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A note on mutually absolutely continuous belief systems
*
Kin Chung Lo
Department of Economics, York University, 4700 Keele Street, North York, Toronto, Ontario, Canada M3J 1P3
Received 2 October 1998; received in revised form 21 September 1999; accepted 27 October 1999
Abstract
This paper shows that in a strategic game, if players’ beliefs are mutually absolutely continuous, then epistemic results that are established using the notion of belief can be reformulated using the standard notion of knowledge. 2000 Elsevier Science S.A. All rights reserved.
Keywords: Belief with probability one; Partitional information structure; Epistemic conditions for Nash equilibrium; Finitely repeated prisoner’s dilemma
JEL classification: C72
1. Introduction
In a very interesting paper, Aumann and Brandenburger (1995) provide epistemic conditions for Nash equilibrium. The model that they use to discuss epistemic matters is called an interactive belief system (or simply belief system). To guarantee that Nash equilibrium arises in games with n.2 players, they assume that players’ beliefs come from a common prior. Stuart (1997) also uses the framework of belief system to provide epistemic conditions for defection in every period of the finitely repeated prisoner’s dilemma. Instead of assuming that players’ beliefs come from a common prior, he assumes that the belief system is mutually absolutely continuous. Roughly speaking, this assumption says that if a player believes that a given state is possible, then all players believe that the state is possible. As Stuart points out in his paper, if the belief system has a common prior, then it is mutually absolutely continuous.
Both of the above papers adopt the notion of belief. That is, a player believes an event E if he assigns probability one to E. In this paper, we show that if the belief system is mutually absolutely continuous, then epistemic results that are established using the notion of belief can be reformulated
*Tel.:11-416-736-2100, ext. 77032; fax: 11-416-736-5987. E-mail address: kclo@yorku.ca (K.C. Lo)
1
using the standard notion of knowledge. An advantage of such a reformulation is that standard intuition about knowledge can be applied to make the results more transparent. At the end of the paper, we demonstrate this point by reformulating the results of Aumann and Brandenburger (1995) and Stuart (1997).
2. Belief system and partitional structure
Fix a strategic game form A5A13 ? ? ? 3A , where A is a finite set of strategies for player i.n i
Throughout, the indices i, j and k vary over distinct players inh1,2, . . . ,nj. Unless specified otherwise or emphasis is necessary, the quantifier ‘for all such i, j, and k’ is to be understood. Define
A2i5 3j±iA . Elements in A and Aj i 2i are denoted by a and a , respectively. A game is defined asi 2i called an event and any element t[T is called a state of the world.
ˆ
A conjecturefi of player i is a probability measure on A . The marginal probability measure of2i fi on A is denoted by margj Afi. If player i’s type is t , then i’s conjecturei fi(t ) is given byi
Say that player i of type t is rational ifi
ˆ ˆ
a (t )i i [arg max
O
u (t )(a ,ai i i 2i)fi(t )(ai 2i). ai[Aia2i[A2iˆ
That is, player i of type t is rational if his strategy a (t ) maximizes expected payoff when the payoffi i i
ˆ
function is u (t ) and the conjecture isi i fi(t ).i
1
˜ ˆ ˜
Given any event E, say that player i believes E at t ifht[T :p (t;t )i i .0j7E. Define B (E ) as thei
1 1
set of all those t at which i believes E. Set B (E )5B (E )1 >? ? ? >B (E ). That is, B (E ) is the eventn 1
that all players believe E. If t[B (E ), say that E is mutually believed at t. For any positive integer m,
m11 1 m ` m
define B (E )5B (B (E )). Define CB(E )5>m51B (E ). If t[CB(E ), say that E is commonly believed at t.
ˆ
A probability measure p on T is called a common prior of the belief system if for all i and all t ,i
ˆp(ti3T2i).0 and
Proposition 1 below is due to Aumann and Brandenburger (1995, p. 1168, Theorem B).
Proposition 1. Consider a game u, an n-tuplef of conjectures, and a belief system with a common
˜ ˜
and s is a Nash equilibrium of u.
ˆ
Say that a belief system is mutually absolutely continuous if for any player i, p (t;t )i i .0 implies
ˆ
that for all players j, p (t;t )j j .0 (Stuart, 1997, p. 136). It is immediate that if the belief system has a
ˆ ˆ
common prior, then it is mutually absolutely continuous. Suppose n52 and the game (u (t ),u (t ))1 1 2 2
at every state t[T is the N-period repeated prisoner’s dilemma (where N is any positive integer).
Proposition 2 below is due to Stuart (1997, p. 139).
Proposition 2. Consider a mutually absolutely continuous belief system description of the N-period
˜ ˆ
repeated prisoner’s dilemma. Suppose there is a state t such that p (t,t ).0 and the event ht[T :
i i
˜
every player i of type t is rationalj is commonly believed at t. Then at state t, both players defect in
i
every period of the game.
2.2. Partitional structure
A partitional structure for the game form A is denoted ashV,H ,p ,u ,ai i i ij. Each of its components is explained as follows.
• All the players are facing a finite setV of states of the world. Any subset ofV is called an event.
• Player i’s information structure is represented by a partition H ofi V. For every state v[V,
H (i v)[H is the partitional element which containsi v.
• At each v[V, i’s beliefs over V are represented by a probability measure p (i ?;v) with the
Note that p , u and a are functions oni i i V and they are assumed to be measurable with respect to H .i At every state v, i’s conjecture fi(v) is given by
fi(v)(a2i)5p ([a ];i 2i v) ;a2i[A ,2i (1)
˜ ˜
where [a ]2i ; hv[V:a (v2i )5a2ij. An n-tuple (f1(v), . . . ,fn(v)) of conjectures is denoted by
f(v). Similarly, an n-tuple (u (v), . . . ,u (v)) of payoff functions is denoted by u(v).1 n Say that player i is rational at v if
a (iv)[arg maxa[A
O
u (i v)(a ,ai 2i)fi(v)(a2i).i ia2i[A2i
That is, i is rational atv if his strategy a (v) maximizes expected payoff when the payoff function isi u (iv) and the conjecture is fi(v).
of i’s partition cells). An event that is simultaneously self-evident to all the players is called a public
event. Aumann (1976) provides the following convenient characterization of commonly known
events.
Proposition 3. An event F is commonly known atv if and only if there exists a public event E such
that v[E#F.
A probability measure p on V is called a common prior of the partitional structure if for all i and all v, p(H (v))i .0 and
Note that if the partitional structure has a common prior p, then i’s conjecture fi(v) defined in (1) will be the conditional probability measure which is derived from p as follows:
p([a ]2i >H (i v))
]]]]]
fi(v)(a2i)5 p(H (v)) ;a2i[A .2i (2)
i
It is well known that conditional probabilities satisfy the sure-thing principle. That is, given any two disjoint events F and G, if the probability of an event is equal toa conditional on F, and similarly, if the probability of the event is alsoa conditional on G, then the probability of the event conditional on
3
F<G is alsoa. Therefore, for any nonempty event E that is self-evident to i, if E#hv[V:fi(v)5
fij, then (2) and the sure-thing principle imply that
3
p([a ]2i >E )
]]]]
fi(a2i)5 ;a2i[A .2i (3)
p(E )
Recall that the function a is measurable with respect to H . This implies that for all ai i i[A , the eventi
˜ ˜
Note that (3) and (4) are also valid for i’s conjecture over any subset of his opponents. That is, [a ]2i in the two equations can be replaced by >j[)[a ], wherej ) is any subset of h1, . . . ,i21,i1
1, . . . ,nj. The equivalence between (3) and (4) is a key to the proof of Proposition 5 in the sequel. Finally, say that a partitional structure has full support if for all i and all v, p (vi ;v).0.
3. Decision-theoretic transformation
ˆ ˆ ˆ
Fix a mutually absolutely continuous belief system hT ,p ,u ,ai i i ij. Define
ˆ
S5ht[T :p (t;t )i i .0j.
Since the belief system is mutually absolutely continuous, the definition of S does not depend on i.
ˆ ˆ ˆ
in addition, if the belief system has a common prior p, then the partitional structure also has a
ˆ
common prior p such that for all t[S, p(t)5p(t).
Proposition 4 below provides some useful relationship between a mutually absolutely continuous belief system and its decision-theoretic transformation. Its main message is that belief of rationality and conjectures in a belief system can be restated as knowledge of rationality and conjectures in its decision-theoretic transformation.
ˆ ˆ ˆ
Proposition 4. Fix a mutually absolutely continuous belief system hT ,p ,u ,ai i i ij and its decision-theoretic transformation hV,H ,p ,u ,ai i i ij. Then
ˆ
Proof. Parts (a), (b), (c) and (d) follow from the definition of decision-theoretic transformation, and (f) follows from (e). To establish (e), note that for all t[T and for all E#T,
Proposition 4 implies that Proposition 1 can be reformulated as Proposition 5 below. We also provide a proof of Proposition 5 which is relatively more transparent than the proof of Theorem B in Aumann and Brandenburger (1995, pp. 1168–1169).
Proposition 5. Consider a game u, an n-tuple f of conjectures, and a full support partitional ˜
structure with a common prior p. Suppose there is a state v such that hv[V: every player i is
˜ ˜ ˜ ˜ ˜
rational atvj andhv[V:u(v)5ujare mutually known atv, andhv[V:f(v)5fjis commonly known atv. Then there is an n-tuples5(s1, . . . ,sn), wheresi is a probability measure on A , suchi
that fi5 3j±isj and s is a Nash equilibrium of u.
˜ ˜
Proof. The hypothesis that hv[V:f(v)5fj is commonly known at v implies agreement of marginal conjectures: margAifj5margAifk (Aumann, 1976, p. 1237). This enables us to define,
If there exists j±1 such that [a ]j >E55, then it is obvious that both sides of (7) are zero. Suppose
implying that the right hand side of (7) can be simplified to
p([a ]2 >[a ]3 >E ) p([a ]4 >E ) p([an21]>E ) p([a ]n >E )
]]]]]] ]]]]? ? ?]]]]] ]]]]. (9)
p(E ) p(E ) p(E ) p(E )
Simplifying (9) and its subsequent expressions using the same argument as many times as desired, the right hand side of (7) will eventually be transformed to its left hand side.
All the knowledge hypotheses in the proposition imply that for every i and for every j±i,
˜ ˜ ˜ ˜ ˜ ˜
H (jv)7hv[V:player i is rational atvj>hv[V:u (vi )5uij>hv[V:fi(v)5fij.
˜ ˜
That is, every ai[ha (i v):v[H (j v)j maximizes i’s expected payoff when the payoff function is ui and the conjecture is fi, where fi has been shown to be equal to 3j±isj. Since the partitional structure has full support and si is j’s marginal conjecture about i at v, the support of si is
˜ ˜
actuallyha (i v):v[H (jv)j. Therefore, s is a Nash equilibrium of u. h
Proposition 4 also implies that Proposition 2 can be reformulated as Proposition 6 below.
Proposition 6. Consider a full support partitional structure description of the N-period repeated
˜ ˜
prisoner’s dilemma. Suppose there is a statev such that hv[V: every player i is rational at vj is
commonly known at v. Then at state v, both players defect in every period of the game.
˜
Proof. According to Proposition 3, there exists a public event E such that v[E#hv[V: every
˜ ˜ ˜ ˜
player i is rational at vj. Since the partitional structure has full support, p (v;vi ).0 for all v[E. ˜
These two facts imply that for all v[E, the hypotheses stated in Claim (a) below are satisfied. The ˜
claim establishes that at everyv[E, both players defect in the last period of the game. Given Claim ˜
(a), Claim (b) below can be applied repeatedly to conclude that at everyv[E, both players defect in
every period of the game. We provide a proof of Claim (b), which can be easily simplified to prove Claim (a).
Proof of Claim b. Suppose, to the contrary, there exists K$M11 such that i does not defect in
˜ ˜
˜
that if i follows a (vi ) only in the first N2(M11) periods and then switches to defection in the last
˜ ˜
M11 periods, his payoff at every state in H (vi ) will not decrease and that at vwill strictly increase.
˜ ˜ ˜
This and p (v;vi ).0 imply that his expected payoff atv will also strictly increase, contradicting the
˜
hypothesis that i is rational at v. h
Acknowledgements
I especially thank a referee for providing valuable and detailed suggestions. I also thank Larry Epstein for valuable discussions and the Social Sciences and Humanities Research Council of Canada for financial support.
References
Aumann, R.J., 1976. Agreeing to disagree. Annals of Statistics 4, 1236–1239.
Aumann, R.J., Brandenburger, A., 1995. Epistemic conditions for Nash equilibrium. Econometrica 63, 1161–1180. Brandenburger, A., Dekel, E., 1987. Common knowledge with Probability 1. Journal of Mathematical Economics 16,
237–245.
