Download Common p-Belief: The General Case

GAMES AND ECONOMIC BEHAVIOR
ARTICLE NO.
18, 73]82 Ž1997.
GA970513
Common p-Belief: The General Case
Atsushi Kajii* and Stephen Morris†
Department of Economics, Uni¨ ersity of Pennsyl¨ ania
Received February 22, 1995
We develop belief operators for information systems where individuals have an
uncountable number of possible signals, and we give a general version of Monderer
and Samet’s Ž1989. theorem relating ‘‘iterative’’ and ‘‘fixed point’’ notions of
common p-belief. Journal of Economic Literature Classification Numbers: D82.
Q 1997 Academic Press
1. INTRODUCTION
An event is ‘‘common p-belief’’ if everyone believes it with probability at
least p, everyone believes with probability at least p that everyone
believes it with probability at least p, and so on ad infinitum. Monderer
and Samet Ž1989. whereafter, MSx provide a characterization of common
p-belief which relates this iterati¨ e definition to the following fixed-point
definition:1 An event is said to be p-evident if whenever it is true, everyone
believes it with probability at least p. An event E is common p-belief at
state v if and only if v is an element of some p-evident event F with the
property that everyone believes E with probability at least p whenever F
is true. Since common 1-belief is essentially equivalent to the usual notion
of common knowledge, this result is a generalization of Aumann’s classic
result 2 giving a fixed point characterization of common knowledge.
Common p-belief has been shown to be a natural notion of ‘‘almost
common knowledge’’: for p sufficiently close to one, economic outcomes
*Financial support from the Center for Operations Research and Econometrics ŽCORE.
fellowship is gratefully acknowledged. Currently at the Institute for Socio-Economic Planning, University of Tsukuba.
†
Financial support from an E.E.C. Human Capital and Mobility Program institutional
fellowship for a visit to C.O.R.E. is gratefully acknowledged.
1
Stinchcombe Ž1988. independently introduced an alternative notion of almost common
knowledge.
2
Aumann Ž1976..
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Copyright Q 1997 by Academic Press
All rights of reproduction in any form reserved.
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KAJII AND MORRIS
are similar to outcomes under common knowledge.3 Unfortunately the MS
result requires each information set of each individual to have positive
probability and thus each individual to have at most a countable number of
possible signals. These assumptions remove the indeterminacy of conditional probability at particular states and so make it possible to define a
belief operator which specifies at which states a given event is believed with
probability p. This simplifies both the technical analysis and the interpretation, but excludes important applications.
In an analysis of common 1-belief, Nielsen Ž1984. identified sets which
differ only by zero measure sets. In this note, we follow this approach and
define belief operators on classes of equivalent events Žrather than on
events directly.. We prove a straightforward analogue of the MS result
using such belief operators. This approach has the attractive feature that
all definitions and results are independent of any particular version of
conditional probability: there is a unique representation of belief operators
despite the multiplicity of conditional probabilities. However, we emphasize that an alternative interpretation of our results is that we are confirming that it does not matter Žalmost surely. if a particular conditional
probability is fixed.
The note is organized as follows. After introducing some measure
theoretic properties in Section 2, we then define class based belief operators Žin subsection 3.1. and use them to define and characterize common
p-belief Žin subsection 3.2.; this characterization includes an uncountable
signals version of Monderer and Samet’s theorem. In Section 4, we discuss
the application of the approach to a simple example.
2. SETUP
Throughout the paper we will fix a set V, a s-field F on V, a
probability measure P on w V, F x, a finite collection of individuals I , with
sub s-fields representing the information of each individual i, Fi 4i g I . We
will identify events which differ only by zero probability events. We first
summarize some measure theoretic facts, which are versions of standard
results that can be found in, for instance, Halmos Ž1950.. Although our
terminology is slightly different, we are following Nielsen’s Ž1984. work on
knowledge Ž1-belief., who in turn attributes the idea to Koopman Ž1940.
and Halmos Ž1944..
Say two measurable sets E, F are equi¨ alent Žunder P . if P Ž ED F . s 0,
where ED F denotes the symmetric difference of E and F. We shall write
3
This has been shown for games of incomplete information ŽMS., no disagreement results
ŽMS., no trade results ŽSonsino, 1995., and learning ŽMonderer and Samet, 1995..
COMMON
p-BELIEF
75
E ; F if E and F are equivalent. It is easy to show that the relation ; is
an equivalence relation on F. Let E denote the equivalent class of events
that contains event E. Then E ; F if and only if E s F. Denote the set of
equivalent classes by F˜; that is, F˜s Fr; .
We write E ; F if there are measurable sets E9 g E, F9 g F such that
E9 ; F9. It is readily verified that ; is a reflexive and transitive relation
on F˜. Notice that the relation ; is not the class inclusion and that it
depends on the measure P.
Binary relations on F˜ analogous to union and intersection are defined
as follows:
E j F s E9 j F9 : E9 g E, F9 g F4 ;
E l F s E9 l F9 : E9 g E, F9 g F4 .
It can be readily verified that E j F and E l F are equivalence classes,
containing E j F and E l F, respectively. The relative complement is
also defined analogously: E _ F s E9 _ F9 : E9 g E, F9 g F4 , which is an
equivalence class, containing E _ F. The union and intersection of countably many classes are also well defined:
for E n , n s 1, 2, . . . ,
`
D En s
ns1
`
F En s
ns1
`
½F
ns1
`
½D
EnX : EnX g E n ,
5
ns1
EnX : EnX g E n .
5
It is easy to show that D `ns 1 E n Žresp. F `ns1 E n . is an equivalence class,
containing D `ns 1 En Žresp. F `ns1 En .. To sum up, the relations l, j, and
_ work exactly as their set theoretic counterparts as long as we are dealing
with countable operations on F˜ and the choice of representative element
in each class does not matter.
We say F g F is essentially Fi-measurable if there is F9 g Fi such that
F ; F9. By construction, if F is essentially Fi-measurable, then the class F
consists of essentially Fi-measurable sets, and we say F is Fi-measurable. It
is easy to see that Fi-measurable classes are closed under countable
operations of j, l, and _. A real-valued F-measurable function f on V
is said to be essentially Fi-measurable if f coincides with an Fi-measurable
function almost surely.
Let E g F. A conditional probability function of event E g F with
respect to Fi is an essentially Fi-measurable function f such that
HA f Ž v . dP s HA 1 E Ž v . dP Žs P Ž E l A.. for any A g Fi , where 1 E is the
indicator function of E. Since functions that are identical almost every-
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KAJII AND MORRIS
where have identical integrals, a conditional probability function is essentially unique in the sense that f is a conditional probability of E if and
only if any f 9 that coincides with f almost everywhere is a conditional
probability of E.
Conditional probability is countably additive in the following sense:
Fact 1. Let E1 , E2 , . . . , En , . . . g F be a sequence of disjoint sets.
Suppose for each n, f n is a conditional probability function of En . Then
`
the function Ý ns
1 f n is well defined and it is a conditional probability
`
function of D ns 1 En .
DEFINITION 2. The posterior probability of a class E with respect to Fi
Ždenoted by P i ŽE.. is the collection of all conditional probability functions
of all sets E9 g E, with respect to Fi .
To summarize the above discussion, we have:
Fact 3. f g P i ŽE. if and only if f is Fi-measurable and HA f Ž v . dP s
HA 1 E9Ž v . dP for any E9 g E and A g Fi . Therefore, in particular, f g
P i ŽE. if and only if any f 9 that coincides with f P-almost surely belongs to
P i ŽE..
3. p-BELIEF OPERATORS: A GENERAL APPROACH
3.1. Indi¨ idual Belief Operators
For any function f : V ª w0, 1x, let
F p Ž f . s v : f Ž v . G p4 .
Then from Fact 3, the following can be readily verified:
Fact 4. For any f, f 9 g P i ŽE., F p Ž f . ; F p Ž f 9.. If A g F satisfies
A ; F p Ž f . for some f g P i ŽE., then there exists g g P i ŽE. such that
F p Ž g . s A.
The following describes the class of events where the individual pbelieves E.
B˜ip Ž E. s F p Ž f . : f g P i Ž E. 4 .
The following proposition summarizes basic properties of the p-belief
operator B˜ip , which are completely analogous to the finite or the countable
case Žas studied by MS., except that we are now dealing with classes of
events.
COMMON
p-BELIEF
77
PROPOSITION 5. The following properties hold:
ŽB0. B˜ip ŽE. is a class;
ŽB1. B˜ip ŽE. is Fi-measurable;
ŽB2. if E is Fi-measurable, then E s B˜ip ŽE.;
ŽB3. if E ; F, then B˜ip ŽE. ; B˜ip ŽF.;
ŽB4. if E is Fi-measurable, then B˜ip ŽE l F. s E l B˜ip ŽF.;
ŽB5. if F n is a decreasing sequence of classes Ž i.e., F nq 1 ; F n for all
n s 1, 2, . . . ., then B˜ip ŽF `ns1 F n . s F `ns1 B˜ip ŽF n ..
Proof. ŽB0. By fact 4.
ŽB1. By construction.
ŽB2. If E is Fi-measurable, then there exists E9 g E with E9 g Fi ;
thus 1 E g P i ŽE. and the result follows.
ŽB3. Let E ; F and f g P i ŽE.. Choose E9 g E and F9 g F such that
E9 : F9. Choose a conditional probabilities Žwith respect to Fi . f of E9
and g of F9. Then by definition, g G f on E9 almost surely, thus B˜ip ŽE. ;
B˜ip ŽF..
ŽB4. Choose any E9 g E, F9 g F. Recall that if E9 is Fi-measurable,
then 1 E9 is a conditional probability of E9. Let f be a conditional
probability function of F9. Then g Ž v . s minŽ1 E9Ž v ., f Ž v .. is a conditional
probability of E9 l F9. Indeed, for any A, HA g s HA l E9 f s HA l E9 1 F 9 s
HA 1 E9l F 9. Since g Ž v . G p if and only if v g E9 and f Ž v . G p, ŽB4. holds.
ŽB5. We can choose a decreasing sequence of measurable sets F n g
F , and set F s F `ns 1 F n. F g F `ns1 F n by construction. Let f n, f be
conditional probability functions of F n, F, respectively. Since F n is decreasing, applying fact 1 to E n s F n _ F nq 1 , we have that f n converges
monotonically, almost surely to f. In particular, v : f Ž v . G p4 ;
F `ns 1 v : f n Ž v . G p4 ; that is, B˜ p ŽF. s F `ns1 B˜ p ŽF n ., as desired. B
n
3.2. Common p-Belief Operator
We now want to consider belief operators for many individuals and
formulate the concept of common p-belief in this general framework.
Define the ‘‘everyone believes’’ operator by the rule
˜ p Ž E. s
B#
F B˜ip Ž E. .
ig I
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KAJII AND MORRIS
LEMMA 6.
˜ p Ž B#
˜ p ŽE.. ; B#
˜ p ŽE..
B#
Proof.
˜ p Ž B#
˜ p Ž E. . s
B#
˜ p Ž E. . 4
F B˜ip Ž B#
ig I
s
F B˜ip F B˜jp Ž E.
½ ½
F½
ig I
s
jg I
B˜ip Ž E. l B˜ip
,
so by Ž B1 . and Ž B4 . ,
žF ˜ Ž ./5
Bjp E
,
so by Ž B3 . ,
j/i
ig I
;
55
F B˜ip Ž E. 4
ig I
˜ p Ž E. . B
s B#
Define ‘‘common p-belief operator’’ C˜ p by the rule
˜ p Ž E. l B#
˜ p Ž B#
˜ p Ž E. . l B#
˜ p Ž B#
˜ p Ž B#
˜ p Ž E. . . l ??? .
C˜ p Ž E. s B#
Because C˜ p ŽE. is a countable intersection of classes, it must itself be a
class. In this framework, it no longer makes sense to ask if an event or
class of events is common p-belief at a particular state v , unless the
singleton set v 4 occurs with positive probability. We rather define when a
class E is common p-belief at some other class F.
DEFINITION 7. Class E is common p-belief at F if F ; C˜ p ŽE..
This iterati¨ e definition will be related to the following fixed-point
characterization.
˜ p ŽE..
DEFINITION 8. Class E is p-evident if E ; B#
The following result is thus an uncountable version of MS’s result, which
is in turn a generalization Žfrom 1-belief to p-belief. of Aumann’s Ž1976.
characterization of common knowledge.
THEOREM 9. The class E is common p-belief at A if and only if there is a
˜ p ŽE..
p-e¨ ident class F such that A ; F ; B#
We can clarify the theorem by restating it in two parts. Pick any class E;
˜ p ŽE., we have
then Ža. for any p-evident class F such that F ; B#
F ; C˜ p Ž E. .
COMMON
p-BELIEF
79
Conversely, Žb. for any class A such that A ; C˜ p ŽE., there exists a
˜ p ŽE..
p-evident class F such that A ; F ; B#
Proof. Now the ‘‘if’’ statement follows from Ža., and the ‘‘only if’’ part
follows from Žb..
˜ p ŽE.. By ŽB3.,
Proof of Ža.: Suppose a p-evident class F satisfies F ; B#
n
˜ p . Ž F. ; Ž B#
˜p .
Ž B#
nq1
Ž E. ,
for n s 0, 1, 2, . . .
and so
n
˜ p . Ž F.
C˜ p Ž F. s F `ns 1 Ž B#
n
˜ p . Ž F.
; F `ns 0 Ž B#
since F is p-evident,
n
˜ p . Ž E.
; F `ns 1 Ž B#
s C˜ p Ž E. .
˜ p ŽF. and thus, by
By the definition of p-evident, we also have F ; B#
ŽB3.,
n
˜ p . Ž F. ; Ž B#
˜p .
Ž B#
nq1
Ž F. ,
n
˜ p . Ž F. ,
and F ; Ž B#
for n s 0, 1, 2, . . . ,
for n s 1, 2, . . .
n
˜ p . Ž F. s C˜ p Ž F. ; C˜ p Ž E. .
So F ; F `ns 1 Ž B#
˜ p ŽE..
Proof of Žb.: Let F s C˜ p ŽE.. By construction of C˜ p , A ; F ; B#
0
So it is enough to show that F is p-evident. Let E s E, and define
˜ p ŽE ky1 . iteratively. By construction, F s F `ks 1 E k.
E k s B#
By Lemma 6, E k , k s 1, 2, . . . , is a decreasing sequence of classes.
Therefore, by ŽB5., B˜ip ŽF. s F `ks1 B˜ip ŽE k ., so
˜ p Ž F. s F i g I B˜ip Ž F. ,
B#
using the relation above,
s F i g I F `ks1 B˜ip Ž E k . 4 ,
and exchanging the order,
s F `ks 1 F i g I B˜ip Ž E k . 4
˜p Ž E k . .
s F `ks 1 B#
˜ p ŽE k . s
On the other hand, since E k is decreasing, F `ks 1 B#
`
p
k
`
k
p
˜ ŽE . s F ks1 E s F. Therefore, we have F s B#
˜ ŽF., so in parF ks 2 B#
ticular F is p-evident as desired. B
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KAJII AND MORRIS
4. DISCUSSION
4.1. A Fixed Conditional Probability Approach
Nielsen Ž1984. gave a characterization of common 1-belief for uncountable state spaces using the class based approach which we pursue here.
Brandenburger and Dekel Ž1987. pursued the alternative approach Žfor
common 1-belief. of fixing a particular conditional probability, and continuing to analyze common 1-belief about e¨ ents defined at states. We believe
that while the former approach Žwhich we have pursued here. is mathematically more elegant, it is often useful in applications to use a natural
conditional probability. One interpretation of the results presented here is
that as long as we interested in probability 1 statements, there is no loss of
generality in fixing a conditional probability. We therefore conclude by
briefly summarizing how to do so.
For each i, fix a regular conditional probability Pi : F = V ª R: that is,
Pi Ž E < ? . g P i ŽE. for all i and E g F , and Pi Ž?< v . is a probability measure
with probability one wsuch regular conditional probabilities exist if V is a
separable metric space and F is the Borel fieldx. For these fixed Pi , define
pŽ .
event belief operators Bip Ž E . s v : Pi Ž E < v . G p4 , B#
E s F ig I
pŽ .
pŽ .
pŽ .
p Ž p Ž ..
p Ž p Ž p Ž ...
Bi E , and C E s B# E l B# B# E l B# B# B# E l ??? .
Then by ŽB1. of Proposition 5 and construction, it can be readily verified
˜ p ŽE. holds if and only if B#p Ž E . ; E9, and so E
that for any E g F , E9 g B#
is p-evident if and only if there is E9 g E such that for all i, Bip Ž E9. _ E9
is a null set. So we say that an event E is p-evident if Pi Ž E < v . G p holds
for almost every v g E. Also C˜ p ŽE. is the class containing C p Ž E .. Thus if
we say that E is common p-belief at v if v g C p Ž E ., we can rephrase
Theorem 9 as follows:
COROLLARY 10. For any e¨ ent E, the following statement is true for
almost e¨ ery v : E¨ ent E is common p-belief at v if and only if there exists a
pŽ .
p-e¨ ident e¨ ent F such that v g F ; B#
E.
In the remainder of this note, we will informally present an example
which illustrates the usefulness of using fixed conditional probabilities to
characterize common p-belief.
4.2. The Noise Example
The following example was analyzed by Morris et al. Ž1993.. There are
two individuals 1 and 2. Individual 1 observes a number x 1 g w0, 1 . and
individual 2 observes a number x 2 g w0, 1 .. So let V s w 0, 1. 2 with its
Borel sets and Fi be the collection of the sets of the form Ž x 1 , x 2 .: x i s x 4 ,
for some x g w0, 1 .. By convention, we will identify any number with its
decimal part, so that if a ) b , w a , b x s w a , 1. j w0, b x and < a y b < s
COMMON
p-BELIEF
81
min a y b , 1 y Ž a y b .4 . We consider probability measures indexed by
« g w 0, 12 . : P « is the uniform measure on the set Ž x 1 , x 2 .: < x 1 y x 2 < F « 4 ;
in particular, P 0 is uniform on the diagonal set D s Ž x 1 , x 2 .: x 1 s x 2 4 .
Thus the marginal distribution of x i is uniform on the interval w0, 1 ..
Given x i , individual j observes a signal x j which is distributed uniformly
on the interval w x j y « , x j q « x.
Let F * be the collection of events which exclude some open interval on
the diagonal, i.e.,
F* s
½
there exists x g 0, 1 . and d ) 0
E g F:
such that E ; Ž x 1 , x 2 . : 5 Ž x 1 , x 2 . y Ž x, x . 5 ) d 4
5
.
We will show that for any p ) 12 , any event E in F * is never common
p-belief for all sufficiently small « ) 0. However, if « s 0, such events are
p-evident and thus common p-belief whenever they are true, for any
p ) 0. We will show how the analysis of this paper can be used to make
these claims precise and independent of the choice of conditional probability.
Consider first the case with « g Ž0, 12 .. The conditional density is
fi Ž x j < x i . s
¡1
¢0,
~ 2« ,
if x j g w x i y « , x i q « x
,
otherwise
so, writing l for the Lebesgue measure on w0, 1x, the ‘‘natural’’ regular
conditional probability is
Pi Ž E < x i . s
1
2«
l Ž xXj : Ž x i , xXj . g E and < xXj y x i < F « 4 . .
The associated event belief operators are denoted by Bip Ž E .. Now suppose
E g F *, with E ; Ž x 1 , x 2 . : 5Ž x 1 , x 2 . y Ž x, x .5 ) d 4 . Consider any p ) 12
and « g Ž0, dr3. and let j / i; now if x i g Ž x y « , x q « ., individual i
assigns probability 1 to x j being in the interval Ž x y 2 « , x q 2 « .. But
'
5Ž x i , x j . y Ž x, x .5 F « 2 q Ž 2 « . 2 s Ž'5 . « - d if x i g Ž x y « , x q « .
and x j g Ž x y 2 « , x q 2 « .; so individual i assigns probability 0 to Ž x i , x j .
being an element of E if x i g Ž x y « , x q « . and so
Bip Ž E . ; Ž x 1 , x 2 . : x i f Ž x y « , x q « . 4
p
B#
Ž E . ; Ž x 1 , x 2 . : x i f Ž x y « , x q « . for i s 1, 2 4
w B#p x Ž E . ; Ž x 1 , x 2 . :
n
½
x i f Ž x y g Ž « , p, n . , x q g Ž « , p, n . .
,
for i s 1, 2
5
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KAJII AND MORRIS
where g Ž « , p, n. s « Ž1 q Ž n y 1.Ž2 p y 1... Thus C p Ž E . s B. Thus for
any E g F *, C p Ž E . s B for all p ) 12 , for « sufficiently small.
Now consider the case where « s 0, and thus probability is distributed
uniformly on the diagonal. The ‘‘natural’’ conditional probability has
individual i observing x i assigning probability 1 to individual 2 observing
x i ; the corresponding event belief operator is therefore
Bip Ž E . s Ž x 1 , x 2 . : Ž x i , x i . g E 4 ,
for all p g Ž0, 1 x. Thus any E is p-evident and C p Ž E . s D l E for all
E g F and all p g Ž0, 1 x.
The above analysis Žlike that of Morris et al., 1993. fixed the conditional
probabilities. But the analysis of this note shows that the choice of
conditional probability does not matter. So if « s 0, we have C˜ p ŽE. s E
for all p g Ž0, 1 x and all classes E; but if « is positive but sufficiently
small, C˜ p ŽE. is the null class for all E g F * and all p g Ž 12 , 1 x.
This extreme sensitivity of common p-belief to noise is important in a
number of applications; see, for example, Carlsson and van Damme Ž1993..
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