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
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 and K a preference ordering over . Given any event E S, say that E is null if for all f, f 9, g [ ,
f v
if v [ E
f 9 v
if v [ E
| .
3
F G F
G
g v
if v [
⁄ E g
v if
v [ ⁄ E
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 pE 5 0 ; 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 s x,
x if
s [ D x
if s [ E
F G
K
F G
. 5
x if
s [ ⁄ D
x if
s [ ⁄ E
1
Alternative definitions of null events have been proposed for the Choquet expected utility and multiple priors models. See Ryan 1998 for a summary. If an event is null in the sense of 3, then it must also be null
according to all the alternative definitions. The converse is not true. Therefore, the preference based condition for stochastic independence defined in 10 below will be weaker if any of those definitions is adopted.
K .C. Lo Mathematical Social Sciences 39 2000 207 –234
213
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 pD min pE .
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 s D
min s E
k k
s [marg F
s [marg F
k A
i k
A i
k k
if and only if min
s D min
s E. 9
k k
s [marg F
s [marg F
k A
j k
A j
k k
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 ,E A and any two
j j
j 3
events D ,E A that are non-null from i’s point of view,
k k
k
i believes that D 3 D is more likely than E 3 D
j k
j k
if and only if i believes that D 3 E is more likely than E 3 E . 10
j k
j k
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 ,E A and D ,E A such that
j j
j k
k k
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 s x,
x if
s [ E x
if s [ D
K .
6
F G F
G
x if
s [ ⁄ E
x if
s [ ⁄ D
That is, the decision maker prefers to bet against E rather than against D. If K is represented by the multiple priors model, then D is more likely than E according to 6 if and only if max
pD max pE . In this
p [3 p [3
paper, we adopt 5. However, all the results are valid if we adopt 6.
3
The notation D 3 D in 10 refers to the event that j chooses an action from D and k chooses an action from
j k
j
D . Similarly for E 3 D , D 3 E and E 3 E .
k j
k j
k j
k
214 K
.C. Lo Mathematical Social Sciences 39 2000 207 –234
min s D . 0 and
min s E . 0,
k k
k k
s [marg F
s [marg F
k A
i k
A i
k k
min f D 3 D min f E 3 D
i j
k i
j k
11
f [F f [F
i i
i i
if and only if min f D 3 E min f E 3 E .
i j
k i
j k
f [F f [F
i i
i i
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:
marg F 5 marg F .
12
A i
A j
k k
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 A 5
hL, Rj,
3
marg F 5
hs [ MA : 0.4 s L 0.6j and
A 1
3 3
3
3
marg F 5
hs [ MA : 0.3 s L 0.7j. 13
A 2
3 3
3
3
Clearly, marg F ± marg F and therefore 12 is violated. However, both players
A 1
A 2
3 3
believe that L and R are equally likely. In fact, the sets of probability measures in 13 can be rewritten as
marg F 5
h1 2 e s 1 e s : s [ MA j and
A 1
1 3
1 3
3 3
3
marg F 5
h1 2 e s 1 e s : s [ MA j,
A 2
2 3
2 3
3 3
3
where s 5 L, 0.5;R, 0.5, e 5 1 5 and e 5 2 5. That is, marg F and marg F
3 1
2 A
1 A
2
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:
marg F marg F ± 5.
14
A i
A j
k k
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 A 5
hL, Rj,
3
marg F 5
hs [ MA : 0 s L 0.6j and
A 1
3 3
3
3
marg F 5
hs [ MA : 0.6 s L 1j.
A 2
3 3
3
3
The probability measure L, 0.6; R, 0.4 is contained in both marg F and marg F .
A 1
A 2
3 3
K .C. Lo Mathematical Social Sciences 39 2000 207 –234
215
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 that
F satisfies the following independent product property:
i
for all events E A and E A ,
j j
k k
min f E 3 E 5
min s E
min s E .
15
i j
k j
j k
k f [F
s [marg F
s [marg F
i i
j A
i k
A i
j k
It is well known that if F is a singleton, then F is completely determined by the right
i i
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 E A 3 A with
uproj E u . 1 and uproj E u .
2i j
k A
2i A
2i
j k
1, f [ ME , and e [ 0,1 such that F 5
h1 2 e f 1 e f : f [ ME j. Then
i 2i
i i
i i
i i
i 2i
F does not satisfy the independent product property.
i
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 set
V 5 hv , v , v , v j of states of the world,
1 2
3 4
where every state
v [ V contains the following specification of actions a v [ A and
2 2
a
v [ A taken by players 2 and 3, respectively.
3 3
v v
v v
1 2
3 4
a U
D U
D
2
a L
R R
L
3
Suppose player 1’s beliefs about V are represented by the set of probability measures
L 5 h1 2 e m 1 e m : m [ MV j,
16
1 1
1 1
1 1
where m 5 v , 0.25; v , 0.25; v , 0.25; v , 0.25 and e [ 0,1. Given the action
1 1
2 3
4 1
functions a and a , every probability measure in L induces a probability measure on
2 3
1
4
It follows that although Eichberger and Kelsey 1994 and Mukerji 1995 require all the marginal beliefs marg
F to be contaminated, the overall beliefs F are not contaminated.
A i
i j
216 K
.C. Lo Mathematical Social Sciences 39 2000 207 –234
hU,Dj 3 hL,Rj in the obvious manner. If we do it for every probability measure in L , we
1
obtain the set of probability measures F 5 1 2 e
P
s 1 e f : f [ M hU, Dj 3 hL, Rj ,
17
1 1
j 1
1 1
H J
j 52,3
where s 5 U, 0.5; D, 0.5 and s 5 L, 0.5; R, 0.5.
2 3
Given F in 17, 1’s marginal beliefs about 2 and 3’s action choices are represented
1
by marg
F 5 h1 2 e s 1 e s : s [ MhU, Djj and
A 1
1 2
1 2
2
2
marg F 5
h1 2 e s 1 e s : s [ MhL, Rjj, 18
A 2
1 3
1 3
3
3
respectively. Note that F , marg F and marg F are all contaminated with the same
1 A
1 A
1
2 3
degree of contamination e . The reason for this is as follows. The number e represents
1 1
player 1’s degree of uncertainty about V. Note that the action choices of players 2 and 3
are both functions on V. 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 by e . In
1
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’ e . For example, take the actions U and L. Let C be any set of probability
1 1
measures on A 3 A such that marg C 5 marg F , marg C 5 marg F , and the
2 3
A 1
A 1
A 1
A 1
2 2
3 3
independent product property is satisfied. Then we have min
c U 3 L 5
min s U
min s L
1 2
3 s [marg
F s [marg
F c [C
2 A
1 3
A 1
1 1
2 3
2
5 1 2 e 0.25,
1
which is strictly less than min
f U 3 L 5 1 2 e 0.25.
1 1
f [F
1 1
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
F in 17 does not satisfy the independent product property, it has the
1
following features of ‘stochastic independence’: the benchmark probability measure
P
s is a product measure and the contamination is defined on the product space
j 52,3 j
hU,Dj 3 hL,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 F to contain at least one product measure.
19
i
This definition is compatible with contaminated beliefs. For instance, suppose A 5 hU,
2
D j, A 5 hL, Rj and
3
F 5 h0.5f 1 0.5f : f [ MA 3 A j,
1 1
1 1
2 3
K .C. Lo Mathematical Social Sciences 39 2000 207 –234
217
where f 5 UL, 0.5; UR, 0; DL, 0; DR, 0.5. Then F satisfies 19 because it contains
1 1
the product measure UL, 0.25; UR, 0.25; DL, 0.25; DR, 0.25. However, we have min
f UL 5 min f DR 5 0.25 and
min f UR 5 min f DL 5 0.
1 1
1 1
f [F f [F
f [F f [F
1 1
1 1
1 1
1 1
That is, player 1 believes that UL is more likely than DL, but UR is less likely than DR. This demonstrates that, even if
F is contaminated, 19 does not imply the preference
i
based condition for stochastic independence in 10.
4. Epistemic conditions
