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Lattice structure of knowledge and agreeing to disagree
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JOURNAL OF
Mathematical ECONOMICS
ELSEVIER
Journal of Mathematical Economics _,~" (I 9971 389-410
Lattice structure of knowledge and agreeing to disagree Takashi Matsuhisa, Kazuyoshi Kamiyama Department O! A rt~ and 5( ~em'e~. Ihar~ k' \'c tiop ~ [ ('o/h',,,e (!( Tc~ hn dok'y .,Voka~w 8,56. Hitachtnaka. lharaki 312. ,lap,in
Submitted October 1995: accepted Max Ig96
Abstract This paper is a contribution to the studx of the underlying mathematical structure of common-knowledge, which gives the aell-known result of Aumann about the impossibility of "agreeing to disagree'. We present the Bayesian subiective probability model with player's belief; i.e. a triple (--~i- ~9-,- /x), in which i is a player. -~i is a lattice in the field of sets of a state space -Q. , 7 is a correspondence assigning to each state o) a filter ,7-(co) in ~ i , and g is a common-prior. For this model, we impose none of the important restrictions on the information structure in the A u m a n n - B a c h a r a c h model: axiom of knowledge K~. axiom of transparency K, and axiom of wisdom K~. We can extend both the disagreement theorem of Aumann and the agreement theorem of Geanacoplos and Polemarchakis under the assumption that each ~ , is an Artinian lattice. ./EL c/assi/~cotion." C72: (:'73: D82:D83
Knov, ledge ,tructure: Belief of person: ('ommon-bclicf: Bayesian sub ecme probability model: Information structure: Common-prior assumption: Posterior probability Kevuords:
1. I n t r o d u c t i o n T h e idea of c o m m o n - k n o w l e d g e is central to the g a m e t h e o r y a n d the econ o m i c s o f i n f o r m a t i o n . For e x a m p l e , the a n a l y s i s o f a g a m e with c o m p l e t e i n f o r m a t i o n starts with the a s s u m p t i o n that the structure o f the g a m e is c o m m o n k n o w l e d g e a m o n g the players. I n t u i t i v e l y speaking, two p e r s o n s (1, 2) are said to h a v e c o m m o n - k n o w l e d g e o f an e v e n t if b o t h k n o w it. 1 k n o w s that 2 knox, s it. 2 03t)4-4068/"97.,'SI7.00 i~ It)97 Else,.icr Science S.,\. All right,, rescrxed PII S 0 3 0 4 - 4 0 6 8 ( 9 7 ) 0 0 7 8 5 9
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knows that 1 knows it. 1 knows that 2 knows that 1 knows it. and so on. Despite recent progress in a theory of games, the important matter remains more or less obscure: a formal structure of common-knowledge, especially that of player's knowledge. This paper intends to address this problem. Aumann (1976) took the first step in the direction to solve this problem. He introduced the formal notion of common-knowledge and showed that persons who have the same prior cannot agree to disagree: i.e., if their posteriors for a given event are common knowledge, then these must be equal, even though they are based on different information. The information structure of a person in Aumann's model is given by' partitions. Bacharach (1985) found that Aumann's model was equivalent to the modal logic S s. He introduced an event-based model with knowledge operator and obtained an extension of the disagreement theorem of Aumann. The information structure is given not by partitions but by the knowledge operator of a person, and it involves three properties: K t, persons can only really know' that something has happened if it actually has happened; K : , they cannot know that something has happened without knowing that they' know it; K ~, if they do not know that they, do not know something, then they know, it. He posed the underlying question (among others) of how strong the rationality assumptions of economic analysis ought to be. He also pointed out the widely held view that even the weakest systems of modal logics are unrealistically strong, even to the extent that they are at variance with the normal meaning of "i knows that'. and that systems similar to that of his models that contain the axioms, especially, K 3, are 'objectionable" in this regard. Binmore and Brandenburger (1990) also directed suspicion at axiom K~. Consequently, we would like to develop a theory of information structures without the three axioms. In fact. Samet (1990) succeeded in extending Aumann's theorem to a proposition-based model whose information structure involves only, the two properties K~ and K , , together with the assumption that every person's knowledge is finitely generated. Furthermore, he showed (Samet, 1992) that the theorem does not generally hold without this assumption. This paper follows the same line of in~,estigation. The purpose is to show, that the information structures of a person's knowledge (K~ and K , ) are superfluous for ruling out "agreeing to disagree'. We shall show' that the nature of the disagreement theorem of Aumann and of the agreement theorem of Geanakoplos and Polemarchakis (1982) is not in the restrictions on the information sets, or in the restrictions on the knowledge operators, but is in the lattice structures of a person's knowledge, especially in its finitely' generated structure. To describe the idea of the lattice structure of knowledge, we shall consider the three frameworks of belief. These all involve a set ,Q of possible states of the world, a field .'7- of subsets of J'), a set 1 of persons and, for each person i, a
T. Mat.~uhisa, K. Kumi~amo / J o t t r n a l ¢!/ Malhemotico/ Ecommliu.~ 27 ( 1997J 389 410
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lattice ~ ' r To accomplish this. we shall identify, each ~/i with a sublattice of ,~7. Indeed, in recent work by Bacharach (1985) and Samet (1990). the elements of a lattice have been taken to be sentences (precisely. equivalence classes of sentences), with conjunction and disjunction determining the meet and join operators. We shall define the following three kinds of structure. (1) A "knowledge structure" of person i is :7;. in which ,7-i is a mapping of .(2 to the family 2-~, of all subclasses of J i that assign a filter of .~i to each state co. The intended interpretation of ~7~(co) is as the set of sentences that. i believes to be possible at co. To specily that i's 'beliefs to be possible" are a filter is to suppose that i believes that the conjunction of two sentences is possible whenever i believes each of them to be possible separately. One also supposes that i believes every, logical implication of a sentence that i believes, where the lattice-ordering is taken to represent implication. (2) A "possibility correspondence" of person i is P,. in which P, is a mapping of .(2 to .-~i that assigns a non-void element of ~ , to each co in .Q. The set of /',(co) is an element of .e'i to be interpreted as the set of all the states of nature that i believes to be possible at co. (This is the Aumann set-up.) (3) An "epistemic model' of person i is K . in ,xhich K i is a knowledge operator of ~'Jl to .~- such that K,(AAB)=K~(A)~K~(B) for all sentences A. B: the set K ( A ) will be interpreted as the set of all the states for which i believes A to be possible at co. (This is the Bacharach set-up.) Our main results are as folio,as. In the case where each 2"i is an Artinian lattice (i.e. it satisfies the descending-chain condition), we can first sho,a that the three frameworks of belief are equivalent to one other. Therefore. with a knowledge structure ,)--, are uniquely associated the knowledge operator K~ and the possibility correspondance P~: for each T of ~(co). K,(T) will be interpreted as the set of all the states l\~r which i believes T to be true at co. and K,P,(co) will be taken as the smallest belief at co that is the minimal set in all sets K,(T) li)r T in /-~(~o). We say that a sentence A is "common-belief" at co if it includes the smallest set in all the sets of states that the persons, as a whole, believe to be true there. We shall define by q,,,, = / ~ ( X [ K P,(co)) i s posterior of event X at a state co. Thus. it will be interpreted as the posterior probability of X. given the belief of / a t co. In these circumstances, we can also show that. if ~,c impose a common-prior for each person, then the disagreement theorem is xalid: if each posterior q,,, is common-belief among the persons, then q .... q,, for each i. /. In other words. agreeing to disagree is still impossible, exen though the associated knowledge operators of a person satisfy none of K~. K~ or K ~. This paper is organized as follows. Section 2 describes the characterization of Artinian lattices in terms of their filters, which is prerequisite for the study of finitely generated knowledge structures in the later sections. In Section 3. we
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present the notion of knowledge structure and establish the connection between knowledge structures and possibility sets. In Section 4, we introduce the two types of epistemic model, i.e. CK 0- and K0-epistemic models, and define the D"iequivalence relation on them. In Sections 5 and 6, we define complete and finitely generated knowledge structures. In Section 7, we establish the equivalence theorem of the three frameworks. In Section 8, we present the notion of common-belief. Section 9 describes the epistemic model of Bacharach in our framework. We remark that. in his model, A u m a n n ' s notion of common-knowledge is the same as our notion of common-belief. In Section 10, we define Bayesian subjective probability models. In Sections 11 and 12, we establish the common-belief results: the generalization of A u m a n n ' s theorem and that of Geanacoplos-Polemarchakis" in finitely generated knowledge structure models.
2. Foundations We recall that a poser is a set with a binary relation which satisfies refllexiuity, antisvmmetrv and transitit'itx. We let .Q be a non-void set. called a 'state space', and let 1 be a set of finitely many persons, and ,'7 be a field of subsets 2 ~*, i.e. the poser of all subsets of ~Q. Each member of ,~- is called an 'event" and each element of O called a 'state'. 2.1. Semi-complete lattices
By a lattice of ,'7 we shall mean a non-void subposet of a field ,'7 that is closed under set theoretical intersection and union of each two members. A lattice is said to be semi-complete provided that it is closed under intersection, i.e. the intersection of each non-void subclass of the lattice belongs again to it. The void-lattice {~i} is exceptional in various (mostly trivial) respects. Throughout this paper we assume that a lattice is not the void-lattice. 2.2. A,fllrer m a lattice J '
By this, one means a non-void subclass V of a lattice J ' such that Ft: the intersection of two members of V always belongs to V, and F~: i f T ~
V and T c E ~ Z / , then E ~ V
A filter is said to be proper if it is a proper subclass of a lattice ~ ' . Plainly. the void-set never belongs to every proper filter. If P is a non-void member of ,~/', then the subclass {T ~ 2 / 1 P c T} forms a proper filter: it is written ( P ) . We shall call it the principal filter generated by P and P is the generator.
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2..7. Artinian lattices One says that a lattice ..~' is Artinian if every descending chain of members of .~L i.e. T I >T,>T,>
...
>T,
>T,,,.
I >
...
such that T,, +~ T,. i (it is called a strictly descending chain) is finite. We shall prove the characterization theorem of Artinian lattices in terms of the filters considered in later sections. It is more or less well known and the proof will be included for completeness. Pt'oposition I. Let (.2 be a state space alul+7 a field (~/subsets. Suppose t/tat ~/' i,s a lattice d , 7 . T/ten. the.fbllowin? three statemenLs are equiz'alent: ( a) a lattice ~ is Artinian: (bY e+'erv n(m-t'+Ud subclass ~>~ q f z~ /',~osses<~,,s +l mi#limal member: (c) e r e o f i ' l t e r in W ~ is principal. Proqf. F r o m (a) to (h). Let T, be a m e m b e r of . v . If ~ is not minimal, then it properly contains a m e m b e r T~ of ,J'. If T~ is not minimal, then it properly contains a m e m b e r T, of t . Inductively. if we ha~e found Z],, that is not minimal. then it contains properly a m e m b e r T,,÷ ~ of / . In this ,xay. we could construct an infinite chain, which is impossible. Thus. there exists a natural number N such that T~ is a minimal member. F r o m ( D ) g ~ ( c ) . Let g be a filter in .-/ and 71, a m e m b e r in V. If V was not identical to (T.). then there exists a m e m b e r T I in V that does not belong to (T0). such that Tt is properly contained in 7,,. Proceeding inductively, we can find an ascending chain of principal filters in "_,". i.e.
The subclass {71, ] n - (), I. 2. 3 . . . . } of ~ possesse~ a minimal member, say Tu. It is then clear that the principal filter (77~) must be equal to V. as required. F r o m (c) to (a). Suppose that we have a descending chain of members in J . i.e. T, > T , > T , >
... >T, D T
i>...
Let V,, be the principal filter generated by 7-,,. Then. the subclass {V,, In G N} is an ascending chain of filters in .~/. W e shall denote by V the union of all the filters ~,. It is easil} verified that g is a filter: consequently, it is principal and generated by a m e m b e r T in ..~. Hence. there exists a natural n u m b e r N such that T ~ g v, and it follo,xs immediately that V = V~. Hence, one obtains that 77~ = T~. i = ...=7". []
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In view of parts (a) and (b) in Proposition 1. the following corollary is readily proved. Corollary 1. Ererv Artinian lattice is semi-complete.
3. Knowledge structures and possibility sets In this section, we shall present the concepts of knowledge structures and possibility sets, and we shall observe the connection between them. 3.1. A knowledge structure qf person i By this. we shall mean a pair ( ~ i . ;7-) in which ~ i is a lattice of(, is a lattice of J . and P, is a correspondence of [1 into a class of non-void events of .Lv¢~.,.We shall call its image P;(a~) a possibility set of person i at a state ~o. A system of possibility correspondences is a family of all possibility correspondences ( ~ i . P,) for person / i n I. 3.3. The connection between knowledge structures and possibility correspondences We state the result in this section, which is easily verified in view of the definitions above. Proposition 2. Let [) be a state space, .<7 a.field qfsubsets and i a person. There exists a bijection ( ~ i , 57,) ~ ( 2 i . Pi) hetween the class qfl all knowledge structures of person i and the class qfl all possibili(v correspondences (qf that person. This bijeetion assigns to each (J',..9"- i) the pair (.5/i. P,) in which P, is the correspondence that assigns to each state oo the generator (Tf a filter
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Corollary 2. Let ~1 be a state .space a~td 1 a set o f persons. There exists a bijection between the system o f knowledge structures and the systems Oflpossibilio" correspondet~'es.
4. Epistemic models In this section, we shall define epistemic models and shall establish the connection between knowledge structures and these models. 4.1. K~,- and CK~,-epistemic models
By an "epistemic model" of person i we shall mean a pair (,~-. K i) in which K, is an operator on .'7 such that the axiom EM:
K,~C)=f) and K i O = G
is valid. This K~ is called a "knowledge operator" of person i. The event K i E will be interpreted as the set of states of nature for which i believes E to be possible. An epistemic model (,9-. K,) is said to be a K0-epistemic model if the axiom Ko:
For each E. F ~ , Y - . K i ( E I " F )
=K.E'~KiF:
is true. It is also called a Ko-knowledge operator of person i. An epistemic model (:~-. K,) is said to be a complete K0-epistemic model (or simply CKo-epistemic model) if the axiom
is valid. It is called a CK0-knowledge operator of ]person i. It should be noted that each CK0-knowledge operator is always a K0-knowledge operator. A system of Ko-epistemic models (or of CK~,-epistemic models) is a family that consists of all CK~- (or K,,-)epistemic models ( S . Kz) for person i in I. 4.2. Ez'em CK~-knowledge operator is order 1,'eserting
In fact. viewing axiom CK,b. we can easily prove the following. For each events E. F in :7. we have OP:
If E c = f . t h e n
K I£cKf
Remark. Every, K0-knowledge operator is also order preserving. 4.3. The Si-equivalence relation
We shall define the equivalence relation on the class of epistemic models as follows. Given a lattice ~.71 of J associated with person i, we shall say that two
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epistemic models ( j - , K,) and ( J . K I) are J)-equivalent (or simply say that K, and K I are ~i-equivalent), provided that K, and K I are identical on ~ i ; i.e. KiT= KIT for each event T of ~'i. We shall denote by [(.'7, K,)] the equivalence class of the epistemic model ( J . K,).
4.4. Minimal CK~,-epistemic models Given a lattice .S, of .'7, we shall say that a CK{)-(Ko-)epistemic model K~) is minimal in its ~i-equivalence class if an event KiE is contained in KIE for every event E of ,~. and for every C K , - (K0-)knowledge operator K I that is 2G-equivalent to K,. (J.
Proposition 3. Suppose that J~ is a lattice of :7 associated with person i. Then, there exists a unique minimal CKc~- (or K{}-)episternic model. Furthermore, the correspondence (,~. Ki ~ ) ~ [ ( j , K / )] is a bijection between the class q{ all minimal CKII- (or Ko-)epistemic models and the class of all ..~,-equicalence classes o[ the CK o- (or Ko-)epistemic model. Proq/2 We prove for CK0-epistemic models (:7. K~). Let K / be the operator on .'7 that assigns to each event E the intersection of all events KIE. i.e.
X/£=
nx;£ A"
where (,'7, K I) runs through all the CK0-epistemic models that are ~i-equivalent to (.Y, K~). We shall f r s t note that the pair (,7. K i ) is actually a CKo-epistemic model. In fact, it is easy to observe that K / fulfills axiom EM; furthermore, we shall prove that the knowledge operator K, ~ satisfies axiom CK o. This is because, viewing the definition of K / and axiom CK o for K I, one obtains that, for every subclass {/&[ A ~ A} of 5 -
A~ ~
K I A~ I
,',~
The converse is verified as follows. It is evident from OP that. if K I is also __~,-equivalent to Kp then KI(["I A IE~)_-cKi/£a for each A. Consequently, one obtains K / ( l " l ,~ ~Ea) ~ ('1Ae ~Ki'EA, and the required result follows. In view of the definition of K / , it is readily verified that (,'7, Ks*) is ~.V~-equivalent to ( J , Ki). This will prove our proposition. Indeed, if it is the case, then, by the definition of K / , it follows plainly that the CKo-epistemic model ( J , Ki x ) is minimal, and it also follows that the mapping [ ( J , Ki)] ~ (,~, K / ) gives a bijection; its inverse assigns to each minimal CKo-epistemic model (,Y, Ki :~) the 2i-equivalence class [ ( J , Ki* )] in completing the proof. []
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5. Complete knowledge structure In this section, we shall introduce the notion of complete knowledge structures and establish the connection between CK~-epistemic models and these knowledge structures. 5. I. A complete kmm'ledge structure ¢~/l~ers(m i By this. we shall mean a knowledge structure ( J , . . / - z ) of person i with which the associated lattice .J, is semi-complete. A system of complete knowledge structure is a family that consisting of all complete knowledge structures for i in 1. We shall prove the following. Proposition 4. Let -(2 be a stale space a l u l , 7 a fieht oj subsets. Suppose that 2/I ix a semi-complete lattice associated with i)ers(m i. T/tell. there exists a b~jection (2/i..7-) ~ [(.7. K [ )] between the clas.~ o[all complete knowledge structures qf person i and tile class olall .y)-equicale+lce classes O/the CK~-epistemic model Of person i. This bijection ass&ns to each knowledee structure (z/i, 7 , ) with possibilio" sets P , ( o ) the .:/i-equiz'alence class (g the CK~Tepistemic model [ ( ~ . K [ )]. ill which K,' is" ttle operator on :7 t/tat assigns to each ezent E the euent K [ E t/tat consists c~[ all the states co such that P(oo) is contailwd in E. Its im'erse assigns to each ,J ,-equiualence class ¢~!t/le CK~-epislenffc model [(,/-. K;)] the knowledge structure ( . ~ , J i ) , in which ,/- is the correspondence that assigns to each state ~o the fiher ,7i(co) that consisl.s o/all the members T (~/ -/i such that eo belongs to KiT.
Before proceeding to the proof, we shall first prove the following. Lemma I. Gizen a semi-complete lattice ,,/, we .find a mappillg (,~i, Pi)~-~ ( J . K[ ) qf the class 0/' all po~sibili O corre.w~mdenee ofperson i to the class of" all minimal ['K~-epistemic models o! l)ersoJt i. This mapping assiglts to each (,L/i, Pi) the pair (,~-. K[ ). in which K , i~ the operator Oil ,~ d
We first note that the pair C7. K ) is a CK~-epistemic model. Indeed. it is easy to obser,,e that K[,Q = ~2. On noting that P,(eo) is not void. we obtain that K / ~ = ;~. so that the operator K [ satisfies axiom EM. Furthermore. in view of the definition of K [ . it is easy to verify that K, satisfies axiom CK 0` and the required result follows.
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It remains to be shown that the CK,~-epistemic model (.'7, K [ ) is actually minimal. Suppose that (.Y-, K I) is the arbitrary CK0-epistemic model that is 2/~ equivalent to (.~-. K,* ). For each E in ,~-, if ~ is an element of K / E , and noting that ~: belongs to K[ Pi(~) = KIPi( ~-), then we can easily observe that Pi(~ ) is contained in E. From OP, we obtain that ~ K i & ( ~ ) ~ K IE. []
5.2. Proqf of Proposition 4 Suppose that ( J , K i) is a CK0-epistemic model. We shall define .Ui to be the correspondence that assigns to each state ~o in .(2 the subclass J i ( ° ~ ) that consists of all the members T of ~', such that ~o belongs to KiT, i.e.
J-,(,.,,) = {:r
I ,o c
We first note that the pair (~'{.i, ,'7,) is actually' a complete knowledge structure of person i. Indeed, it is first noted that .Ti(w) is a filter in ~ i . This is because, viewing axioms CK 0 and OP, we can plainly verify, that ,U,(w) satisfies the conditions F~ and F,. On nothing that ,(2 belongs to ,~7-~(~o) by axiom EM. it follows immediately that J , ( w ) is not void. We shall denote by Pi(w) the intersection of all the members of J-i(w). P,(w) is the generator of a filter J , ( w ) , because, viewing axiom CK 0, we can easily observe that P,(~o) is the minimal member and it is not void, since the void-set does not belong to J ~ ( w ) by axiom EM. The correspondence P, that assigns P,(~o) to each w is plainly a possibility correspondence into ~ } \ { Q } . Hence. we have shown that ( ~ i - 47,) is a complete knowledge structure with possibility sets P,(~o). Thus, we have defined the correspondence
@: ( J , x,)--, ( J ; , ,,7) of the class of all CK0-epistemic models into the class of all complete knowledge structures. Furthermore, in view of the definition of ..t'~-equivalence classes, it follows that the correspondence @ induces the injection of the class of all ~,¢i-equivalence classes of CK0-epistemic models into the class of all complete knowledge structures. To verify that the induced injection actually makes a surjection, it suffices to prove that @ does also. Suppose that (2s I. ,U,) is a complete knowledge structure of person i. According to Proposition 2. we shall denote by P,(w) the possibility set that corresponds to this structure. In view of Lemma I, one obtains the minimal CK0-epistemic model (J-. K [ ) that corresponds to (4/i, <7i). in which the CK0-knowledge operator K [ is defined in the same way as in Proposition 4: i.e. Ki* E = {w ~ ~Q I Pi(w) c=E} for each E of ,-7. We shall show that ( ~ i , J , ) is precisely the image of (
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minimal member of .:/i(w) and the required result immediately' follows. This shows that the correspondence g' is a surjection. []
5.3. Further discussion Proposition 5. Let ,(2 be a ,state space, :7 a,field Of subsets atzd i a person, lf C7-, K,) is a CKo-epistemic model, then there exists a unique complete knowled~g,e ,~trueture ( ~ i , "7-,) .sueh that J i is the subclass q/" ,7 that consists q/" all the erents T contained m K T, and ( J , K i) i,v Ji-equiz'alent to the minimal CKo-epistemie model that c'orre,v~onds to (W~, , 7 ), Proo/. We first note that ~ i is a semi-complete lattice of ~-. In fact, it follows plainly from axiom EM that ~'i is not the void-lattice. Viewing OP. we observe that, for each ft, T, of ,--~i, we have
T / QT~_cK,T 1 ; q K , T : ~ K , ( T ~ , Q T : ) Furthermore, viewing axiom C K . , we obserxe that, for each subclass {Ta I k ¢ ,1} of -~i, we have
N TA~ n
K~=Ki
as required. The complete knowledge structure (c/I. 7 ) will be defined such that .7, is a correspondence that assigns to each state ~o ill .(2 the filter J , ( ~ o ) that consists of all the events T of : / i such that w belongs to KiT. It follows plainly from axiom CK 0 that each filter , 7 ( w ) is principal. The generator P~(w) is the intersection of all members of ,'£i(~o) but is not a void-set, since, in the view of Axiom EM, the void-set does not belong to ,Y(o~). According to Lemma 1. let ( Y , Ki* ) be the minimal CK~-epistemic model that corresponds to (.-.~i. ,Zi ). It can be verified that two knowledge operators K i and K i are _/, equivalent. In fact. for each T of J i - each state w belongs to K i T if and only if P,(~o) is contained in T, which is equivalent to the condition that T belongs to ,'f~(~o). i.e. w belongs to K,T. Consequently,, viewing Proposition 4, we obtain that ( J , , .~:i) corresponds uniquely to (~7, K,) up to ~-equivalence. []
6. Finitely generated knowledge structures In this section, we shall introduce the notion of finitely generated knowledge structures. By "a finitely generated knowledge structure' of person i, we shall mean a knowledge structure of i with which the associated lattice is Artinian. A system of finitely generated knowledge structures is a family of all finitely generated
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knowledge structures for persons i in /. It should be noted that each finitely generated knowledge structure is always complete, which immediately follows from Corollary 1. The proposition below is analogous to Proposition 4. Proposition 6. Let ~2 be a state space and ,'7 a field of subsets. Suppose that ~ , is an Artinian lattice associated with person i. Then, there exists a bijection (,~,,, J , ) ~ [ ( J - , K [ )] bem'een the class of all finitely generated knowledge structures ofl person i and the class of all 2Yl-equil'alence classes of the Ko-epistemic model of person i. This bijection is defined in the same way as in Proposition 4. Proof. We can prove in the same way as Proposition 4, and shall omit the details. It should be noted that, for each K0-knowledge operator K,, the subclass . J , ( w ) = {Te~5~ ! ~o ~ K,T} is a principal filter in J',. This is because it is easy to observe that 3,(~o) makes a filter in ~ i ; furthermore, since ~ i is Artinian, it follows from Proposition 1 that the filter has the minimal member P~(aJ). i.e. the generator of .%(o0). []
7. Trinity of belief models We are now in a position to state the equivalence theorem of the three frameworks of belief: the class of all systems of knowledge structures for I, that of all minimal epistemic model for 1, and that of all possibility correspondences for 1. We shall say that two frameworks are equivalent if there exists a bijection between these classes, and shall say that two systems are equivalent if there exists a bijection between them. Main Theorem. Let ~ be a state space, :7 a.field qf" subsets 2 ~ and let 1 be a set of persons. Then, the jollowing two statements are true: (i) The jollowing three frameworks are equit'alent to one other: the class of all systems of complete knowledge structures f o r 1: that q f all systems of minimal CKo-epistemic models for 1: and that qf all systems qJ'possibility correspondences for I. (ii) Gilen a family o f Artinian lattices z? I q [ , 7 associated with each person i in 1. the fi)llowing three systems are equicalent: the system of finitely generated knowledge structures ,]'or 1: that of minimal Ko-epistemic models f o r 1; and that of possibility correspondences.fi~r 1. Proof. For (i). The equivalence between the class of all systems of complete knowledge structures and that of all systems of minimal CK0-epistemic models, through all the .~-equivalence classes of them, is established by the family of bijections in Propositions 4 and 5, together with Proposition 3. The equivalence
7~ Matsuhi~a, K. Kami~ama / Joutvml q/ Maltlemati~ al Ec~mcmm's 27 (1997J 389 -410
401
between the class of all systems of complete knowledge structures and that of all systems of possibility correspondences is also established by the family of bijections in Corollary 2. For (it). This is an immediate consequence of Proposition 6, together with Corollary 2.
Remark. In view of the Main theorem, the partitional model of Aumann (:~-~, P,) can be characterized as the complete knowledge structure (.~..<7-,) such that each .~,(w) makes a maximal filter and each state to belongs to the generator of
J,(to). From now to the end of this paper, we suppose that (._1". <'7-,) is a complete (or finitely generated) knowledge structure, and that (--:/i. P,) is the associated possibility correspondences to it. where ~ i is a semi-complete (or an Artinian) lattice of J - associated person i. We suppose that ( J . K,) is a CK 0- (or K0-)epistemic model in the .Jcequivalence class of (.7. K . ). where (<'5-. K, * ) is the minimal CK~-epistemic model that corresponds to (..~,...7,) according to the Main theorem.
8. Belief and common-belief In this section, we present the notion of belief and that of common-belief in our framework.
8.1. The beliE(to) as the set of states believed to be true there by the community of persons as a whole. We have to note the folh)wing lemmas.
Lemma 2. Each belie[ base .~,( to) is depe#ldem only on the knowledge structure ( J i. <7-). It is independent ~?[the choices ~?[representatices o[ the "~i-equhalence
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class ~f K,. Moreot'er. the same applies.fi)r each common-belief base ,~( w ) only on the system {( ~ i , .7-i) [ i ~ 1}. Proof. This immediately follows from the definition of the _~g-equivalence relation. [] Lemma 3. Each belie/" base ,Z~i(oo) is closed under intersection and it possesses the minimal element K, Pi( °9 )" Consequently, each common-belief base 2g( w ) is also closed under intersection. Proof. Since J i ( w ) is the filter generated by P,(o9). and K i satisfies axiom CK 0` the first assertion follows immediately, from OP. The remaining assertion is clear. {Z. 8.2. Belief and common-belief We shall say that person i believes event E to be true at w if there exists a member M,(~o) in ~',(~o) such that Mi(w) is contained in E. An event E is said to be "common-belief" to be true among l at oJ if there exists a member M(o~) in .Zd(w) such that M ( w ) is contained in E. Remark. The two concepts above are dependent only on the system of knowledge structures, i.e. they are independent of the choices of representatives of the J',-equivalence class of (,~-. K,) for each i. In fact, this immediately follows from Lemma '
9. Epistemic model of Bacharach In this section, we describe the epistemic model of Bacharach in our knowledge structure framework. 9.1. An epistemic model of Bacharach By' this, we shall mean an epistemic model ( J , axioms, together with axiom CK 0` are satisfied: K I"
KiEC=E
K~"
KiEC=K, KiE.
K,) such that the following
for each E, F in ,7. A system of epistemic models of Bacharach is a family {(J-, K i ) l i ~ I} in which (,'7, K,) is an epistemic model of Bacharach.
T. Mat,~utU.~'a, K. Katnivatna / J o u r n a l o / M a t h e m a l i { a ! E~ om~mi~ s 27 ¢1997) 389 410
403
9.2. The knowledx, e structure that corresponds to the Bacharach mode/ We shall now prove the following.
Proposition 7. Suppose that { ( ~ . K,)i i e 1} is a ,vstem of epistemic models ~/ Bacharach. Let {( .~i. <*)-i)t i e I} be a system ~?/"pairs ( ..7 i, J i ) in which k~I i.'~ the subclass of <'~- t/tat consists (~/"all dte elents T such that KiT = T, and )-, is the corre.g~ondence that assigns to each state to the subclass .7-/(02) o / all the el'ents in ,Z/i to which to belongs. Then, {( c/I. .7i)l i ~ I} is a svstenl ~?['complete kmm'ledge structures to which {(.U. K,)] i e l} is the correspondint4 system of the minimal CK~-epistemic model.s. Viewing part (i) of the Main theorem, we have readily' verified that the system of epistemic models of Bacharach is equivalent to the system of complete knowledge structures above.
ProofqfPropositio#7 7. In view of Proposition 5. together with axioms K 1 and K~. it is readily verified that the pair ( J i - , U , ) is actually a complete knowledge structure. It should be noted that ~ , = {K,E[ E e + 7 } and S , ( 0 2 ) = {K k.[ t o e K i E. E e ,9- } {or every i in 1: this follows immediately from axioms K i and K~. Now. we shall show that, for each E in ~7. K i E - { t o e (_,). I Pi( 02) c= E}. To prove it. suppose that to is any state in .(2. If P/(to) is contained in E, then it follows from OP that 02 ~ Pi(w) = K P,(02) c= K E. The converse is clear: indeed, in view of the above points about .<7, and
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10. Bayesian subjective probability models In this section, we shall define the Bayesian subjective probability model that possesses knowledge structure and shall introduce the notion that a posterior is common-belief. We define the conditional probability / x ( E I F ) by
/*( E ~ F ) #(ElF)
-
I~( F )
for each event E, F of , ~ such that p,(F) 4= 0. 10.1. A Bayesian subjectire probabili O" model that possesses knowledge structure By this, we shall mean a family {(~-~'i, "/,, # i ) l i ~ I} in which a pair (2VI. ,'2-,) is a knowledge structure of i with possibility sets P,(~o), and ~, is a probability measure on a o-field ~ such that. for each i in I. :~, includes all the belief bases ~>Y~(~o) for w in /2: K, is the knowledge operator that assigns to each E of :Y- the event K~E that consists of all the states o~ such that P ( w ) is contained in E. The model will be said to possess complete (or finitely generated) knowledge structure if each associated knowledge structure is complete (or finitely generated). It will be said to be with 'countable range" if the range of each correspondence , ~ is countable. A Bayesian subjective probability model will be said to satisfy the "commonprior assumption', provided that p_~ = #j for every i, j in 1. 10.2. Posterior probability q,~o Let X be an event in ~ , . Suppose that co is a state in ,Q such that I~,(K,P,(w)) ~ O. By the "posterior probability" of event X. given the belief of person i at w, we shall mean the conditional probability
q,,o = #,( X I K , P,( w ) ) Remark. Each posterior q,., is independent of the choice of representatives of the Jl-equivalence class of (:7. Ki). It should be noted that. in the epistemic model of Bacharach, the posterior q,~ coincides with the ordinal one: /z(XI P;(~o)). This is the posterior, given the knowledge of person i. 10.3. A person i belieres h i s / h e r posterior probability We shall say that person i believes the posterior probability of event X, given the belief at ~, if person i believes the event E.,~ to be true at ~, where Ei,o is defined by
E,,. = { ~
~21 #,( X I K,P,( ~ )) - q,,,. # , ( K I P , ( ~ ) ) 4- 0}
T. Matsuhisa, K. Kamiyama / Journal o[Mathemalical Ecotlomicv 27 !1997) 389 410
405
In other words, there exists a member Mi(to) in ,3~i(to) such that Mi(to) is contained in E,,,.
Remark. The concept that a person i believes h i s / h e r posterior is dependent only on the Bayesian subjective probability model that possesses knowledge structure. In fact, in view of Lemma 2, it follows plainly that both events M,(to) and E,,, with inclusion relation are all independent of the choice of representatives of the _Ti-equivalence classes of (,'7. K). 10.4. The posterior prohabilit3." is commo~>heli~q We shall say that the posterior probability of event X. given the belief of person i at to, is common-belief, if the event E~,,, (as above) is common-belief to be true at to. In other words, there exists an event M(to) in ,~(to) such that M(to) is contained in E,~,.
Remark. In view of the preceding remark in Subsection 10.3. we have readily verified that this concept is also dependent only on the Bayesian subjective probability model that possesses knowledge structure.
10.5. Samet decompo.sitiotz We shall say that two states (to and to') are K,-equivalent. provided that
X P,(to) = X,P,(to') Obviously this is an equivalence relation on a state space _(2. We shall say that an event possesses the 'Samet decomposition" for i if it can be decomposed into a disjoint union of K,-equivalence classes. The following properties can be easily verified. Suppose that ( - / i - , ' 7 , ) is a finitely generated (or complete) knowledge structure. that (--~i. Pi ) denotes the possibility correspondences associated with it. and that (:7. K i) is a corresponding K,,- (or C K , - ) epistemic model. Then. we obtain the following properties: KPI: For each t o ~ .(2. t o e K,Pi(to) KP, : For each to ~ f2. if ( ~ K i P(to). then K, P,( .~ ) ,c K i Pi(w) SD: For each t o e .(2. every member of the belief base .7/(to) of person i possesses the Samet decomposition for i. i.e. for every T of Y)(~o)
KiT= L A / ( ( )
(disjoinl union)
where A , ( ( ) is the K~-equivalence class of ~ and ( runs through a set
K,T Remarlc In view of the definition of the .v-equivalence class, it should be noted that the concept of Samet decomposition is dependent only on a knowledge
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structure. In other words, it is independent of the choice of representatives of the Si-equivalence class of (,~-. K,).
11. Agreeing to disagree In this section, we shall prove a generalization of the disagreement theorem of Aumann (1976). Before proceeding, we shall show the following lemmas.
Lemma 4. Suppose that ( ~ i , K,) is a K.-epistemic model that corresponds to a .finitely generated knowledge structure (,Ji. , 7 ). Every non-l'oid subclass J ) of the class {KiT IT¢,'S,} possesses the minimal element. Consequently, el'era" stricth' descending chain in { KiT IT ~,,7)} is finite. Proof. Let J ' , denote {KJa I Ta e ~ i . A ~ ~1}. where ~l is an index set. Then. by Proposition 1. it follows plainly that the subclass {TAiTa eL/i . A e A} has a minimal element T0. Hence, by OP, we obtain that KJo is contained in KiTa for each a ~ A: it is minimal in J } , as asserted. Let {KIT,, IT,, <~i, ,7 e N} be a strictly decending chain in { K J I T ~ : / i } . Then. by the result above, it follows plainly that the chain has a minimal element K,T v for some natural number N. Hence. it must be finite. [] The following result is the key to proving our Disagreement theorem and Agreement theorem.
Fundamental lemma. Suppose that {(,Ji. ,~,. #)[ i ~ I} is a Bayesian subjectil'e probabilio" model that possesses finitely generated knowledge structure with countable range, and that it satisfies the common-prior assumption. Let X. M and N be i.t-measurable events. Suppose that ~o is a state such that the probability #( K iP,( w) (~ N) is non-xero. Let qi denote the posterior probabiIio': #( X I K , Pi( to ) (~ N ). lf M is a member of the belief base /Y, i( o9) of i with which M is contained in the el'ent E that consists qf all the states E such that #( XI K,P,( ~) C ' X ) = q , and #( KiP,( E ) (~U) 4~O then we obtain the equali O" ~ ( X ] M C , N ) =qr Proof We shall observe the two points made above: first, that the class {K~P¢(E) ¢ 3 N I E ~ . Q } is countable: secondly, that the event M possesses the Samet decomposition into countably many tt-measurable components. In fact. since each
T. Matsuhisa, K. Kamiyama /Journal :~['Mathematical E¢ onomics 27 ¢] 997) 389-4 I0
407
knowledge structure has countable range, it is easily obse~'ed that the number of possibility sets is countable and that the class of events K,P,(-6 ) for all sc in 12 is also countable. Furthermore, in view of properties KP~ and KP 2, it can be verified that, for each -6. A,(-6) is written by the relative complement of the countable union
c_ with respect to K,P,(,~). Therefore, A~(6) belongs to the o--field .~,. In view of the definition of K, equivalence, we can easily observe that the correspondence Ai(-6) ~ KiPi(:~) makes an injection of the class of all Ki-equivalence classes into the class of all K,P,(-6) for ~ in .(2: the former class of the components A,(s~) is also countable. Therefore, it follows from SD that each member of , ~ ( o J ) can be decomposed into a countable union of /,-measurable components A~(~); from this, the second assertion follows. Consequently, in view of KP 1 and KP~. it follows plainly from Lemma 3 that the class { K , P , ( - 6 ) a N I -6~ .(2} makes a finitely nested reflexive and transitive information structure in the sense of Samet (1992). Hence, similarly to Samet (1990, Proof of Theorem 7, pp. 200-202), we can prove that # ( X [ A i ( - 6 ) C " N) = qi for every component A,(s~) of the Samet decomposition of M, and that t x ( X I M C ~ N ) = q , f o r e v e r y i. []
Disagreement theorem. Suppose that { ( ~ ' . ,7~,. /x)l i ~ I} is a Bayesian subjectil'e probability model that possesses finitely generated knowledge structure with countable range, and that it satisfies the common-prior assumption. Let q~o denote the posterior probabili~., ol~ecent X. gicen the belie[" of person i at a state w. lf .fi)r ecer 3' person i in I, the posterior probabili U q,,, is common-belief among I at oo, then the persons cannot agree to disagree, i.e. q,,, = q,o .fi>r each i, j in I. Proofi Suppose that (,:7, K~) is a K 0-epistemic model that corresponds to (Be.". J - ) , and let A,(-6) denote the K,-equivalence class of a state £. Let E denote the event {-6~ .(2 ] tx(X] K : P ~ ( - 6 ) ) = q , , , , # ( K , P , ( - 6 ) ) e:0} In view of the assumption of the theorem, we first note that there exists an event M(co) in . ~ ( w ) such that M(w)__c f " l , :E,¢,,. Indeed, for every i in I, there exists N:(w) in , ~ ( w ) that is contained in E,,. Then the intersection 0 i~ :N,(w), denoted by M(oo). is also contained in f"l ,~ IE,,,. It is evident from Lemma 3 that M ( w ) belongs to ~>£(w) and the required result follows. In view' of the Fundamental lemma, one obtains that, for every i in 1. the following equality holds: /,(X I M ( w ) ) = q,,. Therefore. we obtain that q,o, = %,, for every j.
408
?2 M a t s u h i s a , K. K a m i y a m a / J o u r n a l o f M a t h e m a t i c a l Economic~ 2 7 (1997) 3 8 9 - 4 1 0
12. Dialogue and agreement theorem In this section, we shall present the communication process between two persons and shall show a generalization of the agreement theorem of Geanakoplos and Polemarchakis (1982). Throughout, we suppose that {(.5¢I, Y,. /x)[i = 1, 2} is a Bayesian subjective probability model that possesses complete knowledge structure and that it satisfies the common-prior assumption. Let ( ~ i , P) denote the possibility correspondences of person 1 and let (J.~, Q) denote that of person 2. Suppose that {C~, K,)li = l. 2} is a corresponding system of CK0-epistemic models. Let X be a /,-measurable event.
12.1. A dialogue on ecent X between two persons l and 2 By this, we shall mean the process ~" of revision of the posteriors of an event X at ~o by persons 1 and 2 defined stepwise as follows.
Step 1. Let q,~obe the posterior /X(X I K I P(to)) and let E~ denote the event
{ ~ Ill MXl K~P( ~)) = q',o, /X( K~P( ~)) :~ 0} which is assumed to be a member of the belief base .Sfl(w). Thus. the event E 1 will be interpreted as the information that person l believes the posterior q~ to be true at ~o. Person 1 announces E~l,,to person 2 and then person 2 revises h i s / h e r posterior r~o=/X(XJ K~.O(w) N E Io). Let Fo~ denote the event
{~I2L
/X(XIK:Q(~))=r',,
/ X ( K z Q ( ~ ) ) =#0 }
which is assumed to be a member of the belief base ,X,(co). The event F~ will be interpreted as the information that person 2 believes the revised posterior r~, to be true at ¢o. Person 2 announces F~ to person 1.
Step t. Having received person 2"s announcement, person 1 revises h i s / h e r posterior
qL=#{XlK~P(w)N ,
1~'2 t=1
Let E~, denote the event
(
~121
in,'t
/X X , K t P ( ~ ) N
Fi
=
/
=
(
q;,. # K,P(s~)C~ ,
#:0 t
.
}
which is assumed to be a member of the class .~/~(o)). Person 1 announces this to person 2 and then person 2 revises h i s / h e r posterior
(
',)
,-';=/x XlK:Q(,o) n n e , ,,
k=l
,
12 Matsuhisa, K. Katni3ama / Journal q/Mathematical Econontics 27 (1997) 389-410
409
Let F,r, denote the event
/~.(21p.
XlK:O(~)r~
I
,
* =,-,~,,a J(
/
K_,(2(s~)r>
E.~ ~0
,
which is assumed to be a member of the class ,Z',(w). Person 2 announces this to person 1. We shall sap' that the dialogue ~r converges at step T if the equality qi'o = rio holds for all t >~ T.
12.2. The dialogue is determined only by the system of knowledge structures This means that the dialogue is independent of the choice of representatives of the Ji-equivalence class of ( J . K,) for each i = 1. 2. In fact. on noting that E~ is a member of ,Z/i(~o) and F~ a member of A"~(~o) for every k. 1. we can easily observe that these events and the posteriors qi'o and r,"o are all independent of the choice of representatives of the Yl-equivalence class of K~, The result immediately follows as required. 12.3. The agreement theorem Agreement theorem. Suppose that {(L/i, ,/-,, # ) I i = 1, 2} is a Bavesiwl subjecti~e probability model that possesses .finitely gellerated knowledge structure with countable range, amt that it satL~fies the common-prior assumption. Then, the dialogue ,'7 defined as above conrerges in.finitely mal~v steps, i.e. there exists a natural number T such that q~,, = rio ./or l > T. Proqfi Let E~I] and hi'' be the same as in Subsection 12.1. We first note that
t k
I
u~
• "- . . . .
is a descending chain in ,/~(~o) and that
[ ' 'k/9F';Iz=l'23 I
] .... )I'
is in ~,(~o). In view of Lemma 4, it follows plainly that there exists a natural number T such that I
f
t
N E:o N C -
k-]
k
]
and
N l-I
t
O<
-
/
I
for each t~> T. We shall denote 0 : ,E,~ by Eand shall denote f " l r F,: by F. By Lemma 3, it is readily verified that E is el member of ,~(~o) and F is a
410
T. Matsuhisa. K. Kamiyama/ Journal (~fMathema#cal Economic~27 (1997) 389-410
m e m b e r of ,7#~(w). T h e r e f o r e , it f o l l o w s plainly that there exists a sufficiently large natural n u m b e r T s u c h that, for e a c h t >~ T, E is c o n t a i n e d in
{ ~
u(XIKIP(~)NF)=q~,
tx(K,P(~)f3F)4:O}
a n d F is in
{~~ ~
tx( X I K2Q( ~ ) N E) = ;;. #( K:Q( ~ ) N E) 4: O}
T h e r e f o r e , m v i e w of the F u n d a m e n t a l l e m m a , we o b t a i n that
q2= p,( Xl E N F ) :r;) for each t > / T .
[]
Remark.
In general, n e i t h e r E r n o r F T m a k e s c o m m o n - b e l i e f be true at w, e v e n if a d i a l o g u e ~ c o n v e r g e s at step T.
Acknowledgements T h e a u t h o r s t h a n k the e d i t o r a n d the a n o n y m o u s referees for p o i n t i n g o u t r e d u n d a n c i e s of the o r i g i n a l m a n u s c r i p t a n d for t h e i r helpful c o m m e n t s to i m p r o v e the paper.
References Aumann, R.J.. 1976. Agreeing to disagree, Annals of Statistics 4. 1236-1239. Bacharach. M.. 1985, Some extensions of a claim of Aumann in an axiomatic model of knowledge, Journal of Economic Theory 37. 167-190. Binmore. K. and A. Brandenburger, 1990, Common knowledge and game theory, in: K. Binmore, Essays on the foundation of game theory (Basil Blackwell, Oxford) 105-150. Geanakoplos, J.D. and H.M. Polemarchakis, 1982. We can't disagree forever, Journal of Economic Theory 28. 192-200. Samet, D.. 1990, Ignoring ignorance and agreeing to disagree, Journal of Economic Theory 52, 190-207. Samet, D., 1992. Agreeing to disagree in infinite information structures. International Journal of Game Theory 21, 213-218.
JOURNAL OF
Mathematical ECONOMICS
ELSEVIER
Journal of Mathematical Economics _,~" (I 9971 389-410
Lattice structure of knowledge and agreeing to disagree Takashi Matsuhisa, Kazuyoshi Kamiyama Department O! A rt~ and 5( ~em'e~. Ihar~ k' \'c tiop ~ [ ('o/h',,,e (!( Tc~ hn dok'y .,Voka~w 8,56. Hitachtnaka. lharaki 312. ,lap,in
Submitted October 1995: accepted Max Ig96
Abstract This paper is a contribution to the studx of the underlying mathematical structure of common-knowledge, which gives the aell-known result of Aumann about the impossibility of "agreeing to disagree'. We present the Bayesian subiective probability model with player's belief; i.e. a triple (--~i- ~9-,- /x), in which i is a player. -~i is a lattice in the field of sets of a state space -Q. , 7 is a correspondence assigning to each state o) a filter ,7-(co) in ~ i , and g is a common-prior. For this model, we impose none of the important restrictions on the information structure in the A u m a n n - B a c h a r a c h model: axiom of knowledge K~. axiom of transparency K, and axiom of wisdom K~. We can extend both the disagreement theorem of Aumann and the agreement theorem of Geanacoplos and Polemarchakis under the assumption that each ~ , is an Artinian lattice. ./EL c/assi/~cotion." C72: (:'73: D82:D83
Knov, ledge ,tructure: Belief of person: ('ommon-bclicf: Bayesian sub ecme probability model: Information structure: Common-prior assumption: Posterior probability Kevuords:
1. I n t r o d u c t i o n T h e idea of c o m m o n - k n o w l e d g e is central to the g a m e t h e o r y a n d the econ o m i c s o f i n f o r m a t i o n . For e x a m p l e , the a n a l y s i s o f a g a m e with c o m p l e t e i n f o r m a t i o n starts with the a s s u m p t i o n that the structure o f the g a m e is c o m m o n k n o w l e d g e a m o n g the players. I n t u i t i v e l y speaking, two p e r s o n s (1, 2) are said to h a v e c o m m o n - k n o w l e d g e o f an e v e n t if b o t h k n o w it. 1 k n o w s that 2 knox, s it. 2 03t)4-4068/"97.,'SI7.00 i~ It)97 Else,.icr Science S.,\. All right,, rescrxed PII S 0 3 0 4 - 4 0 6 8 ( 9 7 ) 0 0 7 8 5 9
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T. Matsuhisa, K. Kamivanut / Journal ~!t Mathematt¢al t-~conomi¢'s 27 (1997) 389 410
knows that 1 knows it. 1 knows that 2 knows that 1 knows it. and so on. Despite recent progress in a theory of games, the important matter remains more or less obscure: a formal structure of common-knowledge, especially that of player's knowledge. This paper intends to address this problem. Aumann (1976) took the first step in the direction to solve this problem. He introduced the formal notion of common-knowledge and showed that persons who have the same prior cannot agree to disagree: i.e., if their posteriors for a given event are common knowledge, then these must be equal, even though they are based on different information. The information structure of a person in Aumann's model is given by' partitions. Bacharach (1985) found that Aumann's model was equivalent to the modal logic S s. He introduced an event-based model with knowledge operator and obtained an extension of the disagreement theorem of Aumann. The information structure is given not by partitions but by the knowledge operator of a person, and it involves three properties: K t, persons can only really know' that something has happened if it actually has happened; K : , they cannot know that something has happened without knowing that they' know it; K ~, if they do not know that they, do not know something, then they know, it. He posed the underlying question (among others) of how strong the rationality assumptions of economic analysis ought to be. He also pointed out the widely held view that even the weakest systems of modal logics are unrealistically strong, even to the extent that they are at variance with the normal meaning of "i knows that'. and that systems similar to that of his models that contain the axioms, especially, K 3, are 'objectionable" in this regard. Binmore and Brandenburger (1990) also directed suspicion at axiom K~. Consequently, we would like to develop a theory of information structures without the three axioms. In fact. Samet (1990) succeeded in extending Aumann's theorem to a proposition-based model whose information structure involves only, the two properties K~ and K , , together with the assumption that every person's knowledge is finitely generated. Furthermore, he showed (Samet, 1992) that the theorem does not generally hold without this assumption. This paper follows the same line of in~,estigation. The purpose is to show, that the information structures of a person's knowledge (K~ and K , ) are superfluous for ruling out "agreeing to disagree'. We shall show' that the nature of the disagreement theorem of Aumann and of the agreement theorem of Geanakoplos and Polemarchakis (1982) is not in the restrictions on the information sets, or in the restrictions on the knowledge operators, but is in the lattice structures of a person's knowledge, especially in its finitely' generated structure. To describe the idea of the lattice structure of knowledge, we shall consider the three frameworks of belief. These all involve a set ,Q of possible states of the world, a field .'7- of subsets of J'), a set 1 of persons and, for each person i, a
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lattice ~ ' r To accomplish this. we shall identify, each ~/i with a sublattice of ,~7. Indeed, in recent work by Bacharach (1985) and Samet (1990). the elements of a lattice have been taken to be sentences (precisely. equivalence classes of sentences), with conjunction and disjunction determining the meet and join operators. We shall define the following three kinds of structure. (1) A "knowledge structure" of person i is :7;. in which ,7-i is a mapping of .(2 to the family 2-~, of all subclasses of J i that assign a filter of .~i to each state co. The intended interpretation of ~7~(co) is as the set of sentences that. i believes to be possible at co. To specily that i's 'beliefs to be possible" are a filter is to suppose that i believes that the conjunction of two sentences is possible whenever i believes each of them to be possible separately. One also supposes that i believes every, logical implication of a sentence that i believes, where the lattice-ordering is taken to represent implication. (2) A "possibility correspondence" of person i is P,. in which P, is a mapping of .(2 to .-~i that assigns a non-void element of ~ , to each co in .Q. The set of /',(co) is an element of .e'i to be interpreted as the set of all the states of nature that i believes to be possible at co. (This is the Aumann set-up.) (3) An "epistemic model' of person i is K . in ,xhich K i is a knowledge operator of ~'Jl to .~- such that K,(AAB)=K~(A)~K~(B) for all sentences A. B: the set K ( A ) will be interpreted as the set of all the states for which i believes A to be possible at co. (This is the Bacharach set-up.) Our main results are as folio,as. In the case where each 2"i is an Artinian lattice (i.e. it satisfies the descending-chain condition), we can first sho,a that the three frameworks of belief are equivalent to one other. Therefore. with a knowledge structure ,)--, are uniquely associated the knowledge operator K~ and the possibility correspondance P~: for each T of ~(co). K,(T) will be interpreted as the set of all the states l\~r which i believes T to be true at co. and K,P,(co) will be taken as the smallest belief at co that is the minimal set in all sets K,(T) li)r T in /-~(~o). We say that a sentence A is "common-belief" at co if it includes the smallest set in all the sets of states that the persons, as a whole, believe to be true there. We shall define by q,,,, = / ~ ( X [ K P,(co)) i s posterior of event X at a state co. Thus. it will be interpreted as the posterior probability of X. given the belief of / a t co. In these circumstances, we can also show that. if ~,c impose a common-prior for each person, then the disagreement theorem is xalid: if each posterior q,,, is common-belief among the persons, then q .... q,, for each i. /. In other words. agreeing to disagree is still impossible, exen though the associated knowledge operators of a person satisfy none of K~. K~ or K ~. This paper is organized as follows. Section 2 describes the characterization of Artinian lattices in terms of their filters, which is prerequisite for the study of finitely generated knowledge structures in the later sections. In Section 3. we
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present the notion of knowledge structure and establish the connection between knowledge structures and possibility sets. In Section 4, we introduce the two types of epistemic model, i.e. CK 0- and K0-epistemic models, and define the D"iequivalence relation on them. In Sections 5 and 6, we define complete and finitely generated knowledge structures. In Section 7, we establish the equivalence theorem of the three frameworks. In Section 8, we present the notion of common-belief. Section 9 describes the epistemic model of Bacharach in our framework. We remark that. in his model, A u m a n n ' s notion of common-knowledge is the same as our notion of common-belief. In Section 10, we define Bayesian subjective probability models. In Sections 11 and 12, we establish the common-belief results: the generalization of A u m a n n ' s theorem and that of Geanacoplos-Polemarchakis" in finitely generated knowledge structure models.
2. Foundations We recall that a poser is a set with a binary relation which satisfies refllexiuity, antisvmmetrv and transitit'itx. We let .Q be a non-void set. called a 'state space', and let 1 be a set of finitely many persons, and ,'7 be a field of subsets 2 ~*, i.e. the poser of all subsets of ~Q. Each member of ,~- is called an 'event" and each element of O called a 'state'. 2.1. Semi-complete lattices
By a lattice of ,'7 we shall mean a non-void subposet of a field ,'7 that is closed under set theoretical intersection and union of each two members. A lattice is said to be semi-complete provided that it is closed under intersection, i.e. the intersection of each non-void subclass of the lattice belongs again to it. The void-lattice {~i} is exceptional in various (mostly trivial) respects. Throughout this paper we assume that a lattice is not the void-lattice. 2.2. A,fllrer m a lattice J '
By this, one means a non-void subclass V of a lattice J ' such that Ft: the intersection of two members of V always belongs to V, and F~: i f T ~
V and T c E ~ Z / , then E ~ V
A filter is said to be proper if it is a proper subclass of a lattice ~ ' . Plainly. the void-set never belongs to every proper filter. If P is a non-void member of ,~/', then the subclass {T ~ 2 / 1 P c T} forms a proper filter: it is written ( P ) . We shall call it the principal filter generated by P and P is the generator.
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2..7. Artinian lattices One says that a lattice ..~' is Artinian if every descending chain of members of .~L i.e. T I >T,>T,>
...
>T,
>T,,,.
I >
...
such that T,, +~ T,. i (it is called a strictly descending chain) is finite. We shall prove the characterization theorem of Artinian lattices in terms of the filters considered in later sections. It is more or less well known and the proof will be included for completeness. Pt'oposition I. Let (.2 be a state space alul+7 a field (~/subsets. Suppose t/tat ~/' i,s a lattice d , 7 . T/ten. the.fbllowin? three statemenLs are equiz'alent: ( a) a lattice ~ is Artinian: (bY e+'erv n(m-t'+Ud subclass ~>~ q f z~ /',~osses<~,,s +l mi#limal member: (c) e r e o f i ' l t e r in W ~ is principal. Proqf. F r o m (a) to (h). Let T, be a m e m b e r of . v . If ~ is not minimal, then it properly contains a m e m b e r T~ of ,J'. If T~ is not minimal, then it properly contains a m e m b e r T, of t . Inductively. if we ha~e found Z],, that is not minimal. then it contains properly a m e m b e r T,,÷ ~ of / . In this ,xay. we could construct an infinite chain, which is impossible. Thus. there exists a natural number N such that T~ is a minimal member. F r o m ( D ) g ~ ( c ) . Let g be a filter in .-/ and 71, a m e m b e r in V. If V was not identical to (T.). then there exists a m e m b e r T I in V that does not belong to (T0). such that Tt is properly contained in 7,,. Proceeding inductively, we can find an ascending chain of principal filters in "_,". i.e.
The subclass {71, ] n - (), I. 2. 3 . . . . } of ~ possesse~ a minimal member, say Tu. It is then clear that the principal filter (77~) must be equal to V. as required. F r o m (c) to (a). Suppose that we have a descending chain of members in J . i.e. T, > T , > T , >
... >T, D T
i>...
Let V,, be the principal filter generated by 7-,,. Then. the subclass {V,, In G N} is an ascending chain of filters in .~/. W e shall denote by V the union of all the filters ~,. It is easil} verified that g is a filter: consequently, it is principal and generated by a m e m b e r T in ..~. Hence. there exists a natural n u m b e r N such that T ~ g v, and it follo,xs immediately that V = V~. Hence, one obtains that 77~ = T~. i = ...=7". []
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In view of parts (a) and (b) in Proposition 1. the following corollary is readily proved. Corollary 1. Ererv Artinian lattice is semi-complete.
3. Knowledge structures and possibility sets In this section, we shall present the concepts of knowledge structures and possibility sets, and we shall observe the connection between them. 3.1. A knowledge structure qf person i By this. we shall mean a pair ( ~ i . ;7-) in which ~ i is a lattice of
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Corollary 2. Let ~1 be a state .space a~td 1 a set o f persons. There exists a bijection between the system o f knowledge structures and the systems Oflpossibilio" correspondet~'es.
4. Epistemic models In this section, we shall define epistemic models and shall establish the connection between knowledge structures and these models. 4.1. K~,- and CK~,-epistemic models
By an "epistemic model" of person i we shall mean a pair (,~-. K i) in which K, is an operator on .'7 such that the axiom EM:
K,~C)=f) and K i O = G
is valid. This K~ is called a "knowledge operator" of person i. The event K i E will be interpreted as the set of states of nature for which i believes E to be possible. An epistemic model (,9-. K,) is said to be a K0-epistemic model if the axiom Ko:
For each E. F ~ , Y - . K i ( E I " F )
=K.E'~KiF:
is true. It is also called a Ko-knowledge operator of person i. An epistemic model (:~-. K,) is said to be a complete K0-epistemic model (or simply CKo-epistemic model) if the axiom
is valid. It is called a CK0-knowledge operator of ]person i. It should be noted that each CK0-knowledge operator is always a K0-knowledge operator. A system of Ko-epistemic models (or of CK~,-epistemic models) is a family that consists of all CK~- (or K,,-)epistemic models ( S . Kz) for person i in I. 4.2. Ez'em CK~-knowledge operator is order 1,'eserting
In fact. viewing axiom CK,b. we can easily prove the following. For each events E. F in :7. we have OP:
If E c = f . t h e n
K I£cKf
Remark. Every, K0-knowledge operator is also order preserving. 4.3. The Si-equivalence relation
We shall define the equivalence relation on the class of epistemic models as follows. Given a lattice ~.71 of J associated with person i, we shall say that two
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epistemic models ( j - , K,) and ( J . K I) are J)-equivalent (or simply say that K, and K I are ~i-equivalent), provided that K, and K I are identical on ~ i ; i.e. KiT= KIT for each event T of ~'i. We shall denote by [(.'7, K,)] the equivalence class of the epistemic model ( J . K,).
4.4. Minimal CK~,-epistemic models Given a lattice .S, of .'7, we shall say that a CK{)-(Ko-)epistemic model K~) is minimal in its ~i-equivalence class if an event KiE is contained in KIE for every event E of ,~. and for every C K , - (K0-)knowledge operator K I that is 2G-equivalent to K,. (J.
Proposition 3. Suppose that J~ is a lattice of :7 associated with person i. Then, there exists a unique minimal CKc~- (or K{}-)episternic model. Furthermore, the correspondence (,~. Ki ~ ) ~ [ ( j , K / )] is a bijection between the class q{ all minimal CKII- (or Ko-)epistemic models and the class of all ..~,-equicalence classes o[ the CK o- (or Ko-)epistemic model. Proq/2 We prove for CK0-epistemic models (:7. K~). Let K / be the operator on .'7 that assigns to each event E the intersection of all events KIE. i.e.
X/£=
nx;£ A"
where (,'7, K I) runs through all the CK0-epistemic models that are ~i-equivalent to (.Y, K~). We shall f r s t note that the pair (,7. K i ) is actually a CKo-epistemic model. In fact, it is easy to observe that K / fulfills axiom EM; furthermore, we shall prove that the knowledge operator K, ~ satisfies axiom CK o. This is because, viewing the definition of K / and axiom CK o for K I, one obtains that, for every subclass {/&[ A ~ A} of 5 -
A~ ~
K I A~ I
,',~
The converse is verified as follows. It is evident from OP that. if K I is also __~,-equivalent to Kp then KI(["I A IE~)_-cKi/£a for each A. Consequently, one obtains K / ( l " l ,~ ~Ea) ~ ('1Ae ~Ki'EA, and the required result follows. In view of the definition of K / , it is readily verified that (,'7, Ks*) is ~.V~-equivalent to ( J , Ki). This will prove our proposition. Indeed, if it is the case, then, by the definition of K / , it follows plainly that the CKo-epistemic model ( J , Ki x ) is minimal, and it also follows that the mapping [ ( J , Ki)] ~ (,~, K / ) gives a bijection; its inverse assigns to each minimal CKo-epistemic model (,Y, Ki :~) the 2i-equivalence class [ ( J , Ki* )] in completing the proof. []
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5. Complete knowledge structure In this section, we shall introduce the notion of complete knowledge structures and establish the connection between CK~-epistemic models and these knowledge structures. 5. I. A complete kmm'ledge structure ¢~/l~ers(m i By this. we shall mean a knowledge structure ( J , . . / - z ) of person i with which the associated lattice .J, is semi-complete. A system of complete knowledge structure is a family that consisting of all complete knowledge structures for i in 1. We shall prove the following. Proposition 4. Let -(2 be a stale space a l u l , 7 a fieht oj subsets. Suppose that 2/I ix a semi-complete lattice associated with i)ers(m i. T/tell. there exists a b~jection (2/i..7-) ~ [(.7. K [ )] between the clas.~ o[all complete knowledge structures qf person i and tile class olall .y)-equicale+lce classes O/the CK~-epistemic model Of person i. This bijection ass&ns to each knowledee structure (z/i, 7 , ) with possibilio" sets P , ( o ) the .:/i-equiz'alence class (g the CK~Tepistemic model [ ( ~ . K [ )]. ill which K,' is" ttle operator on :7 t/tat assigns to each ezent E the euent K [ E t/tat consists c~[ all the states co such that P(oo) is contailwd in E. Its im'erse assigns to each ,J ,-equiualence class ¢~!t/le CK~-epislenffc model [(,/-. K;)] the knowledge structure ( . ~ , J i ) , in which ,/- is the correspondence that assigns to each state ~o the fiher ,7i(co) that consisl.s o/all the members T (~/ -/i such that eo belongs to KiT.
Before proceeding to the proof, we shall first prove the following. Lemma I. Gizen a semi-complete lattice ,,/, we .find a mappillg (,~i, Pi)~-~ ( J . K[ ) qf the class 0/' all po~sibili O corre.w~mdenee ofperson i to the class of" all minimal ['K~-epistemic models o! l)ersoJt i. This mapping assiglts to each (,L/i, Pi) the pair (,~-. K[ ). in which K , i~ the operator Oil ,~ d
We first note that the pair C7. K ) is a CK~-epistemic model. Indeed. it is easy to obser,,e that K[,Q = ~2. On noting that P,(eo) is not void. we obtain that K / ~ = ;~. so that the operator K [ satisfies axiom EM. Furthermore. in view of the definition of K [ . it is easy to verify that K, satisfies axiom CK 0` and the required result follows.
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It remains to be shown that the CK,~-epistemic model (.'7, K [ ) is actually minimal. Suppose that (.Y-, K I) is the arbitrary CK0-epistemic model that is 2/~ equivalent to (.~-. K,* ). For each E in ,~-, if ~ is an element of K / E , and noting that ~: belongs to K[ Pi(~) = KIPi( ~-), then we can easily observe that Pi(~ ) is contained in E. From OP, we obtain that ~ K i & ( ~ ) ~ K IE. []
5.2. Proqf of Proposition 4 Suppose that ( J , K i) is a CK0-epistemic model. We shall define .Ui to be the correspondence that assigns to each state ~o in .(2 the subclass J i ( ° ~ ) that consists of all the members T of ~', such that ~o belongs to KiT, i.e.
J-,(,.,,) = {:r
I ,o c
We first note that the pair (~'{.i, ,'7,) is actually' a complete knowledge structure of person i. Indeed, it is first noted that .Ti(w) is a filter in ~ i . This is because, viewing axioms CK 0 and OP, we can plainly verify, that ,U,(w) satisfies the conditions F~ and F,. On nothing that ,(2 belongs to ,~7-~(~o) by axiom EM. it follows immediately that J , ( w ) is not void. We shall denote by Pi(w) the intersection of all the members of J-i(w). P,(w) is the generator of a filter J , ( w ) , because, viewing axiom CK 0, we can easily observe that P,(~o) is the minimal member and it is not void, since the void-set does not belong to J ~ ( w ) by axiom EM. The correspondence P, that assigns P,(~o) to each w is plainly a possibility correspondence into ~ } \ { Q } . Hence. we have shown that ( ~ i - 47,) is a complete knowledge structure with possibility sets P,(~o). Thus, we have defined the correspondence
@: ( J , x,)--, ( J ; , ,,7) of the class of all CK0-epistemic models into the class of all complete knowledge structures. Furthermore, in view of the definition of ..t'~-equivalence classes, it follows that the correspondence @ induces the injection of the class of all ~,¢i-equivalence classes of CK0-epistemic models into the class of all complete knowledge structures. To verify that the induced injection actually makes a surjection, it suffices to prove that @ does also. Suppose that (2s I. ,U,) is a complete knowledge structure of person i. According to Proposition 2. we shall denote by P,(w) the possibility set that corresponds to this structure. In view of Lemma I, one obtains the minimal CK0-epistemic model (J-. K [ ) that corresponds to (4/i, <7i). in which the CK0-knowledge operator K [ is defined in the same way as in Proposition 4: i.e. Ki* E = {w ~ ~Q I Pi(w) c=E} for each E of ,-7. We shall show that ( ~ i , J , ) is precisely the image of (
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minimal member of .:/i(w) and the required result immediately' follows. This shows that the correspondence g' is a surjection. []
5.3. Further discussion Proposition 5. Let ,(2 be a ,state space, :7 a,field Of subsets atzd i a person, lf C7-, K,) is a CKo-epistemic model, then there exists a unique complete knowled~g,e ,~trueture ( ~ i , "7-,) .sueh that J i is the subclass q/" ,7 that consists q/" all the erents T contained m K T, and ( J , K i) i,v Ji-equiz'alent to the minimal CKo-epistemie model that c'orre,v~onds to (W~, , 7 ), Proo/. We first note that ~ i is a semi-complete lattice of ~-. In fact, it follows plainly from axiom EM that ~'i is not the void-lattice. Viewing OP. we observe that, for each ft, T, of ,--~i, we have
T / QT~_cK,T 1 ; q K , T : ~ K , ( T ~ , Q T : ) Furthermore, viewing axiom C K . , we obserxe that, for each subclass {Ta I k ¢ ,1} of -~i, we have
N TA~ n
K~=Ki
as required. The complete knowledge structure (c/I. 7 ) will be defined such that .7, is a correspondence that assigns to each state ~o ill .(2 the filter J , ( ~ o ) that consists of all the events T of : / i such that w belongs to KiT. It follows plainly from axiom CK 0 that each filter , 7 ( w ) is principal. The generator P~(w) is the intersection of all members of ,'£i(~o) but is not a void-set, since, in the view of Axiom EM, the void-set does not belong to ,Y(o~). According to Lemma 1. let ( Y , Ki* ) be the minimal CK~-epistemic model that corresponds to (.-.~i. ,Zi ). It can be verified that two knowledge operators K i and K i are _/, equivalent. In fact. for each T of J i - each state w belongs to K i T if and only if P,(~o) is contained in T, which is equivalent to the condition that T belongs to ,'f~(~o). i.e. w belongs to K,T. Consequently,, viewing Proposition 4, we obtain that ( J , , .~:i) corresponds uniquely to (~7, K,) up to ~-equivalence. []
6. Finitely generated knowledge structures In this section, we shall introduce the notion of finitely generated knowledge structures. By "a finitely generated knowledge structure' of person i, we shall mean a knowledge structure of i with which the associated lattice is Artinian. A system of finitely generated knowledge structures is a family of all finitely generated
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knowledge structures for persons i in /. It should be noted that each finitely generated knowledge structure is always complete, which immediately follows from Corollary 1. The proposition below is analogous to Proposition 4. Proposition 6. Let ~2 be a state space and ,'7 a field of subsets. Suppose that ~ , is an Artinian lattice associated with person i. Then, there exists a bijection (,~,,, J , ) ~ [ ( J - , K [ )] bem'een the class of all finitely generated knowledge structures ofl person i and the class of all 2Yl-equil'alence classes of the Ko-epistemic model of person i. This bijection is defined in the same way as in Proposition 4. Proof. We can prove in the same way as Proposition 4, and shall omit the details. It should be noted that, for each K0-knowledge operator K,, the subclass . J , ( w ) = {Te~5~ ! ~o ~ K,T} is a principal filter in J',. This is because it is easy to observe that 3,(~o) makes a filter in ~ i ; furthermore, since ~ i is Artinian, it follows from Proposition 1 that the filter has the minimal member P~(aJ). i.e. the generator of .%(o0). []
7. Trinity of belief models We are now in a position to state the equivalence theorem of the three frameworks of belief: the class of all systems of knowledge structures for I, that of all minimal epistemic model for 1, and that of all possibility correspondences for 1. We shall say that two frameworks are equivalent if there exists a bijection between these classes, and shall say that two systems are equivalent if there exists a bijection between them. Main Theorem. Let ~ be a state space, :7 a.field qf" subsets 2 ~ and let 1 be a set of persons. Then, the jollowing two statements are true: (i) The jollowing three frameworks are equit'alent to one other: the class of all systems of complete knowledge structures f o r 1: that q f all systems of minimal CKo-epistemic models for 1: and that qf all systems qJ'possibility correspondences for I. (ii) Gilen a family o f Artinian lattices z? I q [ , 7 associated with each person i in 1. the fi)llowing three systems are equicalent: the system of finitely generated knowledge structures ,]'or 1: that of minimal Ko-epistemic models f o r 1; and that of possibility correspondences.fi~r 1. Proof. For (i). The equivalence between the class of all systems of complete knowledge structures and that of all systems of minimal CK0-epistemic models, through all the .~-equivalence classes of them, is established by the family of bijections in Propositions 4 and 5, together with Proposition 3. The equivalence
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between the class of all systems of complete knowledge structures and that of all systems of possibility correspondences is also established by the family of bijections in Corollary 2. For (it). This is an immediate consequence of Proposition 6, together with Corollary 2.
Remark. In view of the Main theorem, the partitional model of Aumann (:~-~, P,) can be characterized as the complete knowledge structure (.~..<7-,) such that each .~,(w) makes a maximal filter and each state to belongs to the generator of
J,(to). From now to the end of this paper, we suppose that (._1". <'7-,) is a complete (or finitely generated) knowledge structure, and that (--:/i. P,) is the associated possibility correspondences to it. where ~ i is a semi-complete (or an Artinian) lattice of J - associated person i. We suppose that ( J . K,) is a CK 0- (or K0-)epistemic model in the .Jcequivalence class of (.7. K . ). where (<'5-. K, * ) is the minimal CK~-epistemic model that corresponds to (..~,...7,) according to the Main theorem.
8. Belief and common-belief In this section, we present the notion of belief and that of common-belief in our framework.
8.1. The beliE(to) as the set of states believed to be true there by the community of persons as a whole. We have to note the folh)wing lemmas.
Lemma 2. Each belie[ base .~,( to) is depe#ldem only on the knowledge structure ( J i. <7-). It is independent ~?[the choices ~?[representatices o[ the "~i-equhalence
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class ~f K,. Moreot'er. the same applies.fi)r each common-belief base ,~( w ) only on the system {( ~ i , .7-i) [ i ~ 1}. Proof. This immediately follows from the definition of the _~g-equivalence relation. [] Lemma 3. Each belie/" base ,Z~i(oo) is closed under intersection and it possesses the minimal element K, Pi( °9 )" Consequently, each common-belief base 2g( w ) is also closed under intersection. Proof. Since J i ( w ) is the filter generated by P,(o9). and K i satisfies axiom CK 0` the first assertion follows immediately, from OP. The remaining assertion is clear. {Z. 8.2. Belief and common-belief We shall say that person i believes event E to be true at w if there exists a member M,(~o) in ~',(~o) such that Mi(w) is contained in E. An event E is said to be "common-belief" to be true among l at oJ if there exists a member M(o~) in .Zd(w) such that M ( w ) is contained in E. Remark. The two concepts above are dependent only on the system of knowledge structures, i.e. they are independent of the choices of representatives of the J',-equivalence class of (,~-. K,) for each i. In fact, this immediately follows from Lemma '
9. Epistemic model of Bacharach In this section, we describe the epistemic model of Bacharach in our knowledge structure framework. 9.1. An epistemic model of Bacharach By' this, we shall mean an epistemic model ( J , axioms, together with axiom CK 0` are satisfied: K I"
KiEC=E
K~"
KiEC=K, KiE.
K,) such that the following
for each E, F in ,7. A system of epistemic models of Bacharach is a family {(J-, K i ) l i ~ I} in which (,'7, K,) is an epistemic model of Bacharach.
T. Mat,~utU.~'a, K. Katnivatna / J o u r n a l o / M a t h e m a l i { a ! E~ om~mi~ s 27 ¢1997) 389 410
403
9.2. The knowledx, e structure that corresponds to the Bacharach mode/ We shall now prove the following.
Proposition 7. Suppose that { ( ~ . K,)i i e 1} is a ,vstem of epistemic models ~/ Bacharach. Let {( .~i. <*)-i)t i e I} be a system ~?/"pairs ( ..7 i, J i ) in which k~I i.'~ the subclass of <'~- t/tat consists (~/"all dte elents T such that KiT = T, and )-, is the corre.g~ondence that assigns to each state to the subclass .7-/(02) o / all the el'ents in ,Z/i to which to belongs. Then, {( c/I. .7i)l i ~ I} is a svstenl ~?['complete kmm'ledge structures to which {(.U. K,)] i e l} is the correspondint4 system of the minimal CK~-epistemic model.s. Viewing part (i) of the Main theorem, we have readily' verified that the system of epistemic models of Bacharach is equivalent to the system of complete knowledge structures above.
ProofqfPropositio#7 7. In view of Proposition 5. together with axioms K 1 and K~. it is readily verified that the pair ( J i - , U , ) is actually a complete knowledge structure. It should be noted that ~ , = {K,E[ E e + 7 } and S , ( 0 2 ) = {K k.[ t o e K i E. E e ,9- } {or every i in 1: this follows immediately from axioms K i and K~. Now. we shall show that, for each E in ~7. K i E - { t o e (_,). I Pi( 02) c= E}. To prove it. suppose that to is any state in .(2. If P/(to) is contained in E, then it follows from OP that 02 ~ Pi(w) = K P,(02) c= K E. The converse is clear: indeed, in view of the above points about .<7, and
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10. Bayesian subjective probability models In this section, we shall define the Bayesian subjective probability model that possesses knowledge structure and shall introduce the notion that a posterior is common-belief. We define the conditional probability / x ( E I F ) by
/*( E ~ F ) #(ElF)
-
I~( F )
for each event E, F of , ~ such that p,(F) 4= 0. 10.1. A Bayesian subjectire probabili O" model that possesses knowledge structure By this, we shall mean a family {(~-~'i, "/,, # i ) l i ~ I} in which a pair (2VI. ,'2-,) is a knowledge structure of i with possibility sets P,(~o), and ~, is a probability measure on a o-field ~ such that. for each i in I. :~, includes all the belief bases ~>Y~(~o) for w in /2: K, is the knowledge operator that assigns to each E of :Y- the event K~E that consists of all the states o~ such that P ( w ) is contained in E. The model will be said to possess complete (or finitely generated) knowledge structure if each associated knowledge structure is complete (or finitely generated). It will be said to be with 'countable range" if the range of each correspondence , ~ is countable. A Bayesian subjective probability model will be said to satisfy the "commonprior assumption', provided that p_~ = #j for every i, j in 1. 10.2. Posterior probability q,~o Let X be an event in ~ , . Suppose that co is a state in ,Q such that I~,(K,P,(w)) ~ O. By the "posterior probability" of event X. given the belief of person i at w, we shall mean the conditional probability
q,,o = #,( X I K , P,( w ) ) Remark. Each posterior q,., is independent of the choice of representatives of the Jl-equivalence class of (:7. Ki). It should be noted that. in the epistemic model of Bacharach, the posterior q,~ coincides with the ordinal one: /z(XI P;(~o)). This is the posterior, given the knowledge of person i. 10.3. A person i belieres h i s / h e r posterior probability We shall say that person i believes the posterior probability of event X, given the belief at ~, if person i believes the event E.,~ to be true at ~, where Ei,o is defined by
E,,. = { ~
~21 #,( X I K,P,( ~ )) - q,,,. # , ( K I P , ( ~ ) ) 4- 0}
T. Matsuhisa, K. Kamiyama / Journal o[Mathemalical Ecotlomicv 27 !1997) 389 410
405
In other words, there exists a member Mi(to) in ,3~i(to) such that Mi(to) is contained in E,,,.
Remark. The concept that a person i believes h i s / h e r posterior is dependent only on the Bayesian subjective probability model that possesses knowledge structure. In fact, in view of Lemma 2, it follows plainly that both events M,(to) and E,,, with inclusion relation are all independent of the choice of representatives of the _Ti-equivalence classes of (,'7. K). 10.4. The posterior prohabilit3." is commo~>heli~q We shall say that the posterior probability of event X. given the belief of person i at to, is common-belief, if the event E~,,, (as above) is common-belief to be true at to. In other words, there exists an event M(to) in ,~(to) such that M(to) is contained in E,~,.
Remark. In view of the preceding remark in Subsection 10.3. we have readily verified that this concept is also dependent only on the Bayesian subjective probability model that possesses knowledge structure.
10.5. Samet decompo.sitiotz We shall say that two states (to and to') are K,-equivalent. provided that
X P,(to) = X,P,(to') Obviously this is an equivalence relation on a state space _(2. We shall say that an event possesses the 'Samet decomposition" for i if it can be decomposed into a disjoint union of K,-equivalence classes. The following properties can be easily verified. Suppose that ( - / i - , ' 7 , ) is a finitely generated (or complete) knowledge structure. that (--~i. Pi ) denotes the possibility correspondences associated with it. and that (:7. K i) is a corresponding K,,- (or C K , - ) epistemic model. Then. we obtain the following properties: KPI: For each t o ~ .(2. t o e K,Pi(to) KP, : For each to ~ f2. if ( ~ K i P(to). then K, P,( .~ ) ,c K i Pi(w) SD: For each t o e .(2. every member of the belief base .7/(to) of person i possesses the Samet decomposition for i. i.e. for every T of Y)(~o)
KiT= L A / ( ( )
(disjoinl union)
where A , ( ( ) is the K~-equivalence class of ~ and ( runs through a set
K,T Remarlc In view of the definition of the .v-equivalence class, it should be noted that the concept of Samet decomposition is dependent only on a knowledge
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structure. In other words, it is independent of the choice of representatives of the Si-equivalence class of (,~-. K,).
11. Agreeing to disagree In this section, we shall prove a generalization of the disagreement theorem of Aumann (1976). Before proceeding, we shall show the following lemmas.
Lemma 4. Suppose that ( ~ i , K,) is a K.-epistemic model that corresponds to a .finitely generated knowledge structure (,Ji. , 7 ). Every non-l'oid subclass J ) of the class {KiT IT¢,'S,} possesses the minimal element. Consequently, el'era" stricth' descending chain in { KiT IT ~,,7)} is finite. Proof. Let J ' , denote {KJa I Ta e ~ i . A ~ ~1}. where ~l is an index set. Then. by Proposition 1. it follows plainly that the subclass {TAiTa eL/i . A e A} has a minimal element T0. Hence, by OP, we obtain that KJo is contained in KiTa for each a ~ A: it is minimal in J } , as asserted. Let {KIT,, IT,, <~i, ,7 e N} be a strictly decending chain in { K J I T ~ : / i } . Then. by the result above, it follows plainly that the chain has a minimal element K,T v for some natural number N. Hence. it must be finite. [] The following result is the key to proving our Disagreement theorem and Agreement theorem.
Fundamental lemma. Suppose that {(,Ji. ,~,. #)[ i ~ I} is a Bayesian subjectil'e probabilio" model that possesses finitely generated knowledge structure with countable range, and that it satisfies the common-prior assumption. Let X. M and N be i.t-measurable events. Suppose that ~o is a state such that the probability #( K iP,( w) (~ N) is non-xero. Let qi denote the posterior probabiIio': #( X I K , Pi( to ) (~ N ). lf M is a member of the belief base /Y, i( o9) of i with which M is contained in the el'ent E that consists qf all the states E such that #( XI K,P,( ~) C ' X ) = q , and #( KiP,( E ) (~U) 4~O then we obtain the equali O" ~ ( X ] M C , N ) =qr Proof We shall observe the two points made above: first, that the class {K~P¢(E) ¢ 3 N I E ~ . Q } is countable: secondly, that the event M possesses the Samet decomposition into countably many tt-measurable components. In fact. since each
T. Matsuhisa, K. Kamiyama /Journal :~['Mathematical E¢ onomics 27 ¢] 997) 389-4 I0
407
knowledge structure has countable range, it is easily obse~'ed that the number of possibility sets is countable and that the class of events K,P,(-6 ) for all sc in 12 is also countable. Furthermore, in view of properties KP~ and KP 2, it can be verified that, for each -6. A,(-6) is written by the relative complement of the countable union
c_ with respect to K,P,(,~). Therefore, A~(6) belongs to the o--field .~,. In view of the definition of K, equivalence, we can easily observe that the correspondence Ai(-6) ~ KiPi(:~) makes an injection of the class of all Ki-equivalence classes into the class of all K,P,(-6) for ~ in .(2: the former class of the components A,(s~) is also countable. Therefore, it follows from SD that each member of , ~ ( o J ) can be decomposed into a countable union of /,-measurable components A~(~); from this, the second assertion follows. Consequently, in view of KP 1 and KP~. it follows plainly from Lemma 3 that the class { K , P , ( - 6 ) a N I -6~ .(2} makes a finitely nested reflexive and transitive information structure in the sense of Samet (1992). Hence, similarly to Samet (1990, Proof of Theorem 7, pp. 200-202), we can prove that # ( X [ A i ( - 6 ) C " N) = qi for every component A,(s~) of the Samet decomposition of M, and that t x ( X I M C ~ N ) = q , f o r e v e r y i. []
Disagreement theorem. Suppose that { ( ~ ' . ,7~,. /x)l i ~ I} is a Bayesian subjectil'e probability model that possesses finitely generated knowledge structure with countable range, and that it satisfies the common-prior assumption. Let q~o denote the posterior probabili~., ol~ecent X. gicen the belie[" of person i at a state w. lf .fi)r ecer 3' person i in I, the posterior probabili U q,,, is common-belief among I at oo, then the persons cannot agree to disagree, i.e. q,,, = q,o .fi>r each i, j in I. Proofi Suppose that (,:7, K~) is a K 0-epistemic model that corresponds to (Be.". J - ) , and let A,(-6) denote the K,-equivalence class of a state £. Let E denote the event {-6~ .(2 ] tx(X] K : P ~ ( - 6 ) ) = q , , , , # ( K , P , ( - 6 ) ) e:0} In view of the assumption of the theorem, we first note that there exists an event M(co) in . ~ ( w ) such that M(w)__c f " l , :E,¢,,. Indeed, for every i in I, there exists N:(w) in , ~ ( w ) that is contained in E,,. Then the intersection 0 i~ :N,(w), denoted by M(oo). is also contained in f"l ,~ IE,,,. It is evident from Lemma 3 that M ( w ) belongs to ~>£(w) and the required result follows. In view' of the Fundamental lemma, one obtains that, for every i in 1. the following equality holds: /,(X I M ( w ) ) = q,,. Therefore. we obtain that q,o, = %,, for every j.
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?2 M a t s u h i s a , K. K a m i y a m a / J o u r n a l o f M a t h e m a t i c a l Economic~ 2 7 (1997) 3 8 9 - 4 1 0
12. Dialogue and agreement theorem In this section, we shall present the communication process between two persons and shall show a generalization of the agreement theorem of Geanakoplos and Polemarchakis (1982). Throughout, we suppose that {(.5¢I, Y,. /x)[i = 1, 2} is a Bayesian subjective probability model that possesses complete knowledge structure and that it satisfies the common-prior assumption. Let ( ~ i , P) denote the possibility correspondences of person 1 and let (J.~, Q) denote that of person 2. Suppose that {C~, K,)li = l. 2} is a corresponding system of CK0-epistemic models. Let X be a /,-measurable event.
12.1. A dialogue on ecent X between two persons l and 2 By this, we shall mean the process ~" of revision of the posteriors of an event X at ~o by persons 1 and 2 defined stepwise as follows.
Step 1. Let q,~obe the posterior /X(X I K I P(to)) and let E~ denote the event
{ ~ Ill MXl K~P( ~)) = q',o, /X( K~P( ~)) :~ 0} which is assumed to be a member of the belief base .Sfl(w). Thus. the event E 1 will be interpreted as the information that person l believes the posterior q~ to be true at ~o. Person 1 announces E~l,,to person 2 and then person 2 revises h i s / h e r posterior r~o=/X(XJ K~.O(w) N E Io). Let Fo~ denote the event
{~I2L
/X(XIK:Q(~))=r',,
/ X ( K z Q ( ~ ) ) =#0 }
which is assumed to be a member of the belief base ,X,(co). The event F~ will be interpreted as the information that person 2 believes the revised posterior r~, to be true at ¢o. Person 2 announces F~ to person 1.
Step t. Having received person 2"s announcement, person 1 revises h i s / h e r posterior
qL=#{XlK~P(w)N ,
1~'2 t=1
Let E~, denote the event
(
~121
in,'t
/X X , K t P ( ~ ) N
Fi
=
/
=
(
q;,. # K,P(s~)C~ ,
#:0 t
.
}
which is assumed to be a member of the class .~/~(o)). Person 1 announces this to person 2 and then person 2 revises h i s / h e r posterior
(
',)
,-';=/x XlK:Q(,o) n n e , ,,
k=l
,
12 Matsuhisa, K. Katni3ama / Journal q/Mathematical Econontics 27 (1997) 389-410
409
Let F,r, denote the event
/~.(21p.
XlK:O(~)r~
I
,
* =,-,~,,a J(
/
K_,(2(s~)r>
E.~ ~0
,
which is assumed to be a member of the class ,Z',(w). Person 2 announces this to person 1. We shall sap' that the dialogue ~r converges at step T if the equality qi'o = rio holds for all t >~ T.
12.2. The dialogue is determined only by the system of knowledge structures This means that the dialogue is independent of the choice of representatives of the Ji-equivalence class of ( J . K,) for each i = 1. 2. In fact. on noting that E~ is a member of ,Z/i(~o) and F~ a member of A"~(~o) for every k. 1. we can easily observe that these events and the posteriors qi'o and r,"o are all independent of the choice of representatives of the Yl-equivalence class of K~, The result immediately follows as required. 12.3. The agreement theorem Agreement theorem. Suppose that {(L/i, ,/-,, # ) I i = 1, 2} is a Bavesiwl subjecti~e probability model that possesses .finitely gellerated knowledge structure with countable range, amt that it satL~fies the common-prior assumption. Then, the dialogue ,'7 defined as above conrerges in.finitely mal~v steps, i.e. there exists a natural number T such that q~,, = rio ./or l > T. Proqfi Let E~I] and hi'' be the same as in Subsection 12.1. We first note that
t k
I
u~
• "- . . . .
is a descending chain in ,/~(~o) and that
[ ' 'k/9F';Iz=l'23 I
] .... )I'
is in ~,(~o). In view of Lemma 4, it follows plainly that there exists a natural number T such that I
f
t
N E:o N C -
k-]
k
]
and
N l-I
t
O<
-
/
I
for each t~> T. We shall denote 0 : ,E,~ by Eand shall denote f " l r F,: by F. By Lemma 3, it is readily verified that E is el member of ,~(~o) and F is a
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m e m b e r of ,7#~(w). T h e r e f o r e , it f o l l o w s plainly that there exists a sufficiently large natural n u m b e r T s u c h that, for e a c h t >~ T, E is c o n t a i n e d in
{ ~
u(XIKIP(~)NF)=q~,
tx(K,P(~)f3F)4:O}
a n d F is in
{~~ ~
tx( X I K2Q( ~ ) N E) = ;;. #( K:Q( ~ ) N E) 4: O}
T h e r e f o r e , m v i e w of the F u n d a m e n t a l l e m m a , we o b t a i n that
q2= p,( Xl E N F ) :r;) for each t > / T .
[]
Remark.
In general, n e i t h e r E r n o r F T m a k e s c o m m o n - b e l i e f be true at w, e v e n if a d i a l o g u e ~ c o n v e r g e s at step T.
Acknowledgements T h e a u t h o r s t h a n k the e d i t o r a n d the a n o n y m o u s referees for p o i n t i n g o u t r e d u n d a n c i e s of the o r i g i n a l m a n u s c r i p t a n d for t h e i r helpful c o m m e n t s to i m p r o v e the paper.
References Aumann, R.J.. 1976. Agreeing to disagree, Annals of Statistics 4. 1236-1239. Bacharach. M.. 1985, Some extensions of a claim of Aumann in an axiomatic model of knowledge, Journal of Economic Theory 37. 167-190. Binmore. K. and A. Brandenburger, 1990, Common knowledge and game theory, in: K. Binmore, Essays on the foundation of game theory (Basil Blackwell, Oxford) 105-150. Geanakoplos, J.D. and H.M. Polemarchakis, 1982. We can't disagree forever, Journal of Economic Theory 28. 192-200. Samet, D.. 1990, Ignoring ignorance and agreeing to disagree, Journal of Economic Theory 52, 190-207. Samet, D., 1992. Agreeing to disagree in infinite information structures. International Journal of Game Theory 21, 213-218.