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Epistemic conditions for agreement and stochastic
independence of
e
-contaminated beliefs
* Kin Chung Lo
Department of Economics, York University, Toronto, Ontario, Canada M3J 1P3 Received 1 January 1998; received in revised form 1 August 1998; accepted 1 November 1998
Abstract
This paper studies strategic games in which the beliefs of each player are represented by a set of probability measures satisfying a parametric specialization that is callede-contamination. That is, beliefs are represented by a set of probability measures, where every measure in the set has the form (12e)p*1ep, p* being the benchmark probability measure, p being a contamination, and
ereflecting the amount of error in p* that is deemed possible. Under a suitably modified common prior assumption, if beliefs about opponents’ action choices are common knowledge, then beliefs satisfy some properties that can be interpreted as agreement and stochastic independence. 2000 Elsevier Science B.V. All rights reserved.
Keywords: Strategic games;e-Contamination; Agreement; Stochastic independence
JEL classification: C72; D81
1. Introduction
According to the subjective expected utility model (henceforth the expected utility model), the beliefs of a decision maker are represented by a probability measure. However, the Ellsberg Paradox (Ellsberg, 1961) and related experimental findings (summarized by Camerer and Weber, 1992) demonstrate that in situations where a decision maker only possesses ambiguous information about the set of states of the world, the decision maker is typically averse to ambiguity and the probabilistic representation of beliefs is too restrictive. As a result, generalizations of the expected utility model have been proposed. In the Choquet expected utility model (Schmeidler,
*Tel.: 11-416-736-2100, ext. 77032; fax:11-416-736-5987.
E-mail address: [email protected] (K.C. Lo)
1989), the decision maker’s beliefs are represented by a capacity (that is, a non-additive probability measure). In the multiple priors model (Gilboa and Schmeidler, 1989), beliefs are represented by a set of probability measures. Also see Bewley (1986, 1987, 1988, 1989) for a related model and study of the relevance of ambiguity to various economic problems. Following Gilboa and Schmeidler (1993), we reserve the term ‘ambiguous beliefs’ for beliefs that are representable by either a capacity or a set of probability measures.
Nash equilibrium has been the central solution concept in game theory. Adopting the view that choosing an action in a game is a decision problem under uncertainty, Nash equilibrium presumes that players’ preferences are represented by the expected utility model (see Brandenburger, 1992, pp. 90–92). In order to study the implications of ambiguity in strategic situations, this equilibrium concept has been generalized to allow for preferences to be representable by the Choquet expected utility and multiple priors models. See Dow and Werlang (1994), Eichberger and Kelsey (1994), Klibanoff (1993), Lo (forthcoming, 1996), Marinacci (1995) and Mukerji (1995). The issue in this literature is how the following three features of Nash equilibrium should be generalized.
• Best response property. If an action of player i is in the support of player j’s beliefs,
then the action must be optimal for player i given i’s beliefs.
• Agreement of beliefs. For each player, all the other players hold the same beliefs
about the action choice of that player.
• Stochastic independence of beliefs. Each player believes that the action choices of the
other players are stochastically independent.
The above papers focus attention on generalizing the best response property of Nash equilibrium. The issue of agreement and stochastic independence has been very much neglected. In particular, Dow and Werlang (1994), Lo (forthcoming) and Marinacci (1995) avoid the issue by considering only two person games. Even though the other papers consider n-person games, they typically adopt definitions of agreement and stochastic independence of ambiguous beliefs in an arbitrary manner. This phenomenon can be easily explained. When beliefs are probabilistic, there is one standard and well received definition of agreement and stochastic independence. The definition is as follows. Players’ beliefs about an event agree if they attach the same probability to that event. The beliefs of a player satisfy stochastic independence if beliefs are represented by a product measure. When beliefs are ambiguous, many different definitions of agreement and stochastic independence have been proposed (see Section 3.2 for details). They possess some mathematical properties, but none of them can be clearly character-ized in terms of preference.
Since many definitions of agreement and stochastic independence of ambiguous beliefs have been suggested, the need to carry out the exercise of Aumann and Brandenburger for the case of ambiguous beliefs is even stronger. The only result in this direction appears in Lo (1996, p. 471, Proposition 7). However, that result is very restrictive because players’ beliefs are assumed to satisfy ‘complete ignorance’ (see the end of Section 5.3 for details). In this paper, we extend Lo’s result to an important
parametric class of ambiguous beliefs: e-contamination. Roughly speaking, e
-contami-nated beliefs are represented by a set of probability measures, where every measure in
the set has the form (12e)p*1ep, p* being the benchmark probability measure, p
being a ‘contamination’, and e reflecting the amount of error in p* that is deemed
possible. For brevity, we will simply say in this paper that beliefs are contaminated if
the parametric restriction ofe-contamination is satisfied. This class of beliefs is a special
case of the class of belief functions (Dempster, 1967; Shafer, 1976), which is in turn a special case of convex capacities (Schmeidler, 1989).
There are reasons for studying contaminated beliefs. In the multiple priors model, the set of probability measures is unrestricted except by technical conditions. Therefore, the model may be too general for some settings. The class of contaminated beliefs is not only intuitively appealing, it also possesses sufficient structure so that tractable analysis is often possible. As a result, it has been thoroughly studied and its usefulness is well established in the robust statistics literature (see Berger (1984, 1985), Berger and Berliner (1986) and Wasserman (1990), for example). In the economics literature, many papers adopt the class of contaminated beliefs either as the primary model or at least as an illustrative example. See, for instance, Dow and Werlang (1992, p. 202), Dow and Werlang (1994, pp. 313–314), Eichberger and Kelsey (1994), Epstein (1997, pp. 15–16), Epstein and Wang (1994, pp. 288–289), Epstein and Zhang (1999) and Mukerji (1995). Finally, when beliefs are restricted to be contaminated, various rules for updating ambiguous beliefs proposed in the literature are equivalent (see Section 4.1 for details). As a result, the controversy on which updating rule is more appropriate than the others can be avoided.
More precisely, the aim of this paper is to search for a notion of agreement and stochastic independence of contaminated beliefs by asking the following question:
Suppose that players’ beliefs are contaminated. Under a suitably modified common prior assumption, does common knowledge of beliefs about opponents’ action choices imply that beliefs satisfy any property that we may want to call ‘agreement and stochastic independence’?
We provide a positive answer to the above question. Recall that the key to the fundamental result of Aumann (1976) on agreement of beliefs and also to the result of Aumann and Brandenburger (1995) is that conditional probabilistic beliefs that are updated from a prior using Bayes rule satisfy the sure-thing principle: given any two
disjoint events D and E, if the probability of an event is equal toaconditional on D, and
similarly, if the probability of the event is alsoa conditional on E, then the probability
of the event conditional on D<E is also a. See Geanakoplos (1992, pp. 66–67).
the more general subclasses of ambiguous beliefs do not have such a property, we believe that the result in this paper cannot be extended further.
Although the focus of this paper is agreement and stochastic independence, the main result described in the preceding paragraph has an unexpected corollary which constitutes a separate contribution. The corollary is that all the definitions of support of ambiguous beliefs proposed in the literature coincide with each other. Therefore, under the above assumptions, the controversy on how the support of ambiguous beliefs is defined does not arise. Finally, we demonstrate that if players’ rationality is mutual knowledge, then their beliefs constitute an equilibrium.
The plan of this paper is as follows. Section 2 establishes notation for the paper, provides a review of the multiple priors model and contaminated beliefs, and defines strategic games. Section 3 discusses the meaning of agreement and stochastic in-dependence of beliefs in terms of preference and reviews existing definitions in terms of ambiguous beliefs. Section 4 presents the main result and Section 5 discusses its implications. An appendix contains proofs of all propositions.
2. Preliminaries
2.1. Notation
The following notation applies throughout the paper.
• For any set Y, the cardinality of Y is denoted byuYu. For any set Z, Z#Y means that
Z is a subset of Y, and Z,Y means that Z is a strict subset of Y.
• For any finite set Y, the set of all probability measures on Y is denoted by M(Y ). For
any c[M(Y ), the support of c is denoted by supp c. For any C#M(Y ), supp
C; <c[Csupp c. That is, supp C#Y is the union of the supports of the
probability measures inC.
L
• Suppose Y5 3l51Y . For anyl c[M(Y ), the marginal probability measure ofcon Yl
is denoted by margYlc. For anyC #M(Y ), margYlC; hcl[M(Y ):l 'c[C such that
c 5l margYlcj. That is, margYlC is the set of marginal probability measures on Y asl
one varies over the set of probability measures inC.
L
• Suppose Y5 3l51Y . For any Zl #Y, proj ZYl ; hyl[Y :l 'z[Z such that the lth
co-ordinate of z is ylj. That is, proj Z is the projection of Z on Y .Yl l
2.2. Multiple priors model and contaminated beliefs
This section contains a review of the multiple priors model and contaminated beliefs.
Let S be a finite set of states of the world, X a set of outcomes, and^ the set of acts
from S to X. For notational simplicity, x[X also denotes the constant act that yields x
for all s[S. The decision maker has a preference orderingKover^. According to the
multiple priors model, the representation forKconsists of a von Neuman Morgenstern
(vNM) index u: X→R and a closed and convex set3of probability measures on S. The
U( f );min
O
u( f(s))p(s). (1)p[3 s[S
The intuition of the model is as follows. The decision maker’s beliefs over the state space S may be too vague to be represented by a probability measure and are represented
instead by a set3 of probability measures. The decision maker is averse to ambiguity in
the sense that he evaluates an act by computing the minimum expected utility over the
probability measures in 3.
In this paper, we focus on the following parametric specialization of3: there exist an
event E#S, a probability measure p*[M(E ) and a real number e[[0,1] such that
35h(12e)p*1ep: p[M(E )j. (2)
Say that 3 is contaminated if it satisfies (2). The probability measure p* may be
thought of as the benchmark probability measure governing the relative likelihood of events in S. However, the decision maker is not certain about p* in the sense that it is
contaminated or perturbed with weighte by the probability measures in the set M(E ).
Note that the event E is allowed, but is not required, to be the whole state space S.
Finally, observe that with p* and E fixed, the set 3 increases in the sense of set
inclusion ase increases, modeling increased ambiguity aversion. In particular, ife 50,
then 35hp*j; if e 51, then 35M(E ).
2.3. Strategic games
In this section, we define strategic games in which each player behaves according to the single person decision theory in Section 2.2.
n
A strategic game form is denoted as A; 3i51A , where A is a finite set of actionsi i
for player i. Throughout, the indices i, j and k vary over distinct players in h1, . . . ,nj.
Elements in A andi 3j±iA are denoted by a and a , respectively. Given a strategicj i 2i
n
game form A, a strategic game is denoted as u; hui ij51, where u : Ai →R is a vNM
index representing player i’s preference ordering over A.
Since player i is uncertain about the action choices of the other players, the relevant
state space for i is 3j±iA . Consistent with the above single person decision theory, i’sj
preference ordering over the set of acts defined on 3j±iA is represented by the multiplej
priors model defined in (1). Every action ai[A can be identified as an act overi 3j±iAj
and i’s utility of taking a is equal toi
min
O
u (a , ai i 2i)fi(a2i), fi[Fia2i[3j±iAjwhereFiis a closed and convex set of probability measures on 3j±iA . Consistent withj
*
(2), if Fi is contaminated, then there exist E2i#3j±iA ,j fi [M(E ) and2i ei[[0,1]
such that
*
F 5i h(12ei)f 1 e fi i i:fi[M(E )2i j.
* *
not be a product space and fi may not be a product measure. In particular, supp fi
may be any subset of E .2i
3. Agreement and stochastic independence
In Section 3.1, we present some preference based conditions that can be interpreted as agreement and stochastic independence. They serve as some criteria for evaluating the existing definitions of agreement and stochastic independence of ambiguous beliefs which we present in Section 3.2. We also suggest informally the definition of agreement and stochastic independence of contaminated beliefs which we justify in Section 4. For notational simplicity, assume that a game has three players.
3.1. Conditions in terms of preference
The preference based conditions for agreement and stochastic independence make use of the following two choice theoretic notions from Savage (1954). Recall the single
agent setting in Section 2.2, where S denotes a set of states of the world, ^ the set of
acts defined on S andKa preference ordering over^. Given any event E#S, say that
E is null if for all f, f9, g[^,
f(v) if v[E f9(v) if v[E
| . (3)
F
g(v) if v[⁄ EG F
g(v) if v[⁄ EG
In words, an event E is null if the decision maker does not care about payoffs for states 1
that are in E. If K is represented by the multiple priors model having 3 as the
underlying set of probability measures, then
E is null if and only if p(E )50 ;p[3. (4)
The second choice theoretic notion concerns the decision maker’s beliefs about the
relative likelihood of events. Given any two events D,E#S, say that D is more likely
than E if for all x*,x[X with x*sx,
x* if s[D x* if s[E
F
G
KF
G
. (5)x if s[⁄ D x if s[⁄ E
1
In words, D is more likely than E if the decision maker prefers to bet on D rather than 2
on E. If K is represented by the multiple priors model, then
D is more likely than E if and only if min p(D )$min p(E ). (7)
p[3 p[3
Going back to the context of strategic games, consider the beliefs of players i and j about the action choice of player k. The following is a basic condition for agreement of
beliefs: given any two events D,E#A ,k
i believes that D is more likely than E if and only if j believes that D is more likely
than E. (8)
If i and j’s preferences are represented by the multiple priors model, then (7) implies
that (8) can be rewritten as follows: for all D,E#A ,k
min sk(D )$ min sk(E )
sk[margAFi sk[margAFi
k k
if and only if min sk(D )$ min sk(E ). (9)
sk[margAkFj sk[margAkFj
Consider player i’s beliefs about the action choices of players j and k. The following is a basic condition capturing the intuition that i believes that what j is going to do is
independent of what k is going to do: given any two events D ,Ej j#A and any twoj
3
events D ,Ek k#A that are non-null from i’s point of view,k
i believes that Dj3D is more likely than Ek j3Dk
if and only if i believes that Dj3E is more likely than Ek j3E .k (10)
Again, if i’s preference is represented by the multiple priors model, then we can use (4)
and (7) to rewrite (10) as follows: for all D ,Ej j#A and D ,Ej k k#A such thatk
2
The following way to induce a more likely than relation from preference seems equally legitimate: given any
D,E#S, D is more likely than E if for all x*,x[X with x*sx,
x if s[E x if s[D
K . (6)
F
x* if s[⁄ EG F
x* if s[⁄ DG
That is, the decision maker prefers to bet against E rather than against D. IfKis represented by the multiple priors model, then D is more likely than E according to (6) if and only if maxp[3p(D )$maxp[3p(E ). In this
paper, we adopt (5). However, all the results are valid if we adopt (6).
3
The notation Dj3D in (10) refers to the event that j chooses an action from D and k chooses an action fromk j
min sk(D )k .0 and min sk(E )k .0, sk[margAkFi sk[margAkFi
min fi(Dj3D )k $min fi(Ej3D )k (11)
fi[Fi fi[Fi
if and only if min fi(Dj3E )k $min fi(Ej3E ).k
fi[Fi fi[Fi
3.2. Existing definitions in terms of ambiguous beliefs
Eichberger and Kelsey (1994, p. 15, Definition 4.3), Lo (1996, p. 453, Definition 4) and Mukerji (1995, p. 20, Definition 7) adopt the following definition of agreement of ambiguous beliefs:
margAkF 5i margAkFj. (12)
That is, the beliefs of players i and j about the action choice of player k are represented by the same set of probability measures. Clearly, (12) implies (9) and therefore the preference based condition for agreement in (8). The converse is not true. For instance,
suppose A35hL, Rj,
margA3F 51 hs3[M(A ): 0.43 #s3(L )#0.6j and
margA3F 52 hs3[M(A ): 0.33 #s3(L )#0.7j. (13)
Clearly, margA3F1±margA3F2 and therefore (12) is violated. However, both players
believe that L and R are equally likely.
In fact, the sets of probability measures in (13) can be rewritten as
*
margA3F 51 h(12e1)s 1 e s3 1 3:s3[M(A )3 j and
*
margA3F 52 h(12e2)s 1 e s3 2 3:s3[M(A )3 j,
*
where s 53 (L, 0.5;R, 0.5), e 51 1 / 5 and e 52 2 / 5. That is, margA F1 and margA F2
3 3
share the same benchmark probability measure and contamination, and their difference is
due to the fact that e ±e . In Section 4, we will explain why this phenomenon may
1 2
occur.
Klibanoff (1993) adopts the following definition of agreement:
margAkFi>margAkFj±5. (14)
That is, the set of probability measures representing player i’s beliefs and that representing player j’s beliefs have a nonempty intersection. In contrast to the definition of agreement in (12), the one in (14) does not imply the preference based condition in
(8). For instance, suppose A35hL, Rj,
margA3F 51 hs3[M(A ): 03 #s3(L )#0.6j and
margA3F 52 hs3[M(A ): 0.63 #s3(L )#1j.
Therefore, (14) is satisfied. However, player 1 believes that R is strictly more likely than
L but player 2 believes the opposite. Therefore, (8) is violated.
We now turn to existing definitions of stochastic independence of ambiguous beliefs.
Almost all of them imply thatFi satisfies the following independent product property:
for all events Ej#A and Ej k#A ,k
min fi(Ej3E )k 5 min sj(E )j min sk(E ).k (15)
fi[Fi sj[margAjFi sk[margAkFi
It is well known that ifFi is a singleton, thenFi is completely determined by the right
hand side of (15). Therefore, both Ghirardato (1997, p. 270) and Hendon et al. (1996, p. 97) regard the independent product property as basic that any definition of stochastic independence of ambiguous beliefs should satisfy. The definition proposed by Gilboa and Schmeidler (1989, pp. 150–151), which is adapted to the context of strategic games by Lo (1996, p. 453, Definition 4), also satisfies the independent product property. Similarly for the definitions in Eichberger and Kelsey (1994, p. 14, Definition 4.2) and Mukerji (1995, p. 8, Definition 2).
Clearly, the independent product property implies (11) and therefore the preference based condition for stochastic independence in (10). However, except for a few trivial
4 cases, it is incompatible with contaminated beliefs.
Proposition 1. Suppose there exist E2i#Aj3A withk uproj EAj 2iu.1 anduproj EAk 2iu.
* *
1, fi [M(E ), and2i ei[(0,1) such that F 5i h(12ei)f 1 e fi i i: fi[M(E )2i j. Then
Fi does not satisfy the independent product property.
The following example, which is presented in a setting that will be formally defined in Section 4.1, illustrates the intuition behind Proposition 1.
Example 1. Suppose player 1 is facing a setV 5hv1,v2,v3,v4jof states of the world,
where every statev[V contains the following specification of actions a (2v)[A and2
a (3v)[A taken by players 2 and 3, respectively.3
v1 v2 v3 v4
a2 U D U D
a3 L R R L
Suppose player 1’s beliefs aboutV are represented by the set of probability measures
*
L 51 h(12e1)m 1 e m1 1 1:m1[M(V)j, (16)
*
where m 51 (v1, 0.25; v2, 0.25; v3, 0.25; v4, 0.25) and e1[(0,1). Given the action
functions a and a , every probability measure in2 3 L1 induces a probability measure on
4
hU,Dj3hL,Rjin the obvious manner. If we do it for every probability measure inL1, we obtain the set of probability measures
*
F 51
H
(12e1)P
s 1 e fj 1 1:f1[M(hU, Dj3hL, Rj) ,J
(17)j52,3
* *
wheres 52 (U, 0.5; D, 0.5) and s 53 (L, 0.5; R, 0.5).
GivenF1in (17), 1’s marginal beliefs about 2 and 3’s action choices are represented
by
*
margA2F 51 h(12e1)s 1 e s2 1 2:s2[M(hU, Dj)j and
*
margA3F 52 h(12e1)s 1 e s3 1 3:s3[M(hL, Rj)j, (18)
respectively. Note thatF1, margA2F1 and margA3F1are all contaminated with the same
degree of contaminatione1. The reason for this is as follows. The numbere1 represents
player 1’s degree of uncertainty aboutV. Note that the action choices of players 2 and 3
are both functions onV. Therefore, 1’s degree of uncertainty about the action choices of
2 and 3, or each of their action choices separately, should also be represented bye1. In
contrast, given 1’s marginal beliefs in (18), the independent product property requires player 1, when considering the overall action choices of his two opponents, to ‘double
count’ e1. For example, take the actions U and L. Let C1 be any set of probability
measures on A23A such that marg3 A2C 51 margA2F1, margA3C 51 margA3F1, and the
independent product property is satisfied. Then we have
min c1(U3L ) 5 min s2(U ) min s3(L )
s[marg F s[marg F
c1[C1 2 A2 1 3 A3 1
2
5(12e1) (0.25),
which is strictly less than
min f1(U3L )5(12e1)(0.25).
f1[F1
That is, the independent product property ignores the fact that player 1’s uncertainty
about each of his opponents in fact comes from the same ex ante uncertainty over V.
Although F1 in (17) does not satisfy the independent product property, it has the
following features of ‘stochastic independence’: the benchmark probability measure
*
P
j52,3sj is a product measure and the contamination is defined on the product spacehU,Dj3hL,Rj. In Section 4, we will describe precisely the conditions under which these
features arise. j
The only existing definition of stochastic independence of ambiguous beliefs which do not imply the independent product property is due to Klibanoff (1993). He only requires
Fito contain at least one product measure. (19)
This definition is compatible with contaminated beliefs. For instance, suppose A25hU,
Dj, A35hL, Rj and
*
*
wheref 51 (UL, 0.5; UR, 0; DL, 0; DR, 0.5). ThenF1satisfies (19) because it contains
the product measure (UL, 0.25; UR, 0.25; DL, 0.25; DR, 0.25). However, we have
min f1(UL )5min f1(DR)50.25 and min f1(UR)5min f1(DL )50.
f1[F1 f1[F1 f1[F1 f1[F1
That is, player 1 believes that UL is more likely than DL, but UR is less likely than DR.
This demonstrates that, even ifFi is contaminated, (19) does not imply the preference
based condition for stochastic independence in (10).
4. Epistemic conditions
4.1. Generalization of the common prior assumption
In this section, we first establish a decision theoretic framework for discussing epistemic matters. Then we formulate a generalization of the common prior assumption for contaminated beliefs.
Fix a strategic game form A. The model that we use to discuss epistemic matters is
denoted as hV, H ,i Di, u , ai ij. Each of its components is explained as follows.
• All the players are facing a finite set V of states of the world.
• Player i’s information structure is represented by a partition H ofi V. For every state
v[V, H (iv)[H denotes the partitional element which containsi v.
• Player i’s beliefs atvare represented by a closed and convex setDi(v) of probability
measures on H (iv).
• Player i’s payoff function at v is u (iv): A→R.
• The action taken by player i at v is a (iv)[A .i
Note that Di, u and a are functions oni i V. To respect the partitional information
structure, they are measurable with respect to H . Also note that at everyi v[V,
u(v); hu (1v), . . . ,u (nv)jconstitutes a strategic game.
The above model is essentially the same as the standard model that is used to discuss epistemic conditions for Bayesian solution concepts. See, for instance, Dekel and Gul (1997, p. 17) and Osborne and Rubinstein (1994, p. 76). The only difference is that in
the Bayesian case, the set Di(v) of probability measures at every statev[V and for
every player i is required to be a singleton.
The object that is of ultimate interest is player i’s beliefs about opponents’ action
choices. At every statev[V, i’s beliefs over 3j±iA are represented by the closed andj
convex set Fi(v) of probability measures which is induced fromDi(v) as follows:
Fi(v)5hfi[M(3j±iA ):j 'mi[Di(v) such that for all a2i[3j±iA ,j
fi(a2i)5mi(hv9[H (iv): a (2iv9)5a2ij)j, (20)
where a (2iv9); ha (1v9), . . . ,ai21(v9),ai11(v9), . . . ,a (nv9)j. Denote the n-tuple
Say that player i is rational at v if
a (iv)[arg max min
O
u (iv)(a ,ai 2i)fi(a2i).ai[Aifi[Fi(v)a [3 A
2i j±i j
That is, i is rational atvif his action a (iv) maximizes utility when the payoff function is
u (iv) and when beliefs are represented by Fi(v).
Let m be a probability measure on V with the property that m(H (iv)).0 for all
v[V and for all i. Let m(?uH (iv)) be the probability measure on H (iv) which is
derived from m using Bayes rule. The common prior assumption can be stated as
n
follows: the n-tuple hDi ij51 satisfies the common prior assumption if there exists
m[M(V) such that
Di(v)5m(?uH (iv)) ;v[V ;i.
Consider the following generalization of the common prior assumption.
n
Definition 1. The n-tuple hDi ij51 satisfies the assumption of common prior with
n
contamination if there exist m[M(V) and a collection hei ij51 of functions, where ei:
V→[0,1] is measurable with respect to H , such thati
Di(v)5h(12ei(v))m(?uH (iv))1ei(v)mi:mi[M(H (iv))j ;v[V ;i. (21)
Definition 1 is implied, and therefore can be justified, by the following assumption.
Assume that ex ante, players’ beliefs about the set V of states of the world are
contaminated with the same benchmark probability measurem[M(V). That is, i’s ex
ante beliefs are represented by
Li; h(12ei)m 1 eip: p[M(V)j, (22)
whereei[[0,1]. When a statev is realized, player i is informed that the event H (iv) has
occurred. The set Di(v) of probability measures represents i’s updated beliefs. As
suggested by Gilboa and Schmeidler (1993), a natural procedure to derive Di(v) is to
rule out some of the priors inLi and then update the rest according to Bayes rule. Two
updating rules of particular interest are the maximum likelihood updating rule:
Di(v)5hmi[M(H (iv)):'p[Li
ˆ
such that p(H (iv))5max p(H (iv)) andmi(?)5p(?uH (iv))j (23)
ˆ p[Li
and the full Bayesian updating rule:
Di(v)5hmi[M(H (iv)):'p[Lisuch that p(H (iv)).0 andmi(?)5p(?uH (iv))j.
(24)
That is, according to the maximum likelihood updating rule, only those probability
measures inLi that ascribe the maximal probability to H (iv) are updated. On the other
hand, according to the full Bayesian updating rule, every probability measure in Li
measures in (23) must be a subset of that in (24). Proposition 2 below says that if Li
satisfies (22), then the converse is also true. Moreover, Proposition 2 constitutes a
justification for (21). Since Li in (22) can be rewritten as a convex capacity
(Schmeidler, 1989), the maximum likelihood updating rule is also equivalent to the
Dempster-Shafer updating rule for belief functions. See Gilboa and Schmeidler (1993,
p. 42, Theorem 3.3).
Proposition 2. Given the setLi of probability measures defined in(22), the maximum
likelihood and full Bayesian updating rules are equivalent. Moreover, they both imply
that Di(v) satisfies (21) with
ei
]]]]]]
ei(v)5e 1(12e)m(H (v)) ;v[V. (25)
i i i
It is important to note that ei(v) in (25) depends on H (iv). Therefore, even if we
assume that all the players have the same degree of contamination ex ante, they typically do not share the same degree of contamination after receiving their private information. It is also worth pointing out that Definition 1 admits the following special case, where no ex ante stage has to be specified and the existence of a common prior can simply be
viewed as a coincidence. Suppose that when a particular statev is realized, player i is
informed that the event H (iv) has occurred. However, i does not have any extra
information about the relative likelihood of events in H (iv). If i’s beliefs are
probabilistic, it is natural for i’s beliefs to be represented by the uniform probability
measure on H (iv). On the other hand, if i feels completely ignorant, i’s beliefs are
represented by the set of all probability measures on H (iv). If we setm(?uH (iv)) in (21)
to be the uniform probability measure on H (iv), then the set Di(v) of probability
measures in (21) represents a convex combination or ‘compromise’ between these two
cases. If this holds for every v[V and every i, then Definition 1 is satisfied. The
common prior m is the uniform probability measure on V.
Definition 1 has the following variation.
n
Definition 2. The n-tuple hDi ij51 satisfies the assumption of common prior with
n absolutely continuous contamination if there existm[M(V) and a collectionhei ij51 of
functions, where ei: V→[0,1] is measurable with respect to H , such thati
Di(v)5h(12ei(v))m(?uH (iv))1ei(v)mi:mi[M(H (iv)>suppm)j
;v[V ;i.
Definition 2 gets its name because it only allows contamination that is absolutely
continuous with respect to the benchmark probability measure m(?uH (iv)). The
interpretation of this definition is that player i has no ambiguity about events that are
null with respect to m(?uH (iv)). By redefining Li in (22) to be h(12ei)m 1 eip:
4.2. The main result
For any function x defined on V and for any value x, [x] denotes the eventhv[V:
x(v)5xj. For example, [F]; hv[V:F(v)5Fjand [u]; hv[V: u(v)5uj. Adopt
5
the standard definition of knowledge. That is, given any event E, i knows E at v if
H (iv)#E. Note that if i knows E atv, then E is true atv in the sense thatv[E. Say
that E is mutual knowledge at v if everyone knows E atv. Let* be the meet of the
partitions of all the players and*(v) be the element of* which containsv. Say that E
is common knowledge at v if *(v)#E (Aumann, 1976).
We are now ready to state that the assumption of common prior with (absolutely continuous) contamination, together with common knowledge of beliefs about oppo-nents’ action choices, lead to a generalization of a result of Aumann and Brandenburger (1995).
n
Proposition 3. Suppose hDi ij51 satisfies the assumption of common prior with
(absolutely continuous) contamination. Suppose that at a statev*, eitherei(v*)[[0,1)
for all i or ei(v*)51 for all i, and [F] is common knowledge. Then there exists a
n
* *
collection hsiji51 of probability measures, where si [M(A ), such thati
* *
F 5i
H
(12ei(v*))P
s 1j ei(v*)fi:fi[M(3j±isupp sj )J
;i. (26)j±i
The key to the proof of Proposition 3 is as follows. The assumption of common prior
with contamination implies thatFi(v) derived in (20) can be rewritten as
i
Fi(v)5h(12ei(v))f(?uH (iv))1ei(v)fi:
fi[M(ha (2iv9):v9[H (iv)j)j ;v[V, (27)
i
wheref (?uH (iv))[M(3j±iA ) is induced from the posteriorj m(?uH (iv)) in the same
manner as Fi(v) is induced from Di(v). Let D,E[H be two different partitionali
elements for player i. Consider the case whereei(v)[(0,1) for allv[D<E. We have
the following sure-thing principle for contaminated beliefs: if
Fi(vD)5Fi(vE), wherevD[D andvE[E,
then
i i i
f(?uD )5f(?uE )5f (?uD<E ) (28)
and
ha (2iv):v[Dj5ha (2iv):v[Ej5ha (2iv):v[D<Ej. (29)
Therefore, if we require player i’s beliefs about opponents’ action choices conditional on
D<E to satisfy (27) as well, then (28) and (29) imply that i’s beliefs conditional on D<E will still have the same benchmark probability measure and contamination. The
5
proof of Proposition 3 is essentially established by applying the sure-thing principle to
each player’s partitional elements in*(v*).
We also use Examples 2 and 3 below to illustrate Proposition 3. They both make use of the following
3
• strategic game form 3i51A , where Ai 15hN, Sj, A25hU, Dj and A35hL, C, Rj
• set of states of the world V 5hv1,v2, v3, v4,v5,v6j
• information partitions
H15hhv1,v2,v3,v4j,hv5j,hv6jj,
H25hhv1,v3j,hv2,v4j,hv5j,hv6jjand H35hhv1,v4j,hv2,v3j,hv5,v6jj
• action functions
v1 v2 v3 v4 v5 v6
a1 N N N N N S
a2 U D U D U D
a3 L R R L C C
The common prior m will be different in the two examples and therefore will be
specified separately.
3
Example 2. SupposehDi ij51 satisfies the assumption of common prior with
contamina-tion, where the common priorm is the uniform probability measure onV. Suppose that
ei(v)5ei[[0,1] for all v[hv1, v2, v3,v4j. According to (27),
* *
Fi(v)5h(12ei)
P
s 1 e fj i i:fi[M(3j±isuppsj )j ;v[hv1,v2,v3,v4j,j±i
(30)
* * *
wheres 51 (N,1), s 52 (U, 0.5; D, 0.5) and s 53 (L, 0.5; R, 0.5). At every v[hv1,
v2,v3,v4j, [F(v)] is common knowledge. Note that Fi(v) in (30) takes the form of
(26), confirming Proposition 1. On the other hand, consider statev5. According to (27),
*
F3(v5)5h(12e3(v5))f 13 e3(v5)f3:f3[M(hNU, SDj)j,
*
wheref 53 (NU, 0.5; SD, 0.5). SincehNU, SDjis not a product space,F3(v5) does not
satisfy (26). This is attributed to the fact that [F1(v5)] and [F2(v5)] are not common
knowledge at v5, even though [F3(v5)] is. j
Example 3. In Example 2,Fi(v) in (30) takes the form of (26), even if the restriction
‘either ei(v*)[[0,1) for all i orei(v*)51 for all i’ is not imposed. We demonstrate
3
1 2 3
] ] ]
the assumption of common prior with contamination, wherem 5(v1, 10;v2, 10;v3, 10;
2 1 1
] ] ]
v4, 10; v5, 10; v6, 10). Suppose e1(v)5e1[[0,1) and e2(v)5e3(v)51 for all
v[hv1,v2,v3,v4j. Although [F(v)] is still common knowledge at everyv[hv1,v2,
1 2 3
] ] ]
v3,v4j, the benchmark probability measure in F1(v) becomes (UL, 8; DR, 8; UR, 8;
2
]
DL, 8), which is not a product measure. j
It is natural to ask whether Proposition 3 can be further extended to more general subclasses of ambiguous beliefs. Example 4 below suggests that the answer is negative.
Example 4. This example makes use of the following
3
• strategic game form 3i51A , where Ai 15hNj, A25hUj and A35hL, Rj
• set of states of the world V 5hv1,v2, v3, v4j
• information partitions
H15hhv1,v2j,hv3,v4jj,
H25hhv1,v2,v3,v4jj and H35hhv1,v4j,hv2,v3jj
• action functions
v1 v2 v3 v4
a1 N N N N
a2 U U U U
a3 R L L R
Assume that ex ante, the beliefs of all the players are represented by the set of probability measures
1
]
H
J
L 5 p[M(V): p(hv1,v4j)5p(hv2j)5p(hv3j)53 .
Note that Lcan be rewritten as a belief function and therefore also a convex capacity.
Assume that player i’s beliefsDi(v) at every statev[V are updated fromLusing the
6
maximum likelihood updating rule. Again, assume that i’s beliefs Fi(v) about
opponents’ action choices are derived from Di(v) as in (20). In this example, for any
two statesv andv9,F(v)5 F(v9). Therefore, [F(v)] is common knowledge at every
v. However,
6
1
]
H
J
margA3F1(v)5 s3[M(A ):3 s3(L )5s3(R)52 and
2 1
] ]
H
J
margA3F2(v)5 s3[M(A ):3 s3(L )53,s3(R)53 . (31)
According to (31), player 1 believes that it is equally likely for player 3 to play either L or R, but 2 believes that it is strictly more likely that 3 will play L. In fact, (31) also implies that
margA3F1(v)>margA3F2(v)55.
That is, even Klibanoff’s notion of agreement as defined in (14) is violated.
In this example, the ultimate reason for the failure of agreement is that we do not have
any property that looks like the sure-thing principle. Let D5hv1,v2jand E5hv3,v4j.
The setF1(v) represents beliefs conditional on D and on E. The setF2(v) represents
beliefs conditional on D<E. The fact that beliefs conditional on D is the same as that
conditional on E does not have any clear implication for beliefs conditional on
D<E. j
We end this section by pointing out that Proposition 3 is preserved if the notion of 7
knowledge is replaced by the following notion of belief. Given any event E#V, say
that i believes E atv if suppDi(v)#E. That is, i believes E atvifV\E is a null event
for i atv. An event E is mutual belief at v if everyone believes E at v. The notion of
common belief is defined iteratively in the obvious manner. Under the assumption of
common prior with contamination, Proposition 3 is equally valid if the condition ‘[F] is
common knowledge’ is replaced by ‘[F] is common belief’. The reason is as follows.
Unless Di(v) is a singleton (which is the case already taken care of by Aumann and
Brandenburger (1995)), the assumption of common prior with contamination implies
that i believes E atv if and only if i knows E at v. Under the assumption of common
prior with absolutely continuous contamination, Proposition 3 will also be valid ifv*[
suppm and if the condition ‘[F] is common knowledge’ is replaced by ‘[F] is common
ˆ
belief’. The reason is as follows. For everyv[supp m, define H (iv);H (iv)>suppm.
n
ˆ ˆ ˆ
Clearly, H forms a partition of supp m. Define * to be the meet of hHj . Then for
i i i51
ˆ
everyv[suppm, i believes E atv if and only if H (iv)#E, and E is common belief at
ˆ ˆ ˆ
v if and only if*(v)#E. Therefore, by substituting H for H and* for*, the proof
i i
of Proposition 3 under the assumption of common prior with contamination also works for the case where contamination is absolutely continuous.
7
5. Implications of the main result
5.1. Agreement and stochastic independence
We can interpret Proposition 3 as delivering the two properties ‘agreement’ and ‘stochastic independence’ of beliefs as follows.
Agreement: Eq. (26) in Proposition 3 implies that the marginal beliefs of players i and j on the action choice of player k are represented by
* *
margAkF 5i h(12ei(v*))s 1k ei(v*)sk:sk[M(suppsk)j and
* *
margAkF 5j h(12ej(v*))s 1k ej(v*)sk:sk[M(suppsk)j, (32)
*
respectively. They share the same benchmark probability measuresk and the same set
*
M(supp sk) of contaminated measures. Note that (32) implies that for all E#A ,k
*
1 if suppsk #E
min sk(E )5
H
(12e(v*))s*(E ) otherwise. (33)sk[margAkFi i k
The condition ‘eitherei(v*)[[0,1) for all i orei(v*)51 for all i’ in Proposition 3 and
(33) imply (9), which is equivalent to the preference based condition for agreement in (8). Also note that (32) implies
margAkFi#margAkFjif and only ifei(v*)#ej(v*).
Therefore, margAkFi need not be equal to margAkFj, but they must have a nonempty
intersection.
Stochastic independence: According to (26) in Proposition 3, the benchmark *
probability measure
P
j±i sj in Fi is a product measure and the domain of*
contamination is the product space 3j±i supp sj. It is obvious that not all the
*
probability measures in M(3j±i supp sj ) are product measures. However, all those
‘correlated contaminations’ are irrelevant in the following sense. According to the multiple priors model, a decision maker only cares about the minimum expected utility
of acts. Given any action a of player i, there is always a degenerate probability measurei
*
di in M(3j±isupp sj ) such that the minimum utility of a is attained ati di. Of course,
all degenerate probability measures are product measures. Also, (26) implies that for all
Ej#A and Ej 2ij#3k±i, jA ,k
minfi(Ej3E2ij)
fi[Fi
*
1 if3j±isuppsj #Ej3E2ij
5
5
(12e(v*))s(E )P
s(E ) otherwise. (34)i j j k 2ij k±i, j
If ei(v*)[[0,1), then (34) implies (11), which is equivalent to the preference based
5.2. Support of contaminated beliefs
In this section, we explain precisely the sense in which Proposition 3 implies that the controversy on how the support of ambiguous beliefs is defined does not arise.
Conventionally, the support of player i’s beliefs is defined to be the event E2i#A2i
such that A2i\E2i is the largest null event for i. When i’s preference ordering is
represented by the multiple priors model and therefore beliefs are represented by a setFi
of probability measures on A2i, the event which satisfies this definition is suppFi. Dow
and Werlang (1994) argue that this notion of support is too strong. They propose the
ˆ
following definition: an event E2i is a support ofFi if (i) there existsfi[Fisuch that
ˆ
E2i5supp fi and (ii) for allfi[Fiand for all F2i,E ,2i fi(F )2i ,1. Clearly, if E2i
is a support ofFiin the sense of Dow and Werlang, then E2i#suppFi. Variants of Dow
and Werlang’s definition of support can be found in Lo (forthcoming), Marinacci (1995) and Ryan (1998).
To see why the definition of support matters, consider the following two person strategic game.
Player 2
L R
Player 1 U 10,10 210,9
D 9,10 8,9
Suppose we require the beliefs of the players to satisfy the best response property. That is, if an action of player i is in the support of player j’s beliefs, then the action must be optimal for player i given i’s beliefs. Since L is the unique best response for player 2 whatever her beliefs, the best response property implies that the support of player 1’s beliefs must be L. If we adopt the conventional definition of support, then R will be a null event for player 1 and therefore his unique best response will be U. On the other hand, if we adopt Dow and Werlang’s definition of support, R need not be a null event. As a result, player 1 may find it optimal to take the action D.
The ultimate reason for the above controversy is that different probability measures in
Fi may have different supports. For any
*
F 5i h(12ei)f 1 e fi i i:fi[M(E )2i j,
whereei[(0, 1), all the probability measures inFihave the same support if and only if
8
*
supp f 5i E . Therefore, Proposition 3 has the following corollary.2i
n
Corollary of Proposition 3. Suppose hDi ij51 satisfies the assumption of common prior
with(absolutely continuous) contamination. Suppose that at a statev*, ei(v*)[[0, 1)
8
*
The event E2iis the support ofFiaccording to the conventional definition, and suppfi is the support ofFi
for all i, and [F] is common knowledge. Then all the probability measures in Fi have the same support.
It is immediate that if the contamination in Di(v*) is absolutely continuous, then the
contamination in Fi(v*) must also be absolutely continuous. Therefore, common
n
knowledge of beliefs is needed only ifhDi ij51 satisfies the assumption of common prior
with contamination that is not absolutely continuous. We outline the proof of the corollary as follows. For simplicity, consider a game with two players and player 1’s
*
beliefs F 51 h(12e1)f 1 e f1 1 1: f1[M(E )2 j. Suppose F1 is common knowledge at
v*. This implies that at every statev[*(v*), player 1’s beliefs are in fact represented
* *
by F. By definition, supp f #E . Suppose supp f ±E . That is, there exists
1 1 2 1 2
*
a2[E such that a2 2[⁄ supp f1. The fact that a2[E implies2
hv[*(v*): a (2v)5a2j±5. (35)
* *
Since f1 comes from the common prior m, the fact that a2[⁄ supp f1 implies
m(hv[*(v*): a (v)5a j)50. (36)
2 2
Eq. (35) implies that there is a partitional element H (2v)[H such that H (2 2v)#hv[
*(v*): a (v)5a j. However, (36) implies that m(H (v))50, contradicting the
2 2 2
hypothesis that m(H (iv)).0 for allv and for all i.
Example 5 below demonstrates that if [Fi] is not common knowledge at v*, then
different probability measures in Fi may have different supports.
Example 5. Consider the following
2
• strategic game form 3i51A , where Ai 15hU, Dj and A25hL, C, Rj
• set of states of the world V 5hv1,v2, v3, v4,v5,v6j
1 1 1
] ] ]
• common prior m 5(v1, 3; v2, 0; v3, 0; v4, 3; v5, 3; v6, 0).
• information partitions
H15hhv1,v2,v3j,hv4,v5,v6jjand H25hhv1,v2j,hv3,v4j,hv5,v6jj
• action functions
v1 v2 v3 v4 v5 v6
a1 U U U D D D
a2 L L C C R R
2
SupposehDi ij51satisfies the assumption of common prior with contamination, where the
at v3. Unlesse2(v3)50, different probability measures in F2(v3) may have different supports.
5.3. Generalization of Nash equilibrium
Consider the following generalization of Nash equilibrium.
Definition 3. Given a game u, F is a Nash equilibrium with contamination of u if the following conditions are satisfied.
n n
* *
1. There existhsi ji51 andhei ij51, where si [M(A ) and eitheri ei[[0,1) for all i or
e 5i 1 for all i, such that
* *
F 5i
H
(12ei)P
s 1 e fj i i:fi[M(3j±isuppsj )J
;i. j±i2. supp marg F#argmax min o u (a ,a )f(a ) ;i ;j±i.
Ai j ai[Ai fi[Fi a2i[3j±iAj i i 2i i 2i
Definition 3 consists of two conditions. Condition 1 is the property of agreement and stochastic independence of contaminated beliefs that we derive in Proposition 3. The interpretation of condition 2 parallels that of the best response property of Nash equilibrium. That is, player i chooses some definite action, but player j may not know
which one. However, if j knows i’s beliefsFiand that i is rational, then every action ai
in the support of j’s marginal beliefs margAiFj must be a best response for i given i’s
beliefsFi. Therefore, Nash equilibrium with contamination is viewed as an equilibrium
in beliefs.
The event hv[V: every i is rational at vj is denoted as [rationality]. Given
Proposition 3, straightforward adaptation of Theorem A in Aumann and Brandenburger (1995) provides epistemic conditions for Nash equilibrium with contamination, confirm-ing the above interpretation.
n
Proposition 4. Suppose hDi ij51 satisfies the assumption of common prior with
(absolutely continuous) contamination. Suppose that at a statev*, eitherei(v*)[[0, 1)
for all i orei(v*)51 for all i, [rationality] and [u] are mutual knowledge and [F] is
common knowledge. ThenF is a Nash equilibrium with contamination of u.
In our earlier paper (Lo, 1996, p. 453, Definition 4), we propose an equilibrium concept which is called Beliefs Equilibrium with Agreement. The equilibrium concept is a generalization of Nash equilibrium by allowing preferences to be represented by the multiple priors model. Under the assumption that players are completely ignorant, we provide epistemic conditions for Beliefs Equilibrium with Agreement (Lo, 1996, p. 471, Proposition 7). This earlier result is essentially a special case of Proposition 4 in the
following sense. Proposition 4 only requires that either ei(v*)[[0,1) for all i or
ei(v*)51 for all i. On the other hand, Proposition 7 requiresei(v)51 for all i and for
Acknowledgements
I especially thank a referee for providing detailed suggestions and comments that led to substantial improvements. I also thank Larry Epstein for valuable comments and the Social Sciences and Humanities Research Council of Canada for financial support.
Appendix
Proof of Proposition 1
To simplify notation, we restate and prove Proposition 1 in the context of single person decision making. That is, suppose there is a decision maker facing a product state
space S13S .2
Proposition 1. Suppose there exist E#S13S with2 uproj ES1 u.1 and uproj ES2 u.1,
p*[M(E ), and e[(0,1) such that35h(12e)p*1ep: p[M(E )j. Then 3 does not
satisfy the independent product property.
ˆ ˆ
Proof. Fix s1[proj E such that marg p*(s )S1 S1 1 .0. We have
ˆ ˆ
min p (s )1 1 5min p(s13proj E )S2 (37)
p1[margS3 p[3
1
ˆ
5(12e)p*(s13proj E )S2 (38)
ˆ
5(12e)marg p*(s ).S1 1 (39)
ˆ
The equality between (37) and (38) is due to the fact that E≠s13proj E, sinceS2
uproj ES1 u.1. Similarly,uproj ES2 u.1 implies that for all s2[proj E,S2
min p (s )2 2 5(12e)marg p*(s ).S2 2 (40)
p[marg 3
2 S2
Suppose 3 had the independent product property. Then for all s [proj E,
2 S2
ˆ ˆ
min p(s13s )2 5 min p (s )1 1 min p (s )2 2 (41)
p[3 p [marg 3 p[marg 3
1 S1 2 S2
ˆ
5(12e)marg p*(s )(1S1 1 2e)marg p*(s )S2 2 (42)
2
ˆ
5(12e) marg p*(s )marg p*(s ).S1 1 S2 2 (43)
The independent product property implies (41). The equality between (41) and (42) is
ˆ
due to (39) and (40). According to (43), if we sum up minp[3p(s13s ) over all the2
2
ˆ ˆ
O
min p(s13s )2 5O
(12e) marg p*(s )marg p*(s )S1 1 S2 2p[3
s2[projSE s2[projSE
2 2
2
ˆ
5(12e) marg p*(s )S1 1
O
marg p*(s )S2 2s2[projSE
2
2
ˆ
5(12e) marg p*(s ).S1 1 (44)
However, regardless of the independent product property, it is always legitimate to write
ˆ ˆ
O
min p(s13s )2 5O
(12e)p*(s13s )2p[3
s2[projSE s2[projSE
2 2
ˆ
5(12e)
O
p*(s13s )2s2[projSE
2
ˆ
5(12e)p*(s13proj E )S2
ˆ
5(12e)marg p*(s ).S1 1 (45)
ˆ
Since e[(0,1) and marg p*(s )S1 1 .0, (44) and (45) constitute a contradiction.
Proof of Proposition 2
It is obvious that Proposition 2 is true if eithere 5i 0 ore 5i 1. We assume below that
ei[(0,1).
Given the set Li of probability measures defined in (22), the maximum likelihood
updating rule defined in (23) can be rewritten as
Di(v)5hmi[M(H (iv)):'p[Lisuch that
p(H (iv))5e 1i (12ei)m(H (iv)) andmi(?)5p(?uH (iv))j. (46)
Eq. (46) can in turn be rewritten as
(12ei)m(H (iv))m(?uH (iv))1e mi i ]]]]]]]]]]
Di(v)5
H
e 1(12e)m(H (v)) :mi[M(H (iv))J
i i i
5h(12ei(v))m(?uH (iv))1ei(v)mi:mi[M(H (iv))j, (47)
where
ei
]]]]]] ei(v)5e 1(12e)m(H (v)).
i i i
Similarly, (22) implies that the full Bayesian updating rule defined in (24) can be rewritten as
Di(v)5hmi[M(H (iv)):'p[Liand'a[[0,1] such that
p(H (iv))5e a 1i (12ei)m(H (iv)) andmi(?)5p(?uH (iv))j. (48)
(12ei)m(H (iv))m(?uH (iv))1e ami i ]]]]]]]]]]
Di(v) 5 <
H
e a 1(12e)m(H (v)) :mi[M(H (iv))J
a[[0,1] i i i (49)
a a
5 < h(12ei(v))m(?uH (iv))1ei(v)mi:mi[M(H (iv))j,
a[[0,1]
where
e ai
a
]]]]]]] ei(v)5e a 1(12e)m(H (v)).
i i i
Note that
,ei(v) if 0#a ,1 a
ei(v)
H
5e(v) ifa 51. (50)i
Recall the property that given any set 35h(12e)p*1ep: p[M(E )j of probability
measures, 3 increases in the sense of set inclusion as e increases. Therefore, (50)
implies that (47) and (49) are equivalent.
Proof of Proposition 3
We provide below the proof of Proposition 3 under the assumption of common prior with contamination.
The assumption of common prior with contamination implies that Fi(v) defined in
(20) can be rewritten as
i
Fi(v)5h(12ei(v))f(?uH (iv))1ei(v)fi:
fi[M(ha (2iv9):v9[H (iv)j)j ;v[V, (51)
i
where f(?uH (iv))[M(3j±iA ) is derived fromj m(?uH (iv)) in the same manner as
Fi(v) is derived from Di(v). By hypothesis, [Fi] is common knowledge at v*. This
implies thatFi(v*)5Fi and therefore [Fi(v*)] is common knowledge at v*. That is,
F(v)5F(v*) ;v[*(v*). (52)
i i
This in turn implies that
suppF(v)5suppF(v*) ;v[*(v*). (53)
i i
There are two cases to consider.
Case 1. ei(v*)[[0,1) for all i. In this case, it suffices to establish that there exists a
n i
* *
collection of probability measures hsiji51, where si [A , such thati f (?uH (iv*))5
* *
P
j±isj andha (2iv): v[H (iv*)j5 3j±isupp sj.For every Fi(v*) such that uFi(v*)u51, (52) implies that uFi(v)u51 for all
i
v[*(v*). Then (51) implies thatF(v)5f(?uH (v)) for allv[*(v*). Therefore,
i i
i i
f(?uH (v))5f (?uH (v*)) ;v[*(v*). (54)
i i
For every Fi(v*) such that uFi(v*)u.1, (52) implies that uFi(v)u.1 for all v[
*(v*). Then (51) implies that suppF(v)5ha (v9): v9[H (v)jfor allv[*(v*).
i 2i i
Therefore, (53) can be rewritten as
ha (v9):v9[H (v)j5ha (v9):v9[H (v*)j ;v[*(v*). (55)
2i i 2i i
Then (51), (52) and (55) imply that
i i
e(v)5e(v*) andf (?uH (v))5f(?uH (v*)) ;v[*(v*). (56)
i i i i
i
According to (54) and (56), whatever the cardinality of Fi(v*), [f (?uH (iv*)] is
i
common knowledge at v*. Since f(?uH (iv)) comes from a common prior for all
v[V, Theorem B in Aumann and Brandenburger (1995) can be invoked to conclude
n
* *
that there exists a collection of probability measureshsi ji51, wheresi [A , such thati
i
* f(?uH (iv*))5
P
j±isj.i
Since f(?uH (iv*))[M(ha (2iv): v[H (iv*)j) and we have just established that
i
* *
f(?uH (iv*))5
P
j±isj is a product measure, we must have 3j±isuppsj #ha (2iv):*
v[H (iv*)j. Let us establish by contradiction that we actually have 3j±isupp s 5j
ha (2iv): v[H (iv*)j. Suppose that this were not true. That is, suppose there exists
i *
aj[projAjha (2iv): v[H (iv*)j such that aj[⁄ supp sj. Recall that [f (?uH (iv*)] is
i *
common knowledge at v* and that s 5j margAjf(?uH (iv*)) is derived from the
*
common prior m. Therefore, if aj[⁄ supp sj, then
m(hv[*(v*): a (v)5aj)50. (57)
j j
The fact that aj[projAjha (2iv):v[H (iv*)jimplies thathv[*(v*): a (jv)5ajj±5.
By the definition of *(v*), there exists v9[*(v*) such that H (v9)#hv[*(v*):
j
a (jv)5ajj. However, (57) implies that m(H (j
