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
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 , D , u , a j. Each of its components is explained as follows.
i i
i i
• 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 of V. For every state
i
v [ V, H v [ H denotes the partitional element which contains v.
i i
• Player i’s beliefs at
v are represented by a closed and convex set D v of probability
i
measures on H v.
i
• Player i’s payoff function at
v is u v: A →
R.
i
• The action taken by player i at
v is a v [ A .
i i
Note that D , u and a are functions on V. To respect the partitional information
i i
i
structure, they are measurable with respect to H . Also note that at every v [ V,
i
u v
; hu v, . . . ,u vj constitutes a strategic game.
1 n
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
D v of probability measures at every state v [ V and for
i
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 state v [ V, i’s beliefs over 3
A are represented by the closed and
j ±i j
convex set F v of probability measures which is induced from D v as follows:
i i
F v 5
hf [ M 3 A :
m [ D v such that for all a [
3 A ,
i i
j ±i j
i i
2i j ±i
j
f a 5 m hv9 [ H v: a v9 5 a jj,
20
i 2i
i i
2i 2i
where a
v9 ; ha v9, . . . ,a
v9,a v9, . . . ,a v9
j. Denote
the n-tuple
2i 1
i 21 i 11
n
hF v, . . . , F vj by Fv.
1 n
218 K
.C. Lo Mathematical Social Sciences 39 2000 207 –234
Say that player i is rational at v if
a v [ arg max
min
O
u va ,a f a .
i i
i 2i
i 2i
a [ A f [F v
i i
i i
a [
3 A
2i j ±i
j
That is, i is rational at v if his action a v maximizes utility when the payoff function is
i
u v and when beliefs are represented by F v.
i i
Let m be a probability measure on V with the property that mH v . 0 for all
i
v [ V and for all i. Let m ? uH v be the probability measure on H v which is
i i
derived from m using Bayes rule. The common prior assumption can be stated as
n
follows: the n-tuple hD j
satisfies the common prior assumption if there exists
i i 51
m [ MV such that D v 5 m ?
uH v ;v [ V ;i.
i i
Consider the following generalization of the common prior assumption.
n
Definition 1. The n-tuple
hD j satisfies the assumption of common prior with
i i 51 n
contamination if there exist m [ MV and a collection
he j of functions, where e :
i i 51 i
V →
[0,1] is measurable with respect to H , such that
i
D v 5 h1 2 e vm ? uH v 1 e vm : m [ MH vj ;v [ V ;i.
21
i i
i i
i i
i
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 measure
m [ MV . That is, i’s ex ante beliefs are represented by
L ; h1 2 e m 1 e p: p [ MV j,
22
i i
i
where e [ [0,1]. When a state v is realized, player i is informed that the event H v has
i i
occurred. The set D v of probability measures represents i’s updated beliefs. As
i
suggested by Gilboa and Schmeidler 1993, a natural procedure to derive D v is to
i
rule out some of the priors in L and then update the rest according to Bayes rule. Two
i
updating rules of particular interest are the maximum likelihood updating rule: D v 5
hm [ MH v: p [ L
i i
i i
ˆ such that pH
v 5max pH v and m ? 5 p ? uH vj
23
i i
i i
ˆ p [
L
i
and the full Bayesian updating rule: D v 5
hm [ MH v: p [ L such that pH v . 0 and m ? 5 p ? uH vj.
i i
i i
i i
i
24 That is, according to the maximum likelihood updating rule, only those probability
measures in L that ascribe the maximal probability to H v are updated. On the other
i i
hand, according to the full Bayesian updating rule, every probability measure in L
i
which ascribes positive probability to H v is updated. Clearly, the set of probability
i
K .C. Lo Mathematical Social Sciences 39 2000 207 –234
219
measures in 23 must be a subset of that in 24. Proposition 2 below says that if L
i
satisfies 22, then the converse is also true. Moreover, Proposition 2 constitutes a justification for 21. Since
L in 22 can be rewritten as a convex capacity
i
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 set L of probability measures defined in 22, the maximum
i
likelihood and full Bayesian updating rules are equivalent . Moreover, they both imply
that D v satisfies 21 with
i
e
i
]]]]]] e
v 5 ;
v [ V. 25
i
e 1 1 2 e mH v
i i
i
It is important to note that e v in 25 depends on H v. Therefore, even if we
i i
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 state v is realized, player i is
informed that the event H v has occurred. However, i does not have any extra
i
information about the relative likelihood of events in H v. If i’s beliefs are
i
probabilistic, it is natural for i’s beliefs to be represented by the uniform probability measure on H
v. On the other hand, if i feels completely ignorant, i’s beliefs are
i
represented by the set of all probability measures on H v. If we set m ?
uH v in 21
i i
to be the uniform probability measure on H v, then the set D v of probability
i i
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
hD j satisfies the assumption of common prior with
i i 51 n
absolutely continuous contamination if there exist m [ MV and a collection
he j of
i i 51
functions, where e : V
→ [0,1] is measurable with respect to H , such that
i i
D v 5 h1 2 e vm ? uH v 1 e vm : m [ MH v supp mj
i i
i i
i i
i
; 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 v. The
i
interpretation of this definition is that player i has no ambiguity about events that are null with respect to
m ? uH v. By redefining L in 22 to be h1 2 e m 1 e p:
i i
i i
p [ Msupp m
j, the justification for Definition 1 justifies Definition 2 in exactly the same manner.
220 K
.C. Lo Mathematical Social Sciences 39 2000 207 –234
4.2. The main result For any function x defined on
V and for any value x, [x] denotes the event hv [ V:
x v 5 x
j. For example, [F ] ; hv [ V: Fv 5 F j and [u] ; hv [ V: uv 5 uj. Adopt
5
the standard definition of knowledge. That is, given any event E, i knows E at v if
H v E. Note that if i knows E at v, then E is true at v in the sense that v [ E. Say
i
that E is mutual knowledge at v if everyone knows E at v. Let be the meet of the
partitions of all the players and v be the element of which contains v. 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
hD j satisfies the assumption of common prior with
i i 51
absolutely continuous contamination. Suppose that at a state v, either e v [ [0,1
i
for all i or e v 5 1 for all i, and [F ] is common knowledge. Then there exists a
i n
collection hs j
of probability measures , where s [ MA , such that
i i 51
i i
F 5 1 2 e v
P
s 1 e vf : f [ M 3 supp
s ;i.
26
i i
j i
i i
j ±i j
H J
j ±i
The key to the proof of Proposition 3 is as follows. The assumption of common prior with contamination implies that F
v derived in 20 can be rewritten as
i i
F v 5
h1 2 e vf ? uH v 1 e vf :
i i
i i
i
f [ M ha v9: v9 [ H vjj ;v [ V,
27
i 2i
i i
where f ?
uH v [ M 3 A is induced from the posterior
m ? uH v in the same
i j ±i
j i
manner as F v is induced from D v. Let D,E [ H be two different partitional
i i
i
elements for player i. Consider the case where e v [ 0,1 for all v [ D E. We have
i
the following sure-thing principle for contaminated beliefs: if F
v 5 F v , where v [ D and v [ E,
i D
i E
D E
then
i i
i
f ? uD 5 f ? uE 5 f ? uD E
28 and
ha v: v [ Dj 5 ha v: v [ Ej 5 ha v: v [ D Ej. 29
2i 2i
2i
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
The results in this paper are valid if knowledge is replaced by belief. See the end of this section for details.
K .C. Lo Mathematical Social Sciences 39 2000 207 –234
221
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 3
A , where A 5 hN, Sj, A 5 hU, Dj and A 5 hL, C, Rj
i 51 i
1 2
3
• set of states of the world
V 5 hv , v , v , v , v , v j
1 2
3 4
5 6
• information partitions
H 5 hhv , v , v , v j, hv j, hv jj,
1 1
2 3
4 5
6
H 5 hhv , v j, hv , v j, hv j, hv jj and H 5 hhv , v j, hv , v j, hv , v jj
2 1
3 2
4 5
6 3
1 4
2 3
5 6
• action functions
v v
v v
v v
1 2
3 4
5 6
a N
N N
N N
S
1
a U
D U
D U
D
2
a L
R R
L C
C
3
The common prior m will be different in the two examples and therefore will be
specified separately.
3
Example 2. Suppose
hD j satisfies the assumption of common prior with contamina-
i i 51
tion, where the common prior m is the uniform probability measure on V. Suppose that
e v 5 e [ [0,1] for all v [
hv , v , v , v j. According to 27,
i i
1 2
3 4
F v 5
h1 2 e
P
s 1 e f : f [ M 3 supp
s j ;v [ hv , v , v , v j,
i i
j i
i i
j ±i j
1 2
3 4
j ±i
30 where
s 5 N,1, s 5 U, 0.5; D, 0.5 and s 5 L, 0.5; R, 0.5. At every v [ hv ,
1 2
3 1
v , v , v j, [Fv] is common knowledge. Note that F v in 30 takes the form of
2 3
4 i
26, confirming Proposition 1. On the other hand, consider state v . According to 27,
5
F v 5
h1 2 e v f 1 e v f : f [ MhNU, SDjj,
3 5
3 5
3 3
5 3
3
where f 5 NU, 0.5; SD, 0.5. Since
hNU, SDj is not a product space, F v does not
3 3
5
satisfy 26. This is attributed to the fact that [F v ] and [F v ] are not common
1 5
2 5
knowledge at v , even though [F v ] is.
j
5 3
5
Example 3. In Example 2, F v in 30 takes the form of 26, even if the restriction
i
‘either e v [ [0,1 for all i or e v 5 1 for all i’ is not imposed. We demonstrate
i i
3
here that in order to guarantee 26, this restriction is needed. Suppose hD j
satisfies
i i 51
222 K
.C. Lo Mathematical Social Sciences 39 2000 207 –234
1 2
3 ]
] ]
the assumption of common prior with contamination, where m 5 v ,
; v ,
; v ,
;
1 2
3 10
10 10
2 1
1 ]
] ]
v , ;
v , ;
v , . Suppose e
v 5 e [ [0,1 and e v 5 e v 5 1 for all
4 5
6 1
1 2
3 10
10 10
v [ hv , v , v , v j. Although [Fv] is still common knowledge at every v [ hv , v ,
1 2
3 4
1 2
1 2
3 ]
] ]
v , v j, the benchmark probability measure in F v becomes UL, ; DR, ; UR, ;
3 4
1 8
8 8
2 ]
DL, , which is not a product measure.
j
8
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 3
A , where A 5 hNj, A 5 hU j and A 5 hL, Rj
i 51 i
1 2
3
• set of states of the world
V 5 hv , v , v , v j
1 2
3 4
• information partitions
H 5 hhv , v j, hv , v jj,
1 1
2 3
4
H 5 hhv , v , v , v jj and H 5 hhv , v j, hv , v jj
2 1
2 3
4 3
1 4
2 3
• action functions
v v
v v
1 2
3 4
a N
N N
N
1
a U
U U
U
2
a R
L L
R
3
Assume that ex ante, the beliefs of all the players are represented by the set of probability measures
1 ]
H J
L 5 p [ MV : p hv ,v j 5 phv j 5 phv j 5
.
1 4
2 3
3 Note that
L can be rewritten as a belief function and therefore also a convex capacity. Assume that player i’s beliefs
D v at every state v [ V are updated from L using the
i 6
maximum likelihood updating rule. Again, assume that i’s beliefs F
v about
i
opponents’ action choices are derived from D v as in 20. In this example, for any
i
two states v and v9, Fv 5 Fv9. Therefore, [Fv] is common knowledge at every
v. However,
6
For other updating rules, similar examples can be constructed to convey essentially the same message.
K .C. Lo Mathematical Social Sciences 39 2000 207 –234
223
1 ]
H J
marg F
v 5 s [ MA : s L 5 s R 5 and
A 1
3 3
3 3
3
2 2
1 ]
]
H J
marg F
v 5 s [ MA : s L 5 , s R 5 .
31
A 2
3 3
3 3
3
3 3
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
marg F
v marg F v 5 5.
A 1
A 2
3 3
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 D 5 hv , v j and E 5 hv , v j.
1 2
3 4
The set F v represents beliefs conditional on D and on E. The set F v represents
1 2
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 at
v if supp D v E. That is, i believes E at v if V\E is a null event
i
for i at v. 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
D v is a singleton which is the case already taken care of by Aumann and
i
Brandenburger 1995, the assumption of common prior with contamination implies that i believes E at
v 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 if
v [ supp
m and if the condition ‘[F ] is common knowledge’ is replaced by ‘[F ] is common ˆ
belief’. The reason is as follows. For every v [ supp m, define H v
; H v supp m.
i i
n
ˆ ˆ
ˆ Clearly, H forms a partition of supp
m. Define to be the meet of hH j
. Then for
i i i 51
ˆ every
v [ supp m, i believes E at v if and only if H v E, and E is common belief at
i
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
Similar explanation suffices to establish that Proposition 4 below is also preserved if knowledge is replaced by belief.
224 K
.C. Lo Mathematical Social Sciences 39 2000 207 –234
5. Implications of the main result
