210 K
.C. Lo Mathematical Social Sciences 39 2000 207 –234
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 by
uYu. 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 MY . For
any c [ MY, the support of c is denoted by supp c. For any C MY, supp
C ;
supp c. That is, supp C Y is the union of the supports of the
c [C
probability measures in C.
L
• Suppose Y 5 3
Y . For any c [ MY, the marginal probability measure of c on Y
l 51 l
l
is denoted by marg c. For any C MY, marg C
; hc [ MY : c [C such that
Y Y
l l
l l
c 5 marg c j. That is, marg C is the set of marginal probability measures on Y as
l Y
Y l
l l
one varies over the set of probability measures in C.
L
• Suppose Y 5 3
Y . For any Z Y, proj Z ; hy [ Y : z [ Z such that the lth
l 51 l
Y l
l
l
co-ordinate of z is y j. That is, proj Z is the projection of Z on Y .
l Y
l
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 ordering K over . According to the multiple priors model, the representation for K consists of a von Neuman Morgenstern
vNM index u: X →
R and a closed and convex set 3 of probability measures on S. The utility function U :
→ R representing K is given by
K .C. Lo Mathematical Social Sciences 39 2000 207 –234
211
U f ;min
O
u fsps. 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 set 3 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 of 3 : there exist an event E S, a probability measure p [ ME and a real number
e [ [0,1] such that 3 5
h1 2 ep 1 ep: p [ MEj. 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 weight
e by the probability measures in the set ME. 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 as
e increases, modeling increased ambiguity aversion. In particular, if e 5 0, then 3 5
h pj; if e 5 1, then 3 5 ME. 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 ; 3
A , where A is a finite set of actions
i 51 i
i
for player i. Throughout, the indices i, j and k vary over distinct players in h1, . . . ,nj.
Elements in A and 3 A are denoted by a and a , respectively. Given a strategic
i j ±i
j i
2i n
game form A, a strategic game is denoted as u ; hu j
, where u : A →
R is a vNM
i i 51 i
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 3 A . Consistent with the above single person decision theory, i’s
j ±i j
preference ordering over the set of acts defined on 3 A is represented by the multiple
j ±i j
priors model defined in 1. Every action a [ A can be identified as an act over 3 A
i i
j ±i j
and i’s utility of taking a is equal to
i
min
O
u a , a f a ,
i i
2i i
2i f [F
i i
a [
3 A
2i j ±i
j
where F is a closed and convex set of probability measures on 3
A . Consistent with
i j ±i
j
2, if F is contaminated, then there exist E
3 A ,
f [ ME and e [ [0,1]
i 2i
j ±i j
i 2i
i
such that F 5
h1 2 e f 1 e f : f [ ME j.
i i
i i
i i
2i
It should be emphasized that in the absence of further assumptions, the event E may
2i
212 K
.C. Lo Mathematical Social Sciences 39 2000 207 –234
not be a product space and f
may not be a product measure. In particular, supp f
i i
may be any subset of E .
2i
3. Agreement and stochastic independence
