnlR$ for
Viewing $t^{*}ex$ ante.

$\forall n\in N^{6}$$\forall\omega’\in$.

(1)( $1_{R}(d)$ $=1$ if $d$ $\in R$ , and $1_{R}(d)$ $=0$ otherwise.)$\sum_{d\in\Omega}\mu(\{\omega’\})U_{1*}(\omega’,e_{\mathfrak{n}}(\omega’)+t_{\iota}^{*}.(\omega’))$$= \sum_{d\in\Omega}\mu(\{\omega’\})U_{n}(\omega’,e_{n}(\omega’)+t_{n}(\omega’)1_{R}(\omega’))$$= \sum_{d\in\Omega}\mu(\{\omega’\})1_{R}(\omega’)U_{n}(\omega’,e_{n}(\omega’)+t_{n}(\omega’))+\sum_{d\in\Omega}\mu(\{\omega’\})1_{R}\sigma(\omega’)U_{n}(\omega’,e_{\mathfrak{n}}(\omega’))$$= \sum_{d\in R}\mu(\{\omega’\})U_{n}(\omega’,e_{n}(\omega’)+t_{n}(d))+\sum_{d\in R^{G}}\mu(\{\omega’\})U_{n}(\omega’,e_{n}(\omega’))$$\geq\sum_{d\in R}\mu(\{\omega’\})U_{1}.(\omega’,e_{n}(d))+\sum_{d\in R^{C}}\mu(\{\omega’\})U_{*}.(d, e_{n}(\omega’))$$= \sum_{d\in\Omega}\mu(\{\omega’\})U_{n}(\omega’,e_{n}(\omega’)+t_{n}^{*}(\omega’))$(2)

where $p$ denotes the complement of , and the inequalty follows&0m (1).
$\in R$ .
Then theSuppose that the inequalty of (1) is strict for $\exists n\in N$ and. This contradicts our assumption that theinequality of (2) must be strict for
initial endowment is Pareto optimal. Hence,$R$$\ovalbox{\tt\small REJECT}$$\exists n\in N$$e$,$U_{1}.(\omega’, e_{n}(\omega’)+t_{n}(\omega’))=U_{n}(\omega’, e_{\mathfrak{n}}(\omega’))$is also feasible.
\mbox{\boldmath $\tau$}Ea&player at is $ex$ post indifferent between and the zero trade.

$\iota f$$\omega$$t$.

$\forall d\in R,n=:,j^{7}$

(3)

159
Rom

$P\dot{.}(\omega)\subseteq R$

and$P_{j}(\omega)\subseteq R,.(3)$implies;$\sum_{d\in P_{n}(\omega)}\frac{\mu(\{d\})U_{n}(d,e_{n}(d)+t_{n}^{*}(d))}{\mu(\{P_{n}(\omega)\})}=\sum_{d\in P_{\hslash}(\omega)}\frac{\mu(\{d\})U_{n}(\omega,e_{n}(d))}{\mu(\{P_{n}(\omega)\})},n=:,j$Therefore, each player at$\omega$

is interim indifferent between and the zero trade.
$t$.
$\square$Note that common knowledge is not necessary for our theorem, while the original no
trade theorem in Milgrom and Stokey need common knowledge. For this point, our assumption is weaker than the original, since common knowledge means mutual knowledge.
Also note the original needs the word each player interim weakly prefers to $en$ but
does not need the word each player $ex$ post weakly prefers to $e.$ ” For this point, ourassumption is more strong because if aplayer $ex$ post weakly prefers to he interim
weakly prefers to in this case.
$u$

$u$

$t$

$t$

$t$

$t$

$e$

$e$

References
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Chellas, B.: Modal Logic: An Introduction.: Cambridge University Press, 1980.
Hughes, G., and Cresswell, M.: A New Introduction to Modal Logic. :Routledge, 1996.
Milgrom, P.: “An Axiomatic Characterization of Common Knowledge,” Econometrica,
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Milgrom, P., and Stokey, N.: “Information, Thade and Common Knowledge,” Journal of
Economic Theory, 26 (1982), 17-27.
Monderer, D., and Samet, D.: “Approximating Common Knowledge with Common Be
lief,” Games and Economic Behavior, 1(1989), 170190.
Samet, D.: “Ignoring Ignorance and Agreeing to Disagree,” Journal of Economic Theory,52 (1990), 190-207