January 2013
Complete Sequential Equilibrium
Hanjoon Michael Jungyz
The Institute of Economics, Academia Sinica
Abstract
We introduce an improved version of sequential equilibrium in general games. Kreps and Wilson
(1982) introduced the sequential equilibrium in …nite games that allow only a …nite number of types
and strategies. In …nite games, this sequential equilibrium satis…es convex structural consistency
and Nash equilibrium condition. However, it might not satisfy these criteria in general games that
allow a continuum of types and strategies. Hence, we develop a new solution concept referred
to as complete sequential equilibrium. The complete sequential equilibrium always satis…es the
Nash equilibrium condition and, under independent and topologically complete type-state space
assumption, it also satis…es the convex structural consistency in the general games. Moreover, this
complete sequential equilibrium exists in continuous games and it is equivalent to the sequential
equilibrium in …nite games. Its relation to converted versions of perfect equilibrium and perfect
Bayesian equilibrium is discussed.
JEL Classi…cation Number: C72
Keywords: Complete Belief, Complete Sequential Equilibrium, General Game,
Solution Concept, Sequential Convergency, Sequential Equilibrium.
I am grateful to James Jordan, Joel Sobel, Kalyan Chatterjee, Vijay Krishna, and anonymous referees
for their valuable discussions and comments. I would also like to thank Arif Zaman, Byung Soo Lee, Edward
Green, Hans Haller, Ismat Beg, Johannes Horner, Le-Yu Chen, Muhammad Ahsan, Tymo…y Mylovanov,
and Wooyoung Lim for their helpful suggestions.
y
The Institute of Economics, Academia Sinica, 128, Sec. 2, Academia Road, Nangang, Taipei 115, Taiwan
z
Email address: [email protected]
1
Introduction
Kreps and Wilson (1982) introduced sequential equilibrium in the setting of …nite games that
allow only a …nite number of types and strategies. In …nite games, this sequential equilibrium
satis…es criteria of rational solution concepts such as convex structural consistency and the
Nash equilibrium condition. However, it might not satisfy these criteria in general games
that allow a continuum of types and strategies.1 To solve this incapability problem with the
sequential equilibrium, we revise their de…nition and develop a new solution concept referred
to as complete sequential equilibrium.
The sequential equilibrium was de…ned as a pair of a system of beliefs and a strategy
pro…le that satis…es consistency and sequential rationality. According to the de…nition by
Kreps and Wilson, a system of beliefs requires checking the consistency on each of the
information sets separately. However, if we consider each of the information sets separately,
then we might not be able to check the consistency on a measure-zero class of the information
sets. So if we consider a strategy pro…le whose probability vanishes almost everywhere, then
we cannot check the consistency at all. Moreover, to show the consistency, Kreps and
Wilson used a convergent sequence of totally mixed strategies. A convergent sequence of
totally mixed strategies, however, might well de…ne probability distributions only on the
equilibrium path. As a result, o¤ the equilibrium path, we might not be able to check the
consistency through a convergent sequence of totally mixed strategies. Consequently, the
sequential equilibrium might be incapable of satisfying the two criteria of rational solution
concepts in the general …nite-period games.
The complete sequential equilibrium is de…ned the same as the sequential equilibrium
except for the two parts. In the de…nition of the complete sequential equilibrium, a system
of beliefs and a convergent sequence of totally mixed strategies are replaced with a system of
complete beliefs and a sequentially convergent sequence of strategies, respectively. A system
of complete beliefs requires checking the consistency on the whole class of the information
sets in each period. Since every strategy pro…le always well de…nes probability distributions
on the whole class of the information sets, we can check the consistency with respect to any
arbitrary strategy pro…le. Furthermore, a sequentially convergent sequence of strategies can
well de…ne probability distributions o¤ the equilibrium path as well as on the equilibrium
path. So we can properly check the consistency through a sequentially convergent sequence of
1
The perfect Bayesian equilibrium by Fudenberg and Tirole (1991a) might not satisfy the Nash equilibrium condition in the general games either.
1
strategies. Therefore, the complete sequential equilibrium can solve the incapability problem
with the sequential equilibrium in the general games.
In addition, the sequentially convergent sequence of strategies is equivalent to the convergent sequence of totally mixed strategies in …nite games. This equivalence, accordingly,
gives rise to the equivalence between the complete sequential equilibrium and the sequential
equilibrium in …nite games. Moreover, the complete sequential equilibrium exists if a type
space and a state space are complete, totally bounded, and metric and if utility functions are
bounded and continuous. Therefore, we conclude that the complete sequential equilibrium
is an improved version of the sequential equilibrium in the general games.
The rest of the paper is organized as follows. Section 2 formulates general …nite-period
games with observed actions. Section 3 extends the de…nition of the sequential equilibrium
into the general game and illustrates its incapability to satisfy the two criteria of rational solution concepts, namely, the convex structural consistency and the Nash equilibrium
condition. Section 4 introduces new concepts referred to as complete beliefs and sequential convergence, and lays the foundations of the complete sequential equilibrium. Section
5 formally de…nes the complete sequential equilibrium in the general games. Section 6
demonstrates four properties of the complete sequential equilibrium including its existence
condition, and therefore this section shows that the complete sequential equilibrium, as a
version of the sequential equilibrium, indeed improves the sequential equilibrium, by Kreps
and Wilson, in the general games. Finally, Section 7 concludes with the discussion about
how the complete sequential equilibrium is related to the versions of perfect equilibrium and
perfect Bayesian equilibrium that are converted in the general games.
2
General …nite-period games with observed actions
We adopt the “…nite-period games with observed actions”from Fudenberg and Tirole (1991a)
and adapt them to general games that allow in…nite actions and types, but only …nite players.
Hence, like the games in Fudenberg and Tirole (1991a), the general …nite-period games with
observed actions are represented by …ve items: players, a type and state space, a probability
measure on the type and state space, strategies, and utility functions. We present formal
de…nitions of them all. Then, based on this setting of the general games, we extend the
de…nition of the Nash equilibrium. Consequently, this section is devoted to de…ning the
setting of the general …nite-period games with observed actions and the Nash equilibrium.
In the general game, there are a …nite number of players denoted by i = 1; 2; :::; I. Each
2
player i has her type i 2 i and this type is her private information as in Harsanyi (1967–
68). In addition, there exists a state 0 2 0 and the players do not have information about
the actual state. Thus, each player has information about their types i , but no information
about the other players’types and the state i 2
( i0 6=i i0 ). We assume that
i =
0
= Ii=0 i is a non-empty separable metric space. Realizations 2 are governed by a
probability measure on the class of the Borel subsets2 Ii=0 ß( i ) of . For simplicity, we
assume (B) > 0 for every open subset B in .
The players play the game in periods t = 1; 2; :::; T . In each period t, all players simultaneously choose actions, and then their actions are revealed at the end of the period. We
assume, for simplicity, that each player’s available actions are independent of her type so
that each player i’s action space in period t is Ati regardless of her type. In addition, we
assume that At = Ii=1 Ati is a non-empty separable metric space3 for each t. Here, we employ
this setting of the observed actions to de…ne the players’strategies and eventually to de…ne
their expected utility functionals. This setting, however, can be extended to the setting of
partially observed (or unobserved) actions if strategies and expected utility functionals are
well-de…ned, which indeed almost all of the game theory literature take for granted. Therefore, all results in this study can be applied to general …nite-period games with partially
observed actions as long as the players’strategies and their expected utility functionals are
well-de…ned.
A strategy is de…ned as follows. For each i = 1; :::; I and t = 1; :::; T , let ti be a
transition probability4 from i A1
At 1 ß(Ati ) to [0; 1]. Then, a behavioral
strategy i is an ordered list of measures i = ( 1i ; :::; Ti ) such that i) for each ( i ; a1 ; :::; at 1 )
2 i A1
At 1 , ti ( i ; a1 ; :::; at 1 ; ) is a probability measure on ß(Ati ) and ii) for every
0
1
I
B 2 ß(Ati ), ti ( ; B) is ß( i ) ( tt0 =1
(Ati0 )) measurable. The condition i) requires that
i0 =1 ß
1
t 1
each ti ( i ; a1 ; :::; at 1 ; ) specify what to play at each information set
)g.
i f( i ; a ; :::; a
t
The condition ii) requires that i allow a well-de…ned expected utility functional, which is
de…ned later. Hereafter, we simply call a behavioral strategy a strategy. Let i be the set
of strategies for player i and let be the set of strategy pro…les, that is, = Ii=1 i . Note
that these de…nitions originate from Milgrom and Weber (1985) and Balder (1988) and are
2
Given a metric space X, the class of the Borel sets ß(X) is the smallest class of subsets of X such that
i) ß(X) contains all open subsets of X and ii) ß(X) is closed under countable unions and complements.
3
Therefore, the space
A1
AT is a non-empty separable metric space. On this space, expected
utility functionals are well-de…ned. Furthermore, the separability assumption is used to de…ne supports of
the players’strategies, which are, in turn, used to de…ne systems of complete beliefs in Section 4.
4
For information on the transition probability, please refer to Neveu (1965, III), Ash (1972, 2.6), and
Uglanov (1997).
3
adapted to the general game.
A von Neumann-Morgenstern utility function for player i is de…ned as Ui :
A1
AT ! R. We assume that each Ui is bounded above or bounded below and Ii=0 ß( i )
( Tt=1 Ii=1 ß(Ati )) measurable, which guarantees that Ui is integrable. Moreover, we de…ne
an expected utility functional Ei :
! R (= R [ f 1; 1g) as
Ei ( 1 ; :::;
n)
=
R R
A1
R
AT
Ui ( ; a)
T
( ; a1 ; :::; aT
1
; daT )
1
( ; da1 ) (d )
where, for each t, t denotes the product measure of f t1 ; :::; tI g on Ii=1 ß(Ati ), that is, t =
t
t
I . This de…nition of the expected utility functional makes sense according to Ash
1
(1972, 2.6).5 First, since each ti is a probability measure and measurable, so is the product
measure t . Next, since Ui ( ; a) and T ( ; a1 ; :::; aT 1 ; aT ) are Ii=0 ß( i ) ( Tt=11 Ii=1 ß(Ati ))
R
measurable, so is the inner part of the integral AT Ui ( ; a) T ( ; a1 ; :::; aT 1 ; daT ), and thus it
is Ii=1 ß(AiT 1 ) measurable. Furthermore, the inner integral is bounded above or bounded
below, so it is integrable with respect to the measure T 1 . Finally, we can show each inner
part of the integral is integrable likewise, and therefore the whole integral is well-de…ned.
Based on this expected utility functional, the Nash equilibrium by Nash (1951) is extended in the general games. Here, we suggest two conditions for an improved solution
concept of the sequential equilibrium in the general games. One is the Nash equilibrium
condition. The other is convex structural consistency introduced by Kreps and Ramey
(1987). The convex structural consistency is a criterion of consistent relationship between
players’beliefs and players’actual strategies. A formal de…nition of the convex structural
consistency is presented in Section 6.
De…nition 1 A strategy pro…le = (
max 0i 2 i Ei ( 0i ; i ) for each i I.
1
I)
5
is a Nash equilibrium if
satis…es Ei ( ) =
Let zj be a
f ield of subsets of j for each j = 1; :::; n. Let 1 be a probability measure on z1 ,
and, for each (! 1 ; :::; ! j ) 2 1
j , let (! 1 ; :::; ! j ; B), B 2 zj+1 , be a probability measure on zj+1
(j = 1; 2; :::; n 1). Assume that (! 1 ; :::; ! j ; C) is measurable for each …xed C 2 zj+1 . Let = 1
zn .
n and z = z1
(1) There is a unique probability
measure
on z such that for each measurable rectangle A1
An
R R
R
2 z, (A1
An ) = A1 A2
(!
;
:::
;
!
;
d!
)
(!
;
d!
)
(d!
).
1
n
1
n
2
1
1
1
An
R
R
R
(2) Let f : ( ; z) ! (R;ß(R)) be measurable and f 0. Then, f d = 1
f (! 1 ; ::: ; ! n ) (! 1 ; :::
n
; ! n 1 ; d! n )
1 (d! 1 ).
4
3
Example: Incapable sequential equilibrium in a general game
This section shows that an extension of the sequential equilibrium in the general games
might be incapable of satisfying the convex structural consistency and the Nash equilibrium
condition. We …rst extend the sequential equilibrium in the general games. Originally, the
sequential equilibrium by Kreps and Wilson (1982) was de…ned in the setting of …nite games.
However, it has been extended and applied to general games on various economic issues, such
as Auction, Bargaining game, Folk theorem, and Signaling game. Here, we try to present
a universal de…nition of the sequential equilibrium that can be commonly applied to such
general games. Next, we describe the setting of a general game example which is the famous
signaling game by Crawford and Sobel (1982). Then, based on this setting, we show that the
extension of the sequential equilibrium might be incapable of satisfying the convex structural
consistency and the Nash equilibrium condition.
Kreps and Wilson (1982) de…ned a sequential equilibrium as an assessment that is a pair
of a system of beliefs _ and a strategy pro…le such that 1) _ is consistent with and 2)
is sequentially rational with respect to _ . Here, a system of beliefs _ in …nite games was
P
de…ned as a function from a set of all decision nodes to [0; 1] such that x2h _ (x) = 1 for
every information set h. In the general games, therefore, we can extend this de…nition by
generalizing its conditions. First, we extend the de…nition of a system of beliefs as follows.
For each i and t, let _ ti be a measure on i A1
At 1 ( j6=i ß( j )) to [0; 1]. In
addition, for each t, let _ t denote ( _ t1 ; :::; _ tI ). Then, a system of beliefs _ in the general
games is an ordered list of measures _ = ( _ 1 ; :::; _ T ) such that i) for each ( i ; a1 ; :::; at 1 )
2 i A1
At 1 , _ ti ( i ; a1 ; :::; at 1 ; ) is a probability measure on j6=i ß( j ) and ii)
0
1
I
for every B 2 j6=i ß( j ), _ ti ( ; B) is ß( i ) ( tt0 =1
(Ati0 )) measurable.
i0 =1 ß
Next, we extend the two conditions for the sequential equilibrium in general games, which
are the consistency and the sequential rationality. A system of beliefs _ is consistent with if
there exists a sequence of assessments f( _ n ; n )g such that i) each n is totally mixed, ii) _ n
is de…ned by n according to a regular conditional probability,6 and iii) ( _ n ; n ) converges
6
Originally, Kreps and Wilson (1982) employed Bayes’rule to de…ne the consistency condition. There
are three versions of Bayes’ rule, which are a discrete version, a continuous version, and a mixed version.
All of them have limited applications. The regular conditional probability, on the other hand, does not have
a limited application. So we extend their de…nition in the general games by replacing Bayes’rule with the
regular conditional probability. All results in this section, however, would stay true even when employing
Bayes’rule instead of the regular conditional probability. For formal de…nitions of Bayes’rule, please refer
5
weakly to ( _ ; ). In the …rst sub-condition, a strategy pro…le is de…ned to be totally
mixed if it assigns positive probability to every open set of nodes in an information set, that
~ > 0 for every open set A~ At given each ( i ; a1 ; :::; at 1 )
is, for each t, ti ( i ; a1 ; :::; at 1 ; A)
i
1
t 1
2 i A
A . In the second sub-condition, _ n is de…ned by n according to a regular
conditional probability7 if, given each ( i ; a1 ; :::; at 2 ) 2 i A1
At 2 , _ ti satis…es the
R
R
R
1
t 1
1
1
t 2
following functional equation Bi B i A~1
; dat 1 )
~t 1 n ( ; a ; ::: ; a
n ( ; da ) (d )
A
R
R
R
1
t
1
1
t 1
= Bi
; B i ) tn 1 ( ; a1 ; :::; at 2 ; dat 1 )
~1
~t 1 _ i ( i ; a ; :::; a
n ( ; da ) (d ) for
A
i A
every Bi 2ß( i ); B i 2 j6=i ß( j ); A~1 2 Ii=1 ß(A1i ); :::; and A~t 1 2 Ii=1 ß(Ati 1 ). In the
R R
R
third sub-condition, n converges weakly to if
f ( ; a) Tn ( ; a1 ; :::; aT 1 ; daT )
1
A
AT
R R
R
1
1
1
( ; da1 ) (d )
f ( ; a) T ( ; a1 ; :::; aT 1 ; daT )
n ( ; da ) (d ) converges to
A1
AT
for every bounded and continuous real function f on
A1
AT . The weak convergence
of the system of beliefs _ n can be de…ned likewise. Finally, a strategy pro…le is said to
be sequentially rational with respect to _ if, taking _ as …xed, no player can increase her
conditional expected utility in responding to the other players’strategies given any of her
information sets.
Now, we are ready to exemplify the incapability of a sequential equilibrium in a general
…nite-period game with observed actions. Consider the information transmission game introduced by Crawford and Sobel (1982). There are two players, namely, a sender and a receiver.
The sender is assigned a type s that is a random variable from a uniform distribution on
[0; 1] and she makes a signal s 2 [0; 1]2 to the receiver.8 Then, after observing the signal
s, the receiver chooses his action a 2 [0; 1]. The sender has a von Neumann-Morgenstern
utility function U S ( s ; a; b) = ( s (a + b))2 where b > 0, and the receiver has another von
Neumann-Morgenstern utility function U R ( s ; a) = ( s a)2 .
to Papoulis and Pillai (2002, 2.3, 6.6, and 7.2).
7
For more information on a regular conditional probability, please refer to Ash (1972, 6.6).
8
Crawford and Sobel (1982) did not specify the sender’s signal space. They de…ned it as a Borel set
of feasible signals. In this example, we specify it as [0; 1]2 to clearly show the incapability of a sequential
equilibrium in a general game. This speci…cation, [0; 1]2 , can be replaced with other uncountable Borel sets
while preserving the result. For example, suppose that the unit interval [0; 1] is the sender’s signal space.
Then, we can still show the incapability of a sequential equilibrium in this general game as follows. For the
sender’s strategy, consider the Cantor set C that we can construct by continuously deleting middle third
open intervals. Since the set C is uncountable, there is a one-to-one measurable function s : [0; 1] ! C.
Let this function s( ) be the sender’s strategy. Then, this strategy s( ) still fully reveals the sender’s type.
In addition, let the receiver play a strategy a( ) such that a(s( )) = maxf
b; 0g and form his system of
beliefs _ such that _ (s( ); maxf
b; 0g) = 1. Since the Cantor set C is uncountable and of measure zero,
we can construct a sequence of probability density functions as in the example. Consequently, this system
of beliefs and these strategies are a sequential equilibrium. This sequential equilibrium, however, does not
satisfy the convex structural consistency and the Nash equilibrium condition.
6
In this game, consider the following system of beliefs and strategies. Suppose that the
sender plays a strategy s : [0; 1] ! [0; 1]2 such that, for each s 2 [0; 1], s( s ) = ( s ; s )
which denotes that the sender with the type s signals ( s ; s ). That is, the sender truthfully
reveals her type s through her signal ( s ; s ). Responding to this strategy s( ), suppose
that the receiver plays a strategy a : [0; 1]2 ! [0; 1] such that a(s( s )) = maxf s b; 0g
which means that the receiver plays the action maxf s b; 0g after observing the signal s( s )
(= ( s ; s )), which is sent by the sender’s type s . Finally, suppose that the receiver forms
his system of beliefs _ : [0; 1]2 ß[0; 1] ! [0; 1] such that _ (s( s ); maxf s b; 0g) = 1 which
denotes that, given the signal s( s ) (= ( s ; s )), _ assigns the type maxf s b; 0g to the
sender with probability one.
These system of beliefs and strategies are a sequential equilibrium. First, the system of
beliefs _ is consistent with the sender’s strategy s( ) according to the regular conditional
probability. Second, these strategies of the players, s( ) and a( ), are sequentially rational
with respect to the system of beliefs _ . Here, checking the sequential rationality would be
straightforward. So we focus only on checking the consistency between _ and s( ).
To show this consistency, we need to construct a sequence of assessments f( _ n ; n )g that
satis…es the three sub-conditions above. Assessments are pairs of systems of beliefs and
strategies. We construct a sequence of strategies …rst. Consider the following sequence of
probability density functions ffn g on [0; 1]3 such that for every s 2 [0; 1] and n 2 N, i)
R
f ( ; s)ds = 1 and ii) fn ( s ; s) > 0 for almost all s 2 [0; 1]2 . In addition, for each
[0;1]2 n s
type s , suppose that the function fn ( s ; ) converges weakly to the distribution of the type
s ’s action s( s ), which denotes that the signal s( s ) = ( s ; s ) occurs with probability one.
Note that, given any s and n, we can change values of fn ( s ; ) within a set of measure
zero in [0; 1]2 while preserving its distribution. So, without loss of generality, we can assume
that, for each s and n, fn ( s ; s) = 0 for every s 2 s([0; 1]) since the set of signals s([0; 1])
(= f( s ; s ) : s 2 [0; 1]g) is a set of measure zero in [0; 1]2 . Next, we construct a sequence
of systems of beliefs. De…ne a sequence of probability measures f _ n g on [0; 1]2 ß[0; 1] such
that, for each n 2 N, _ n is induced by fn according to the regular conditional probability.
Then, we have constructed a sequence of assessments f( _ n ; fn )g. We use this sequence to
show the consistency between _ and s( ).
It is easy to see that this sequence f( _ n ; fn )g satis…es the …rst two sub-conditions for
the consistency. To check the third sub-condition, note that the sequence of strategies ffn g
does not de…ne conditional probability given signals s in s([0; 1]) according to the regular
conditional probability. This is because we have, for each s and n, fn ( s ; s) = 0 for every s
7
2 s([0; 1]). In this case, the regular conditional probability allows the sequence of systems
of beliefs f _ n g to adopt any arbitrary probability distributions conditional on these signals
s 2 s([0; 1]). As a result, the sequence f( _ n ; fn )g satis…es the third sub-condition for the
consistency, which in turn means that the system of beliefs _ is consistent with the sender’s
strategy s( ). Therefore, this system of beliefs and these strategies, ( _ ; (s( ); a( ))), are a
sequential equilibrium.
This sequential equilibrium, however, is incapable of satisfying the convex structural
consistency and the Nash equilibrium condition. In the scenario of this equilibrium, the
receiver constantly mistakes a true type s for a wrong type maxf s b; 0g. As a result,
the sender’s strategy s( s ) = ( s ; s ) and the receiver’s system of beliefs _ (maxfs b; 0g; s)
= 1 do not induce the same probability distribution even on the equilibrium path which the
players would actually reach if they were to play according to their strategies s( ) and a( ).
Since the convex structural consistency9 requires them both to induce the same probability
distribution at least on the equilibrium path, this sequential equilibrium does not satisfy the
convex structural consistency. Moreover, the receiver’s strategy a(s) = maxfs b; 0g is not
the best response to the sender’s strategy s( s ) = ( s ; s ). So this sequential equilibrium
does not satisfy the Nash equilibrium condition.
This incapability of the sequential equilibrium is caused mainly by the setting that the
sender has a continuum of types and signals. Accordingly, most games having similar settings
can testify that there exist sequential equilibria that violate the convex structural consistency
and the Nash equilibrium condition. Since this setting represents a usual situation, there are
a large class of games including similar settings. Therefore, we conclude that this incapability
is a ubiquitous problem with the sequential equilibrium in the general games. From the next
section, we start to develop a solution concept that is capable of satisfying both the convex
structural consistency and the Nash equilibrium condition in the general games.
4
Complete beliefs and Sequential convergence
The incapability of a sequential equilibrium in the general …nite-period games is mainly due
to the heavy requirement of a system of beliefs. A system of beliefs is a set of probabilistic
assessments about other players’types conditional on reaching each of the information sets.
So, it consists of conditional probability measures on each of the information sets, and each
9
De…nitions 8, 9, and 10 in Section 5 formally de…ne this convex structural consistency in the general
games.
8
conditional probability measure separately denotes players’beliefs about the others’types.
A system of beliefs, therefore, requires us to consider conditional probability on each of the
information sets separately.
In the sequential equilibrium, we use a convergent sequence of totally mixed strategies
to check the consistency between a system of beliefs and a strategy pro…le. However, if
we consider each of the information sets separately, then a convergent sequence of totally
mixed strategies can show the consistency between them at most on almost all information
sets. That is, it might not be able to show the consistency on a measure-zero class of
the information sets. So if a convergent sequence of totally mixed strategies converges to
strategies whose probability vanishes almost everywhere, then it cannot actually show the
consistency between a system of beliefs and a strategy pro…le at all. In this case, any
arbitrary system of beliefs can be considered to satisfy the consistency for a sequential
equilibrium. As a result, some intuitively inconsistent system of beliefs could be part of a
sequential equilibrium, and this system of beliefs could lead the sequential equilibrium to
the incapability in the general games, as shown in the example.
To solve the incapability problem with the sequential equilibrium, we …rst improve it by
replacing a system of beliefs with a system of complete beliefs. A system of complete beliefs
is a set of probability measures de…ned, not on each of the information sets, but on the whole
class of the information sets in each period. A system of complete beliefs, therefore, requires
us to consider probability on the whole class of the information sets in each period together.
Note that all sequences of strategies lead to the whole class of the information sets in each
period with probability one, and thus they can well de…ne probability distributions on the
whole class of the information sets even when their probability vanishes almost everywhere.
For this reason, a system of complete beliefs can improve the sequential equilibrium in the
general games.
In the general games, however, a convergent sequence of totally mixed strategies might
not de…ne probability distributions o¤ the equilibrium path. That is, it might de…ne probability distributions only on the equilibrium path and leave probability distributions o¤ the
equilibrium path unde…ned. Accordingly, o¤ the equilibrium path, we might not be able to
check the consistency between a system of complete beliefs and a strategy pro…le by means of
a convergent sequence of totally mixed strategies. Hence, in our solution concept, complete
sequential equilibrium, we employ a stronger means, which is referred to as a sequentially
convergent sequence of strategies. A sequentially convergent sequence of strategies is a convergent sequence of totally mixed strategies such that the sequence sequentially converges
9
weakly to at most countably many strategies and the union of supports of the probability
measures induced by these strategies is dense in the whole class of the information sets in
each period. Then, through these probability measures, a sequentially convergent sequence
of strategies can de…ne probability distributions o¤ the equilibrium path as well as on the
equilibrium path. As a result, we can properly check the consistency between a system of
complete beliefs and a strategy pro…le by means of a sequentially convergent sequence of
strategies. Therefore, the complete sequential equilibrium, equipped with a system of complete beliefs and a sequentially convergent sequence of strategies, can improve the sequential
equilibrium in the general games.
The de…nition of the complete sequential equilibrium is the same as that of the sequential
equilibrium except for the two replacements. Thus, the complete sequential equilibrium is
de…ned as a pair of a system of complete beliefs and a strategy pro…le such that 1) the
system of complete beliefs is consistent with the strategy pro…le by means of a sequentially
convergent sequence of strategies and 2) the strategy pro…le is sequentially rational with
respect to the system of complete beliefs. In the rest of this section, as the preliminaries to the
study of the complete sequential equilibrium, we examine the de…nitions and the properties
of a system of complete beliefs and a sequentially convergent sequence of strategies in the
general games.
4.1
Complete beliefs
We …rst present a formal de…nition of the system of complete beliefs in the general …niteperiod games with observed actions.
0
1
I
De…nition 2 For each i and t, a probability measure ti on Ii=0 ß
( i ) ( tt0 =1
(Ati ))
i=1 ß
is called a complete belief for player i in period t if ti ( i Bi ) > 0 for every open set
Bi
A1
At 1 . For each t, let t denote ( t1 ; :::; tI ), then a system of complete
i
beliefs is an ordered list of complete beliefs = ( 1 ; :::; T ).
Properties of a system of complete beliefs depend on the properties of the space
A1
AT . Since the space is metric, every system of complete beliefs consists of regular
probability measures10 according to Billingsley (1968, Thm 1.1). If the separable metric
space is also topologically complete, then every system of complete beliefs consists of tight
10
By Billingsley (1968), a probability measure on ß(X) of a metric space X is de…ned to be regular
if for any B 2 ß(X) and " > 0, there exist a closed set G and an open set O such that G
B
O and
(O G) < ".
10
measures11 according to Billingsley (1968, Thm 1). In addition, the condition for a complete
belief, ti ( i Bi ) > 0 for every open set Bi , ensures that, with respect to a complete
belief, we can check sequential rationality of a strategy pro…le in every open class of the
information sets o¤ the equilibrium path as well as on the equilibrium path.
In our solution concept, complete sequential equilibrium, we need to adopt a new approach to de…ne probability distributions for a system of complete beliefs. In the solution
concept of the sequential equilibrium, we de…ne probability distributions for a system of
beliefs. Here, a system of beliefs is a set of conditional probability measures on each of
the information sets. So, when we de…ne probability distributions for a system of beliefs,
we apply a convergent sequence of totally mixed strategies to each of the information sets
separately and derive conditional probability distributions on each of the information sets
separately. A system of complete beliefs, on the other hand, is not a set of conditional probability measures. It is simply a set of probability measures. So we cannot directly adopt the
same separate approach as in the sequential equilibrium to de…ne probability distributions
for a system of complete beliefs.
The complete sequential equilibrium solves this incompatibility problem by adopting
a sequential approach. In the sequential approach, to de…ne probability distributions for
a system of complete beliefs, we sequentially apply a sequentially convergent sequence of
strategies on a class of remaining information sets in each period and derive probability
distributions from the sequence step by step.12 Examples 1 and 2 below present sequences
11
A probability measure on ß(X) of a metric space X is tight if for any B 2 ß(X), (B) is the supremum
of (K) over the compact subsets K of A.
12
To see the di¤erence between the separate approach in the sequential equilibrium and the sequential
approach in the complete sequential equilibrium, consider the extensive game below in which players 1 and
2 sequentially take actions and player 1 has two types A and B and three actions a, b, and c.
Let su¢ ciently small "n > 0 converge to zero. Suppose that, for each n 2 N, player 1 plays the following
totally mixed strategy. When player 1 is of type A, she plays the action a with probability "n , the action
b with probability 1 2"n , and the action c with probability "n . When she is of type B, she plays a with
probability "n , b with probability "n , and c with probability 1 2"n . First, with respect to this sequence
of player 1’s totally mixed strategies, de…ne a system of beliefs according to the separate approach. Then,
we have to consider each of the information sets separately. So, in the information set H1 , we can de…ne a
11
of totally mixed strategies in the signaling game by Crawford and Sobel (1982). With these
sequences of strategies, we illustrate the sequential approach.
Example 1 Consider the game in Crawford and Sobel (1982). For notational simplicity,
let a set of signals E(w1 ; w2 ) ( [0; 1]2 ) for each 0
w1
w2
1 be [w1 ; w2 ] [0; 1].
Then, de…ne the sender’s strategy n as follows. When s 2 [0 ; 0:5], we have n ( s ; S)
10n2
10n
1
= 1+n+n
([0; 1]2 ), and,
2 l(S\E(0; 0:1)) + 1+n+n2 l(S \E(0:2 ; 0:3)) + 1+n+n2 l(S) for every S 2 ß
10n2
10n
when s 2 (0:5 ; 1], we have n ( s ; S) = 1+n+n
2 l(S \E(0:1 ; 0:2)) + 1+n+n2 l(S \E(0:3 ; 0:4))
1
([0; 1]2 ) where l : ß
([0; 1]2 ) ! [0; 1] is a Lebesgue measure.
+ 1+n+n
2 l(S) for every S 2 ß
We can derive probability measures from this sequence f n g1
n=1 for a system of complete
beliefs step by step. For notational simplicity, let a set of nodes E 0 (h1 ; h2 ; w1 ; w2 ) = [h1 ; h2 ]
[w1 ; w2 ] [0; 1] ( [0; 1]3 ) for each 0
h1
h2
1 and 0
w1
w2
1. Then, at
1
the …rst step, we apply f n gn=1 to all the class of the information sets, which is [0; 1]3 .
Then the probability measure 1;n on ß([0; 1]3 ) induced by n converges to the probability
measure 1 such that 1 (B) = 10l0 (B \E 0 (0; 0:5; 0; 0:1)) +10l0 (B \E 0 (0:5; 1; 0:1; 0:2)) for
every B 2 ß([0; 1]3 ) where l0 : ß([0; 1]3 ) ! [0; 1] is a Lebesgue measure. We use probability
distributions induced by 1 for a system of complete beliefs. Note that, in this game, the
receiver needs to know probability distributions on each of the information sets to …nd her
optimal strategy. Since 1 does not de…ne probability distributions on every information
set, we have to …nd on which information sets 1 can de…ne probability distributions. So
R
we marginalize 1 over the sender’s type space s = [0; 1], which is s d 1 , and derive a
R
support13 K1 ( [0; 1]2 ) of s d 1 . Then, we have K1 = [0; 0:2] [0; 1]. Accordingly, at this
…rst step, by using 1 , we can de…ne probability distributions on the class of the information
sets [0; 1] K1 .
At the second step, we apply f n g1
n=1 to the class of all the remaining information sets,
which is [0; 1] (0:2 ; 1] [0; 1] = [0; 1]3 n([0; 1] K1 ). Then the probability measure 2;n
conditional probability distribution that assigns x1 and x4 with probability 21 and 12 , respectively. Likewise,
in the information set H2 , x2 is assigned with probability 1, and, in the information set H3 , x6 is assigned
with probability 1. Next, de…ne a system of complete beliefs according to the sequential approach. Then,
we can consider all the class of the information sets together and de…ne probability distributions step by
step. So, at the …rst step, we can de…ne the probability distribution that assigns x2 with probability 21 , x3
with probability 0, x5 with probability 0, and x6 with probability 21 . At the second step, we can consider
only H1 . Then, x1 and x4 are assigned with probability 12 and 12 , respectively. Finally, to …nish de…ning a
system of complete beliefs, we need to make a convex combination of two probability distributions derived
at each step.
13
A support K of is a (relatively) closed set satisfying 1) (K C ) = 0 and 2) if G is open and G \ K 6= ?,
then (G \ K) > 0.
12
induced by n converges to the probability measure 2 such that 2 (B) = 10l0 (B \E 0 (0; 0:5;
0:2; 0:3)) +10l0 (B \E 0 (0:5; 1; 0:3; 0:4)) for every B 2 ß([0; 1]3 ). Next, to …nd, out of the
class of the remaining information sets [0; 1] (0:2 ; 1] [0; 1], on which information sets 2
can de…ne probability distributions, we marginalize 2 over the sender’s type space s and
R
derive a support K2 = (0:2; 0:4] [0; 1] of s d 2 . Thus, at this second step, by using 2 ,
we can de…ne probability distributions on [0; 1] K2 .
From the third step, we repeat the previous process. So we apply f n g1
n=1 to the class of
all the remaining information sets, [0; 1] (0:4 ; 1] [0; 1] and derive the probability measure
5 0
0
([0; 1]3 ). Then we marginalize
3 such that 3 (B) = 3 l (B \E (0; 1; 0:4; 1)) for every B 2 ß
R
[0; 1] of s d 3 .
3 over the sender’s type space
s and derive a support K3 = (0:4; 1]
Hence, at this third step, by using 3 , we can de…ne probability distributions on [0; 1] K3 .
Since the union of the supports K1 , K2 , and K3 covers the whole of the signals, [0; 1]2 , we
can de…ne the whole of the probability distributions for a system of complete beliefs by using
the probability measures 1 , 2 , and 3 . Note that we need to make a convex combination
of the measures 1 , 2 , and 3 to de…ne probability distributions for a system of complete
beliefs because a system of complete beliefs consists of only one probability measure on the
whole class of the information sets.
Example 2 Consider the game in Crawford and Sobel (1982) again. De…ne the sender’s
1
n2
strategy n as n ( s ; S) = n+1
l(S \ ([0; n1 ]
[0; 1])) + n+1
l(S) for every s 2 [0; 1] and S 2
2
2
ß
([0; 1] ) where l : ß
([0; 1] ) ! [0; 1] is a Lebesgue measure.
We derive probability distributions from this sequence f n g1
n=1 for a system of complete
beliefs step by step. In this example, however, we cannot employ exactly the same approach
as in the previous example because a probability measure 1;n induced by n converges only
weakly, that is, it does not converge strongly.14 As a result, from the second step, we adopt
another approach di¤erent from that in the previous example.
At the …rst step, we can still employ the same approach as in the previous example. That
3
is, we can apply f n g1
n=1 to the whole class of the information sets, which is [0; 1] . Then,
the probability measure 1;n on ß([0; 1]3 ) induced by n converges weakly to the probability
measure 1 such that 1 uniformly assigns its total probability only to the area [0; 1] [0; 0]
[0; 1]. Next, we marginalize 1 over the sender’s type space s and derive a support K1
R
= f0g [0; 1] of s d 1 .
14
A measure
1;n
converges (strongly) to
1
if we have lim sup j
n !1 B
13
1;n (B)
1 (B)
j = 0.
From the second step, however, we cannot apply f n g1
n=1 to the class of all the remaining
3
information sets, which is [0; 1]
(0; 1] [0; 1] = [0; 1] n ([0; 1]
K1 ). This is because
the probability measure 2;n induced by n vanishes on every information set in [0; 1]
(0; 1] [0; 1], and thus 2;n does not converge weakly to any probability measure on [0; 1]
(0; 1] [0; 1]. To solve this problem, we divide the class of all the remaining information
sets into countably many classes of the information sets, and, according to f n g1
n=1 , we
de…ne conditional probability measures on each class of the information sets separately.
For example, for each integer j
2, at the jth step, we can consider only a class of the
1
1
information sets [0; 1]
( j ; j 1 ] [0; 1]. Then, on the set [0; 1]
( 1j ; j 1 1 ] [0; 1], we can
…nd the conditional probability measure j;n induced by n . Obviously, this conditional
probability measure j;n converges to some probability measure j , and hence we can derive
R
a support Kj = ( 1j ; j 1 1 ] [0; 1] of s d j . Since fKj gj2N is a pair-wise disjoint class and it
covers the whole of the signals, [0; 1]2 , we can de…ne the whole of the probability distributions
for a system of complete beliefs by using the probability measures f j gj2N .
4.2
Sequential convergence
In the complete sequential equilibrium, a sequence of strategies that can de…ne probability
distributions for a system of complete beliefs is referred to as a sequentially convergent
sequence of strategies. As shown in Examples 1 and 2, in each step, a sequentially convergent
sequence of strategies induces a probability measure on an open class of the remaining
information sets. Note that every probability measure, which is sequentially induced by the
sequence of strategies, has its support according to Aliprantis and Border (1999, 10.13).15
Then, the union of the supports of these sequentially induced probability measures becomes
dense in the whole class of the information sets. Therefore, we can use these probability
measures to de…ne probability distributions for a system of complete beliefs.
For notational convenience, we de…ne a probability measure with respect to a strategy
pro…le as
(B; ) =
R R
A1
R
At
1
IB ( ; a1 ; :::; at 1 )
t 1
( ; a1 ; :::; at 2 ; dat 1 )
1
( ; da1 ) (d )
0
1
I
for each set B 2 Ii=0 ß( i ) ( tt0 =1
(Ati )) where IB ( ) is an indicator function, that is,
i=1 ß
IB ( ; a1 ; :::; at 1 ) = 1 if ( ; a1 ; :::; at 1 ) 2 B and IB ( ; a1 ; :::; at 1 ) = 0 if ( ; a1 ; :::; at 1 ) 2
= B.
15
Let X be a topological space, and let be a Borel measure on X. If X is second countable, then
a support. Note that a separable metric space is always second countable.
14
has
The probability measure is well-de…ned according to Ash (1972, 2.6) and (B; ) denotes
the probability of B according to the strategy pro…le . Furthermore, given sequences of
t
t
Borel subsets fKi;j
g and open sets fOi;j
g in i A1
At 1 and a strategy pro…le n ,
de…ne a measure j;n as
(B \ (
j;n (B) =
(
i
i
t
t
(Oi;j
n [e<j Ki;e
)); n )
t
t
(Oi;j
n [e<j Ki;e
); n )
(1)
0
1
t
I
n
(Ati )). Then, for each i and t, if we have ( i (Oi;j
for each B 2 Ii=0 ß( i ) ( tt0 =1
i=1 ß
t
); n ) > 0, then j;n would be well-de…ned as a conditional probability measure on
[e<j Ki;e
0
1
t
t
I
I
) since ( ; n ) is a probability
n [e<j Ki;e
(Oi;j
( i ) ( tt0 =1
(Ati )) given
i
i=0 ß
i=1 ß
measure.16
De…nition 3 A sequence of strategy pro…les f n g1
n=1 is sequentially convergent if for
t
gj2Jit and
each i and t, there exist both a sequence of pairwise disjoint Borel subsets fKi;j
1
t
1
t
t
N such
A
with an index set Ji
a sequence of open sets fOi;j gj2Jit in i A
t
that i) Oi;1
= i A1
At 1 ; ii) the conditional probability measure j;n de…ned
by (1) is well-de…ned for every n and converges weakly17 to some probability measure j on
T
0
1
t
I
I
= fGi : Gi is relatively closed in i A1
( i ) ( tt0 =1
(Ati )); iii) Ki;j
i=0 ß
i=1 ß
t
t
is dense. Here, the sequence
and j ( i Gi ) = 1g; and iv) [j2Jit Ki;j
At 1 n [e<j Ki;e
t
g is referred to as sequential supports of f n g for player i in period t.
fKi;j
To resolve the abstractness of this de…nition, we examine how a sequentially convergent
sequence of strategy pro…les f n g operates in the general games. At the …rst step, the
sequence f n g de…nes a sequence of probability measures f 1;n g on the whole class of the
information sets
A1
At 1 in each period, and this sequence f 1;n g converges
R
weakly to a probability measure 1 . Then, the marginal probability measure
d 1 has
i
t
the smallest closed support Ki;1 , whose existence is guaranteed by Aliprantis and Border
t
(1999, 10.13). At the second step, the sequence f n g and the support Ki;1
together de…ne
16
In more detail, we have
t 1
t0 =1
i)
(
I
i=1
A1
At
1
)=
0
(
(
i
i
t
t
(Oi;j
n[e<j Ki;e
);
t n[
t
(Oi;j
e<j Ki;e );
n)
n)
= 1 and
j;n (B)
0 for
ß(Ati )). In addition, for any disjoint countable union of Borel subsets
t
t
t
t
P
(([e2E Be )\(
(Be \(
i (Oi;j n[e<j Ki;e )); n )
i (Oi;j n[e<j Ki;e )); n )
[e2E Be , we have j;n ([e2E Be ) =
=
t
t
t
t
e2E
(
(
i (Oi;j n[e<j Ki;e ); n )
i (Oi;j n[e<j Ki;e ); n )
P
t
= e2E j;n (Be ). Therefore, j;n (B) would denote the conditional probability of B given
(Oi;j
n
i
t
t
t
[e<j Ki;e ) according to n whenever ( i (Oi;j n [e<jRKi;e ); n ) > 0.
R
17
A measure j;n converges weakly to j if limn !1
f ( ; a)d j;n ( ; a) =
A1
At 1
A1
At 1
f ( ; a)d j ( ; a) for every bounded and continuous real function f on
A1
At 1 .
each B 2
I
(
i=0 ß
j;n (
15
a sequence of conditional probability measures f 2;n g given an open class of the remaining
t
t
information sets Oi;2
( i A1
At 1 n Ki;1
) and this sequence f 2;n g converges
R
weakly to a probability measure 2 . Then again, the marginal probability measure
d 2
i
1
t 1
t
t
A nKi;1 . Likewise,
has the smallest support Ki;2 that is relatively closed in i A
j
1
t
ge=1 determine a sequence of
in each step j
3, the sequence f n g and the supports fKi;e
t
conditional probability measures f j;n g, a probability measure j , and the support Ki;j
until
t
[e j Ki;e
becomes dense in i A1
At 1 .
To sum up, a sequence of strategy pro…les f n g is sequentially convergent if f n g together
t
t
g sequentially de…ne a sequence of
g and open sets fOi;j
with its sequential supports fKi;j
conditional probability measures f j;n g such that each sequence f j;n g1
converges weakly
R n=1
to a probability measure j and its marginal probability measure
d j has the smallest
i
t
and relatively closed support Ki;j such that their union is dense.
t
In particular, the condition i) requires that sequences of open sets fOi;j
g, speci…cally the
t
open sets at the …rst step fOi;1
g, cover the whole set i A1
At 1 so that the sequential
t
t
g, can cover all the
g, speci…cally the sequential supports at the …rst step fKi;1
supports fKi;j
information sets on the equilibrium path, and hence a complete sequential equilibrium can
t
satisfy the Nash equilibrium condition. Next, the condition ii) requires that f n g, fKi;j
g,
t
t
and fOi;j
g sequentially de…ne f j;n g and f j g. The condition iii) requires that Ki;j
be the
R
t
smallest and relatively closed support of
d j so that Ki;j is uniquely determined and
i
t
has positive measure according to j . Finally, the condition
every open set in
Ki;j
i
t
t
iv) requires that the union [j2Jit Ki;j
be dense so that [j2Jit ( i Ki;j
) …lls the whole space
A1
At 1 fully enough.
If the space
A1
AT is complete and totally bounded,18 then, given any
convergent sequence of totally mixed strategies f n g1
n=1 , we can always …nd subsequence
1
f n0 gn0 =1 that is a sequentially convergent sequence of strategies. Moreover, in this metric
space
A1
AT , Balder (1988, Theorem 2.3.a)19 proved that the set of strategy
pro…les
is weakly compact if it is complete and totally bounded. Therefore, from any
sequence of totally mixed strategy pro…les, we can …nd its subsequence that is sequentially
convergent. The following algorithm presents a recursive procedure for de…ning sequential
t
t
supports fKi;j
gj2Jit and an open cover fOi;j
gj2Jit according to a given convergent sequence
1
of totally mixed strategies f n gn=1 .
18
A metric space X is totally bounded if, for each " > 0, there exists a …nite subsets fx1 ; ::: ; xn g X
such that a collection of " balls B" (xi ) covers X.
19
If the space A1
AT is a compact metric space, then is weakly compact. Note that a metric
space is compact if and only if it is complete and totally bounded.
16
Note that the following algorithm simply shows that an arbitrary sequence of totally
1
mixed strategy pro…les f n g1
n=1 has its subsequence f n0 gn0 =1 that is sequentially convergent.
However, we can derive this sequentially convergent sequence f n0 g1
n0 =1 from the sequence
t
1
f n gn=1 only by carefully choosing the sequential supports fKi;j gj2Jit and the open cover
t
t
gj2Jit and an arbitrary open cover
gj2Jit . That is, choosing arbitrary supports fK_ i;j
fOi;j
t
_
fOi;j gj2Jit might not result in a sequentially convergent sequence.20
Algorithm Suppose that
A1
AT is complete and totally bounded and f n g1
n=1
is a convergent sequence of totally mixed strategy pro…les as de…ned in Kreps and
t
Wilson (1982). At the …rst step, for each i and t, the open set Oi;1
is de…ned as
1
t 1
the whole class of the information sets i A
A . Then, the probability
measure 1;n induced by n is well-de…ned and converges weakly to some measure
R
d 1 has the smallest closed support
1 . Next, the marginal probability measure
i
t
, whose existence is guaranteed by Aliprantis and Border (1999, 10.13). From
Ki;1
the second step, we de…ne each step recursively. At the jth step for j
2, given
t
the class of the sequential supports fKi;e
ge<j , which were induced at the previous
t
steps, the jth open set Oi;j can be a non-empty open set in i A1
At 1 n O
t
t
,
. Then, conditional on reaching
Oi;j
such that O is a neighborhood of [e<j Ki;e
i
the probability measure j;n induced by n is well-de…ned since n is totally mixed.
Moreover, by choosing a subsequence f j;n0 g of f j;n g if necessary, we can …nd the
probability measure j to which j;n0 converges weakly since the set of the probability
measures is weakly compact according to Balder (1988, Theorem 2.3.a). Next, the
R
t
which is relatively
marginal probability measure
d j has the smallest support Ki;j
i
1
t 1
t 21
closed in i A
A n [e<j Ki;e . We can repeat this recursively-de…ned step
countably many times since each strategy pro…le n is totally mixed. Then, since the
t
space
A1
AT is separable, we can …nd sequential supports fKi;j
gj2Jit with
t
an index set Jit N such that fKi;j
gj2Jit is dense in i A1
At 1 .
In conclusion, Proposition 1 formulates properties of the sequentially convergent sequence
of strategies.
20
The same problem arises when we …nd a one-to-one function from [0; 1] onto R since not every one-to-one
function has R as its range.
21
t
t
t
t
t
Note that we always have Ki;j
\ ([e<j Ki;e
) = ?, but we might not have Ki;j
Oi;j
. That is, fKi;j
g
t
t
is pairwise disjoint. However, Oi;j might not include Ki;j .
17
Proposition 1 Suppose that the separable metric space
A1
AT is complete and
totally bounded.22 Then, any sequence of totally mixed strategy pro…les f n g1
n=1 has its subsequence f n0 g1
n0 =1 that is sequentially convergent. In addition, any sequentially convergent
sequence of strategies can uniquely de…ne a system of complete beliefs.
Proof. The results directly follow from the Algorithm and the de…nition.
5
Complete sequential equilibrium
Now, we present a formal de…nition of the complete sequential equilibrium in the general
games based on the complete beliefs and the sequentially convergent sequence of strategies.
First, we de…ne a complete assessment in the general games.
De…nition 4 An ordered pair of a system of complete beliefs and a strategy pro…le ( ; ) is
a complete assessment.
The complete sequential equilibrium, as an improved version of the sequential equilibrium, is designed to preserve all the conditions for the sequential equilibrium, which are the
consistency and the sequential rationality. As a result, the complete sequential equilibrium
is de…ned as a complete assessment that satis…es both 1) the consistency and 2) the sequential rationality in the general games. De…nition 5 below formally de…nes the …rst condition,
consistency.
De…nition 5 A complete assessment ( ; ) is consistent if there exists a sequentially convergent sequence of strategies f n g such that i) n converges weakly23 to and ii) each
complete belief ti satis…es
t
i (B)
=
P
j2Jit (
1 #fe2Jit :e
)
2
j and e<sup Jit g
j (B)
0
1
I
for every B 2 Ii=0 ß
( i ) ( tt0 =1
(Ati )) where the index set Jit and each probability
i=1 ß
measure j are de…ned according to the same way as in De…nition 3 and #fe 2 Jit : e j
and e < sup Jit g denotes the number of elements e in Jit such that e j and e < sup Jit .
22
Note that a metric space is compact if and only
R R if itRis completeT and 1totallyT bounded.
1
1
A strategy pro…le n converges weakly to if
f ( ; a) n ( ; a ; :::; a 1 ; daT )
n ( ; da )d ( )
A1
AT
R R
R
T
1
1
T 1
T
1
converges to
f ( ; a) ( ; a ; :::; a
; da )
( ; da )d ( ) for every bounded and continuous
A1
AT
real function f on
A1
AT .
23
18
De…nition 5 makes sense according to Proposition 1. Intuitive explanation of this definition is presented in Section 6 when we compare the consistency with convex structural
consistency.
Next, De…nition 6 below formulates the second condition for the complete sequential
equilibrium, sequential rationality, in the general games. Let
be the set of all systems
of complete beliefs. For notational convenience, given i and t, de…ne a conditional expected
0
1
I
utility functional Eit :
ß( i ) ( tt0 =1
(Ati0 ))
! R (= R [ f 1; 1g) as
i0 =1 ß
R
R
R
T
t
t
t
Gi ) Ei ( ; Gi ; ) =
U ( ; a) ( ; a1 ; :::; aT 1 ; daT )
( ; a1 ;
i
t
i(
AT i
i Gi A
:::; at 1 ; dat )d ti ( ; a1 ; :::; at 1 ).
De…nition 6 A strategy pro…le is sequentially rational with respect to a system of
complete beliefs if, for each i and t, we have Eit ( ; Gi ; )
Eit ( ; Gi ; ( 0i ; i )) for every
0
0
1
I
( i ) ( tt0 =1
(Ati0 )) such that ti ( i Gi ) > 0.
i and for every Gi 2 ß
i 2
i0 =1 ß
Here,
Gi denotes a class of player i’s information sets. Thus, the sequential
i
rationality requires, in responding to the other players’strategies i , each player i to play
her best responses i , which induces the greatest expected utility conditional on reaching
the class of the information sets
Gi which has positive probability according to the
i
system of complete beliefs , that is, ti ( i Gi ) > 0. As a result, no player prefers
to change their strategies at any open class of the information sets. Note that Kreps and
Wilson (1982) described the sequential rationality as the condition under which ‘taking the
beliefs as …xed, no player prefers at any information set to change his part of the strategy.’
Therefore, De…nition 6 adapts the sequential rationality from the sequential equilibrium to
the general …nite-period games by replacing ‘any information set’ with ‘any open class of
the information sets.’
Finally, De…nition 7 de…nes the complete sequential equilibrium in the general games.
De…nition 7 A complete assessment ( ; ) is a complete sequential equilibrium if ( ; )
is both 1) consistent and 2) sequentially rational.
6
Properties of the complete sequential equilibrium
This section reveals four properties of the complete sequential equilibrium. The …rst property
is about the existence condition of a complete sequential equilibrium.
19
Theorem 1 Suppose that the separable metric space
A1
AT is complete and totally
bounded and each von Neumann-Morgenstern utility function Ui is bounded and continuous.
Then, there exists a complete sequential equilibrium.
Proof. See Appendix.
The second property is about the relationship between the convex structural consistency
and the consistency for the complete sequential equilibrium. Here, the convex structural
consistency, as a criterion of rational beliefs, is one of the two conditions that we suggest
for an improved solution concept in the general games. Note that Kreps and Ramey (1987)
originally de…ned it in …nite games. So we start by extending their de…nition of convex
structural consistency to the general games in De…nitions 8, 9, and 10.
Kreps and Ramey (1987) de…ned the convex structural consistency as a consistency
criterion under which the players’ beliefs should re‡ect the information on the structure
of a game by means of convex combinations of the players’ strategies. Thus, under this
consistency criterion, if players would be unexpectedly located, they should then form their
beliefs such that a convex combination of strategies can induce their beliefs. Note that the
players have information on their own types, but no information on the actual state 0 2 0 .
Thus, a convex combination of the players’strategies does not a¤ect conditional probability
distributions on the state space 0 given types 0 2
0 , and as a result the players’beliefs
induced by a convex combination of strategies contain almost surely the same conditional
probability distributions on 0 given 0 that are derived from , which is the probability
measure on Ii=0 ß( i ). We regard these conditional probability distributions on 0 given 0
as the information on the structure of a general game. Consequently, the convex structural
consistency24 requires that the players’ beliefs to re‡ect this information. In De…nition 8
below, we …rst formulate a convex combination of the players’strategies in the general games
and refer to it as a convex structurally consistent behavioral measure, which is meant to be a
behavioral measure that can preserve the information on the structure of the general games.
24
In fact, this convex structural consistency is a weak version of the structural consistency in Kreps and
Wilson (1982). Kreps and Wilson de…ned the structural consistency as a consistency criterion under which
the beliefs of the players should re‡ect the informational structure of a game by means of a single strategy
pro…le. Thus, this structural consistency requires players to use only one strategy pro…le to form each
of their beliefs. Because of the strong requirement, however, most of the solution concepts, including the
perfect equilibrium, do not satisfy this criterion even in …nite games. For more information on the structural
consistency and its relation to the consistency and the convex structural consistency, please refer to Osborne
and Rubinstein (1994, 12.2) and Kohlberg and Reny (1997).
20
0
1
I
De…nition 8 For each i and t, a measure ti :
( tt0 =1
(Ati0 )) ! [0; 1] is convex
i0 =1 ß
structurally consistent behavioral if it satis…es the following three conditions. First, for
0
1
I
each 2 , ti ( ; ) is a probability measure on tt0 =1
(Ati0 ). Second, for every B 2
i0 =1 ß
0
t 1
I
( i ) measurable. Finally, there exists a set 0 in
such
(Ati0 ), ( ; B) is Ii=0 ß
t0 =1 i0 =1 ß
0
0
t
1
that ( 0 ) = 1 and, for every B 2 t0 =1 Ii0 =1 ß
(Ati0 ) and ; 2 0 , we have ti ( 0 ; 0 ; B)
I
= ti ( 00 ; 0 0 ; B) whenever 0 = 0 0 2
0 (=
i=1 i ).
The …rst condition requires that each ti ( ; ) specify what to play given each 2 .
The second condition requires that ti allow a well-de…ned expected utility functional. A
convex structurally consistent behavioral measure ti , as a convex combination of the players’
strategies, inherits these two properties from the players’strategies since every strategy has
these two properties. In fact, these two conditions are all requirements for the strategies,
so they make each ti ( ; ) a behavioral measure that simply shows overall behavior by the
players. The third condition requires that, for almost all and 0 , if we have 0 = 0 0 ,
two measures ti ( ; ) and ti ( 0 ; ) be identical. This property of ti comes from the structure
of a general game. In a general game, the players have information on 0 2
0 , but no
information on the actual state 0 2 0 . So, whatever actions they choose, the same actions
must be applied to all decision points associated with a common 0 . For this reason, ti ( ; )
and ti ( 0 ; ) almost surely become identical if we have 0 = 0 0 , and therefore, because
of this property, a convex structurally consistent behavioral measure ti can preserve the
information on the structure of the general games.
Next, in De…nition 9 below, we de…ne a conditional probability measure 0 on ß( 0 )
given Ii=1 ß( i ), which is regarded as the information on the structure of a general game.
We refer to it as a convex structurally consistent informational measure, which is meant to
be an informational measure that re‡ects the structure of the general games. Note that
three conditions in De…nition 9 denote that 0 is simply a regular conditional probability
measure on ß( 0 ) given Ii=1 ß( i ).
De…nition 9 A measure 0 :
ß
( 0 ) ! [0; 1] is convex structurally consistent
0
informational if it satis…es the following three conditions. First, for each 0 2
0,
0
0
( 0 ). Second, for every 0 2 ß
( 0 ), 0 ( ; 0 ) is
0 ; ) is a probability measure on ß
0(
I
( i ) measurable. Finally, for every 0 2 Ii=0 ß
( i ), we have
i=1 ß
( 0) =
where
0
R
0
R
0
I 0(
0; 0) 0(
is the marginal probability measure of
21
0; d 0)
0 (d
0)
obtained by marginalizing
over
0.
Finally, in De…nition 10 below, we de…ne the convex structural consistency in the general
games.
De…nition 10 A system of complete beliefs satis…es convex structural consistency
if, for each i and t, there exists a sequence of pairwise disjoint Borel sets fGti;j gj2Jit in
A1
At 1 with an index set Jit
N such that i) [j2Jit Gti;j is dense and ii) for
i
each j, there exist both a convex structurally consistent behavioral measure ti and a convex
structurally consistent informational measure 0 that together satisfy the functional equation
t
i (E)
=
R
0
R R
0
A1
At
for every Borel set E
Gti;j where
i
measure of ti obtained by marginalizing
1
IE ( ; a) ti ( ; da) 0 (
t
i
0; d 0)
(d
0)
: Ii=1 ß
( i ) ! [0; 1] is the marginal probability
over 0 A1
At 1 .
In short, a system of complete beliefs satis…es the convex structural consistency if it
consists of convex structurally consistent behavioral measures ti and convex structurally
consistent informational measures 0 so that can re‡ect the information on the structure
of the games, which are 0 , by means of convex combinations of the players’ strategies,
which are ti . Note that a system of complete beliefs is de…ned as a set of probability
measures on the overall information sets in each period while a system of beliefs _ is de…ned
as a set of conditional probability measures on each of the information sets. So, because of
this di¤erence, our approach to de…ne the convex structural consistency di¤ers from that in
Kreps and Ramey (1987) in two aspects. First, our de…nition shows the convex structural
consistency on each of the classes of information sets f i Gti;j g, while their de…nition
showed it on each of the information sets separately. Second, our de…nition requires including
convex structurally consistent behavioral measures ti , while their de…nition did not.
Now, we are ready to examine the relationship between the convex structural consistency
and the consistency for the complete sequential equilibrium.
Proposition 2 Suppose that the separable metric space
is topologically complete, and
variables 0 and 0 are independent.25 Then, every system of complete beliefs in a consistent assessment ( ; ) (hence in every complete sequential equilibrium) satis…es the convex
structural consistency.
25
(E0
That is, ß( 0 ) and ß(
E 0 ) = (E0
0)
0)
(
are independent so that, for every E0 2 ß(
E 0 ).
0
22
0)
and E
0
2 ß(
0 ),
we have
Proof. See Appendix.
Proposition 2 means that, if the separable metric space is also topologically complete
and if variables 0 and 0 are independent, then every complete sequential equilibrium
satis…es the convex structural consistency in the general games. Therefore, under these
two assumptions, a system of complete beliefs in a complete sequential equilibrium always
re‡ects the information on the structure of the games by means of convex combinations of the
players’strategies. Here, under the …rst assumption, topological completeness of , regular
conditional probability measures are well-de…ned. Thus, this assumption guarantees that
both convex structurally consistent behavioral measures and convex structurally consistent
informational measures are well-de…ned, both of which are de…ned as regular conditional
probability measures. Furthermore, under the second assumption, the independence of ß( 0 )
and ß( 0 ), the weak convergence of aggregate measures implies the weak convergence
of their partial regular conditional probability measures. Hence, this assumption guarantees
that the weak convergence of systems of complete beliefs implies the weak convergence
of convex structurally consistent behavioral measures and the weak convergence of convex
structurally consistent informational measures, and as a result a system of complete beliefs
in a complete sequential equilibrium satis…es the convex structural consistency.
In Proposition 2, the two assumptions are not only su¢ cient conditions but also necessary
conditions to assure the convex structural consistency of a consistent system of complete
beliefs in the general games. That is, without these assumptions, some consistent systems
of complete beliefs might fail to satisfy the convex structural consistency. We will illustrate
this necessity of the independence assumption in Proposition 2.
We use the following Lemma 1 to explain Proposition 2.
Lemma 1 Let X1 and X2 be metric spaces and let 1n : ß
(X1 ) ! [0; 1] and 2n : ß
(X2 )
! [0; 1] be probability measures, which also means that ß
(X1 ) and ß
(X2 ) are independent.
1
2
In addition, let n be the product measure of n and n on ß
(X1 )
ß
(X2 ) and converge
weakly to a probability measure : ß
(X1 ) ß
(X2 ) ! [0; 1]. Finally, suppose that there exist
probability measures 1 : ß
(X1 ) ! [0; 1] and 2 : X1
ß
(X2 ) ! [0; 1] such that (E)
R R
2
1
= X1 X2 IE (x1 ; x2 ) (x1 ; dx2 ) (dx1 ) for every E 2 ß
(X1 ) ß
(X2 ). Then, 1n converges
weakly to 1 . Moreover, there exists a set X10 2 ß
(X1 ) such that 1 (X10 ) = 1 and, for
each x1 2 X10 , 2n ( ) converges weakly to 2 (x1 ; ), which also means that, if 2n ( ) converges
R R
2
weakly to a probability measure
: ß
(X2 ) ! [0; 1], then we have (E) = X1 X2 IE (x1
2
; x2 ) (dx2 ) 1 (dx1 ) for every E 2 ß
(X1 ) ß
(X2 ).
23
Proof. See Appendix.
In general, without the assumption of the independence between ß(X1 ) and ß(X2 ), the
weak convergence of n (= 1n 2n ) to (= 1 2 ) still guarantees the weak convergence of
1
1
. However, it does not guarantee the weak convergence of 2n to 2 . The following
n to
Example 3 exempli…es the necessity of this independence assumption in Lemma 1.
[0; 1] ! [0; 1] de…ned as 1n (E) = n l(E\(0; n1 ))
Example 3 Let a probability measure 1n : ß
for every E 2 ß
[0; 1] where l : ß
[0; 1] ! [0; 1] is a Lebesgue measure. Next, let a probability
2
ß
[0; 1] ! [0; 1] de…ned as 2n (x1 ; E) = 1 either if x1 2 (0; 1] and E
measure n : [0; 1]
2 ß
[0; 1] contains x2 = 1 or if x1 = 0 and E 2 ß
[0; 1] contains x2 = 0. In addition, let
1
a probability measure
:ß
[0; 1] ! [0; 1] de…ned as 1 (E) = 1 if and only if E 2 ß
[0; 1]
2
contains x1 = 0. Finally, let a probability measure
: [0; 1]
ß
[0; 1] ! [0; 1] de…ned as
2
(x1 ; E) = 1 either if x1 2 (0; 1] and E 2 ß
[0; 1] contains x2 = 0 or if x1 = 0 and E 2
ß
[0; 1] contains x2 = 1.
Then, the product measure 1n 2n converges weakly to 1 2 , and also 1n converges weakly
to 1 . However, for any x1 2 [0; 1], 2n (x1 ; ) does not converge weakly to 2 (x1 ; ). Moreover,
it is easy to see that n is absolutely continuous with respect to the product measure ^
1
1 2
= ^ n ^ n where the probability measure ^ n : ß[0; 1] ! [0; 1] is a marginal probability measure
2
of n obtained by marginalizing n over X2 and ^ n : ß[0; 1] ! [0; 1] is another marginal
probability measure obtained by marginalizing n over X1 . Therefore, Example 3 also shows
that the absolute continuity is not enough to guarantee the outcomes in Lemma 1.
By using Example 3, we can illustrate the necessity of the independence assumption in
Proposition 2 as well. Consider the game in Crawford and Sobel (1982) and revise it by
adding the state space 0 = [0; 1]. So the sender has information about her type s , but no
information about the actual state 0 2 [0; 1]. A convex structurally consistent informational
measure 0 , which is a conditional probability measure on ß( 0 ) given ß( s ) and is regarded
as the information on the structure of this general game, is de…ned according to the same way
as in Example 3. That is, the conditional probability measure 0 is de…ned as 0 ( s ; 00 ) = 1
either if s 2 (0; 1] and 00 2 ß[0; 1] contains 0 = 1 or if s = 0 and 00 2 ß[0; 1] contains
1
0 = 0. Next, for each n, de…ne the sender’s strategy n as, when s 2 (0; n ), n ( s ; )
= l and, when s 2
= (0; n1 ), n ( s ; f(0; 0)g) = 1 where l : ß([0; 1]2 ) ! [0; 1] is a Lebesgue
measure. Then, this sequence of the sender’s strategies is sequentially convergent and, within
two steps, it well de…nes a system of complete beliefs. However, at the second step, this
sequence does not induce the convex structurally consistent informational measure 0 , and
24
thus this consistent system of complete beliefs does not satisfy convex structural consistency.
Therefore, this example proves that the independence assumption in Proposition 2 is a
necessary condition to assure the convex structural consistency of a consistent system of
complete beliefs in the general games.
The third property of the complete sequential equilibrium is that it always satis…es the
Nash equilibrium condition in the general games. The Nash equilibrium condition, as a
criterion of rational strategies, is the other condition that we suggest for an improved solution
concept in the general games.
Proposition 3 Every complete sequential equilibrium is a Nash equilibrium.
Proof. The result follows from the de…nitions.
The …nal property of the complete sequential equilibrium is about its relationship to
the sequential equilibrium in …nite games. For notational simplicity, we de…ne a system of
beliefs26 associated with a system of complete beliefs as _ ( ). That is, _ ( ) is a system of
beliefs inducing the same probability distributions on each of the information sets as . It
is easy to see that, in …nite games, every system of complete beliefs uniquely determines its
associated system of beliefs _ ( ), but not vice versa.
Proposition 4 In …nite games, an assessment ( _ ; ) is consistent if and only if there exists
a consistent complete assessment ( ; ) such that _ ( ) = _ .
Proof. The result follows from the de…nitions.
In Proposition 4 above, the equivalence between the two conditions results from the
equivalence between the sequentially convergent sequence of strategies and the convergent
sequence of totally mixed strategies in …nite games. In …nite games, a sequence of strategies is sequentially convergent if and only if it is a convergent sequence of totally mixed
strategies. Since a sequentially convergent sequence of strategies induces a consistent system of complete beliefs for a complete sequential equilibrium and a convergent sequence of
totally mixed strategies induces a consistent system of beliefs for a sequential equilibrium,
the consistency condition for the complete sequential equilibrium is equivalent to the consistency condition for the sequential equilibrium. Note that, in …nite games, the complete
sequential equilibrium requires the same sequential rationality condition as the sequential
26
Kreps and P
Wilson (1982) de…ned a system of beliefs _ as a function from a set of all decision nodes to
[0; 1] such that x2h _ (x) = 1 for every information set h.
25
equilibrium. Therefore, Proposition 4 also implies that, in …nite games, the complete sequential equilibrium is equivalent to the sequential equilibrium. Theorem 2 below, as a corollary
of Proposition 4, formally presents this equivalence between the two solution concepts in
…nite games.
Theorem 2 In …nite games, an assessment ( _ ; ) is a sequential equilibrium if and only if
there exists a complete sequential equilibrium ( ; ) such that _ ( ) = _ .
To sum up, Proposition 2 guarantees that the complete sequential equilibrium satis…es
the convex structural consistency under the two assumptions, namely, the topological completeness of and the independence of ß( 0 ) and ß( 0 ). In addition, Proposition 3 assures
that it satis…es the Nash equilibrium condition. Note that the convex structural consistency
and the Nash equilibrium condition have been suggested as the criteria of an improved solution concept of the sequential equilibrium in general games. Hence, Propositions 2 and 3
demonstrate that the complete sequential equilibrium indeed improves the sequential equilibrium in the general games. Moreover, Theorem 1 presents the existence condition of the
complete sequential equilibrium and Theorem 2 shows that, in …nite games, the complete
sequential equilibrium is equivalent to the sequential equilibrium, which evidence that the
complete sequential equilibrium can be a version of the sequential equilibrium. Therefore, we
conclude that the complete sequential equilibrium is an improved version of the sequential
equilibrium in the general games.
7
Complements and comments
In …nite games, the perfect equilibrium, introduced by Selten (1975), and the perfect Bayesian
equilibrium, formulated by Fudenberg and Tirole (1991a), are closely related to the sequential
equilibrium in that every perfect equilibrium is a sequential equilibrium and every sequential
equilibrium is a perfect Bayesian equilibrium. These equilibrium concepts can be converted
for the general games. Then, it is natural to ask whether their converted versions maintain
their close relationship in the general games. Hence, as a complement to the previous study,
the current section answers this question and shows they do not.
We …rst de…ne a converted version of the perfect equilibrium.
26
De…nition 11 For " > 0 and given a totally mixed strategy pro…le , an "
-constrained
"
27
equilibrium is a totally mixed strategy pro…le ( ) such that for each i, the strategy "i ( )
solves max i Ei ( i ; " i ( )) subject to i = " i + (1 ") 0i for some 0i 2 i . A strategy pro…le
is a perfect equilibrium if there exists a sequence of "n n -constrained equilibria f "n ( n )g
such that i) "n ( n ) converges weakly to and ii) "n converges to zero.
That is, a strategy pro…le is a perfect equilibrium if there exists a sequence of totally
mixed strategy pro…les f "n ( n )g such that i) the sequence f "n ( n )g converges weakly to
the strategy pro…le and ii) each strategy pro…le "n ( n ) in the sequence constitutes mutual
best responses under some constraint that disappears gradually. According to this de…nition, a perfect equilibrium might not be a complete sequential equilibrium. This is because
convergence of strategy pro…les does not mean convergence of expected utilities if utility
functions are unbounded or discontinuous. As a result, a perfect equilibrium could fail to be
even a Nash equilibrium, and therefore it could fail to be a complete sequential equilibrium.
Next, we de…ne a converted version of the perfect Bayesian equilibrium and call it a
perfect complete equilibrium.
De…nition 12 A complete assessment ( ; ) is a perfect complete equilibrium if ( ; )
is both 1) reasonably consistent and 2) sequentially rational.
In other words, the perfect complete equilibrium is a complete assessment which satis…es
1) that the system of complete beliefs is reasonably consistent with the strategy pro…le and 2)
that the strategy pro…le is sequentially rational with respect to the system of complete beliefs.
The de…nition of reasonable consistency is as follows. For notational convenience, we de…ne
a probability measure with respect to a system of complete beliefs and a strategy pro…le
R
R
as (B; ; ) =
I ( ; a1 ; :::; at 1 ) t 1 ( ; a1 ; :::; at 2 ; dat 1 )d ti 1 ( ; a1 ; :::; at 2 )
A1
At 2 At 1 B
0
1
I
for every set B 2 Ii=0 ß( i ) ( tt0 =1
(Ati )).
i=1 ß
De…nition 13 A complete assessment ( ; ) is reasonably consistent if i) for every i,
we have 1i =
and ii) for each i and t
2, there exists pti 2 (0; 1] such that ti (B)
T
= pti
(B; ; ) for every Borel subset B in
Kit where Kit = fGi : Gi is closed in
i
A1
At 1 and ( i Gi ; ; ) = 1g.
i
27
This "
-constrained equilibrium is named after the “"-constrained equilibrium” in Fudenberg and
Tirole (1991b, 8.4.1)
27
That is, a system of complete beliefs ( ; ) is reasonably consistent if each complete belief
and the strategy pro…le together de…ne the distribution of the next period complete
belief t+1
on the class of the information sets
Kit where Kit is the smallest and closed
i
i
support of the marginal probability measure obtained by marginalizing the product measure
t t
over
i . According to this de…nition, the consistency in the complete sequential equii
librium does not mean the reasonable consistency in the perfect complete equilibrium. This
happens because the consistency and the reasonable consistency place di¤erent restrictions
o¤ the equilibrium path. Therefore, a complete sequential equilibrium might not be a perfect
complete equilibrium in the general games.
Note that De…nition 13 covers only the …rst condition out of the three consistency conditions for the perfect Bayesian equilibrium by Fudenberg and Tirole (1991a). Thus, it does
not cover the “no-signaling-what-you-don’t-know” conditions that restrict beliefs o¤ the
equilibrium path. Basically, these no-signaling-what-you-don’t-know conditions function as
restrictions to make the perfect Bayesian equilibrium close to the sequential equilibrium. In
general games, however, the reasonable consistency, which is a converted version of the …rst
condition for the perfect Bayesian equilibrium, already di¤ers from the consistency, which
is a converted version of the consistency condition for the sequential equilibrium. Thus, nosignaling-what-you-don’t-know conditions do not function as well as they do in …nite games.
Accordingly, these conditions are excluded in De…nition 13 for the sake of simplicity.
t
i
Appendix
Proof of Theorem 1. The proof uses the same approach as in the proof of Theorem 5
in Selten (1975). That is, we …rst show that there exists a Nash equilibrium in every game
with a totally mixed strategy space. Then, we prove that a limit of the Nash equilibria is a
complete sequential equilibrium.
For each player i, since Ui is bounded and continuous, so is the expected utility functional
Ei :
! R. In particular, the expected utility functional Ei is continuous relative to the
weak topology on . Moreover, since Ei is linear and the set of the strategy pro…les is
convex, Ei is quasiconcave. Lastly, let be a totally mixed strategy pro…le and, for each
n 2 N, de…ne a set of strategy pro…les ( ; n) as ( ; n) = f : = n1 + (1 n1 ) 0 for some
0
2 g. Then, since the set of the action pro…les A1
AT is a non-empty compact
separable metric space, is a non-empty compact topological vector space for the quotient
topology according to Balder (1988 Thm 2.3.a), and so is ( ; n) for each n. Moreover
28
is convex, hence so is ( ; n) for each n. Therefore, according to Reny (1999 Thm 3.1),28
for every n, there exists a strategy pro…le n such that, for each i, player i’s strategy ( n )i
solves max i 2 i ( ;n) Ei ( i ; ( n ) i ) where (( n ) i ; ( n )i ) = n .
Next, since the sequence f n g is totally mixed, according to Proposition 1, there exists a
subsequence f n0 g of f n g such that f n0 g is a sequentially convergent sequence of strategies.
Then, since the space is weakly compact, there exists a strategy pro…le (2 ) to which
is sequentially
the subsequence f n0 g converges weakly. Now, it su¢ ces to show that
rational with respect to the system of complete beliefs
that is induced by f n0 g according
to the same way as in De…nition 5. Then, again, it su¢ ces to shows that, for each sequential
t
t
t
; ( i ; i )) for every i 2 i where Eit :
support Ki;j
, we have Eit ( ; Ki;j
; ) Eit ( ; Ki;j
0
1
I
ß( i ) ( tt0 =1
(Ati0 ))
! R is the conditional expected utility functional de…ned
i0 =1 ß
R
R
R
t
t
t
U ( ; a) T ( ; a1 ; :::; aT 1 ; daT )
( ; a1 ;
as i ( i Gi ) Ei ( ; Gi ; ) =
t
AT i
i Gi A
:::; at 1 ; dat )d ti ( ; a1 ; :::; at 1 ).
t
Suppose, by way of contradiction, that there exists a strategy 0i 2 i such that Eit ( ; Ki;j
;
0
t
t
t
Ei ( ; Ki;j ; ( i ; i )). Then, since Ei is continuous relative to the weak topology on , there
exist both a positive real " and an open set 0
such that i) ( 0i ; i ) 2 0 and ii)
t
t
Eit ( ; Ki;j
; ) > Eit ( ; Ki;j
; ) + 2" for every 2 0 . In addition, since the sequence of the
strategy pro…les f n0 g induces the system of complete beliefs and since the strategy pro…le
t
t
;
n [e<j Ki;e
, for su¢ ciently large N 2 N, we have Eit ( ( n00 ); Oi;j
n0 converges weakly to
t
t
) + " for every n00 in f n0 g such that n00
N where ( n00 ) is the
n00 ) < Ei ( ; Ki;j ;
t
system of complete beliefs that is consistent with n00 and Oi;j is the associated open class of
information sets, introduced in De…nition 3. Finally, since the open set 0 includes ( 0i ; i )
and the sequence f n0 g induces , there exist strategy pro…les 00 and 000 in 0 such that i)
00
00
00
00
t
t
t
i ( ; n ) for su¢ ciently large n and ii) Ei ( ( n00 ); Oi;j n [e<j Ki;e ;
i = ( n00 ) i and i 2
00
t
; 000 ) " where (( n00 ) i ; ( n00 )i ) = n00 .
) > Eit ( ; Ki;j
t
t
t
t
Consequently, we have Eit ( ( n00 ); Oi;j
n [e<j Ki;e
; 00 ) > Eit ( ( n00 ); Oi;j
n [e<j Ki;e
; n00 ).
This inequality shows that, responding to ( n00 ) i , player i’s strategy ( n00 )i is not the best
reply under the constraint of his strategy space i ( ; n00 ). That is, he can increase his
expected utility by choosing a strategy that combines the strategy ( n00 )i with the strategy
00
00
i ( ; n ). This contradicts our hypothesis that
i , both of which are in the convex set
the strategy pro…le n00 consists of the best replies under the constraint ( ; n00 ). This
28
For each i, if the set of player i’s strategy pro…les i is a non-empty compact convex topological vector
space and if his utility functional Ei is bounded, quasiconcave, and better-reply secure, then there exists a
Nash equilibrium
2 . Note that, if Ei is continuous, then it is better-reply secure.
29
)<
contradiction completes the proof.
Proof of Lemma 1.
For the …rst assertion, consider an arbitrary bounded and conR R
tinuous real function f1 on X1 . Then, we have limn !1 X1 X2 f1 (x1 ) 2n (dx2 ) 1n (dx1 ) =
R R
R
R
R
limn !1 X1 f1 (x1 ) X2 2n (dx2 ) 1n (dx1 ) = limn !1 X1 f1 (x1 ) 1n (dx1 ) = X1 X2 f1 (x1 ) 2 (x1 ; dx2 )
R
1
(dx1 ) = X1 f1 (x1 ) 1 (dx1 ). Thus, 1n converges weakly to 1 .
For the second assertion, consider arbitrary bounded and continuous real functions f1 on
X1 and f2 on X2 . Then, we have
R R
R R
2
1
f
(x
)f
(x
)
(x
;
dx
)
(dx
)
f (x )f (x ) 2 (dx2 ) 1n (dx1 )
1
1
2
2
1
2
1
X1 X2
X1 X2 1 1 2 2 n
R R
R R
= X1 X2 f1 (x1 )f2 (x2 ) 2 (x1 ; dx2 ) 1 (dx1 )
f (x )f (x ) 2 (dx2 ) 1 (dx1 )
X1 X2 1 1 2 2 n
R R
R R
+ X1 X2 f1 (x1 )f2 (x2 ) 2n (dx2 ) 1 (dx1 )
f (x )f (x ) 2 (dx2 ) 1n (dx1 )
X1 X2 1 1 2 2 n
R
R
2
1
= X1 f1 (x1 ) X2 f2 (x2 )f 2 (x1 ; dx2 )
n (dx2 )g (dx1 )
R
R
1
+ X2 f2 (x2 ) 2n (dx2 ) X1 f1 (x1 )f 1 (dx1 )
n (dx1 )g.
Here, f2 ( ) is bounded and
R
2
f (x )f 1 (dx1 )
n (dx2 )
X1 1 1
lim
n !1
R
1
n
f (x )
X1 1 1
converges weakly to
1
n (dx1 )g = 0. Since
R
f (x )f
X2 2 2
2
1
1
n
(x1 ; dx2 )
. Hence, we have limn
2
n converges weakly to
2
1
n (dx2 )g (dx1 )
= 0.
R
f2 (x2 )
, we have
!1 X2
1 2
(2)
Note that the equation (2) above holds for arbitrary bounded and continuous real functions
f1 on X1 and f2 on X2 . Therefore, there exists a set X10 X1 such that 1 (X10 ) = 1 and for
each x1 2 X10 , 2n ( ) converges weakly to 2 (x1; ). Let 2n ( ) converge weakly to a probability
R R
2
2
measure
: ß(X2 ) ! [0; 1]. Then, we have (E) = X1 X2 IE (x1 ; x2 ) (dx2 ) 1 (dx1 ) for
every E 2 ß(X1 ) ß(X2 ). This completes the proof.
Proof of Proposition 2. Suppose that a complete assessment ( ; ) is consistent. Then,
there exists a sequentially convergent sequence of strategies f n g associated with ( ; ). In
t
addition, for each i and t, there is a sequence of pairwise disjoint Borel sets fKi;j
gj2Jit in
A1
At 1 with an index set Jit N that satis…es the three conditions in De…nition
i
t
3. We can use this sequence fKi;j
gj2Jit to show the convex structural consistency of ti .
0
1
I
For each j 2 Jit , let a measure j on Ii=0 ß( i ) ( tt0 =1
(Ati )) be the probability
i=1 ß
t
measure associated with Ki;j
, which is de…ned according to the same way as in De…nition 3.
Then, to prove this theorem, it su¢ ces to show that there exist both a convex structurally
consistent behavioral measure ti and a convex structurally consistent informational measure
30
0
satisfying the functional equation
j (B)
=
R
0
R R
0
A1
At
1
IB ( ; a) ti ( ; da) 0 (
0; d 0)
(d
0)
for every Borel set B
Gti;j where : Ii=1 ß( i ) ! [0; 1] is the marginal probability
i
measure of j obtained by marginalizing j over 0 A1
At 1 .
Consider the sequence of conditional probability measures f j;n g de…ned according to
t
the equation (1) with respect to fKi;j
gj2Jit and f n g; that is, for each n 2 N,
j;n (B) =
A1
(
t
[e<j Ki;e
; n)
t
At 1 n
[e<j Ki;e
;
i
(B n
i
n)
0
1
I
(Ati )). Here, the probability measure with respect
for each B 2 Ii=0 ß( i ) ( tt0 =1
i=1 ß
R R
R
to a strategy pro…le is de…ned as (B; ) =
I ( ; a1 ; :::; at 1 ) t 1 ( ; a1 ;
1
A
At 1 B
0
1
1
I
:::; at 2 ; dat 1 )
( ; da1 ) (d ) for each B 2 Ii=0 ß( i ) ( tt0 =1
(Ati )). Hence, the
i=1 ß
t
; n ), can be represented by a convex strucnumerator of j;n ( ), which is ( n i [e<j Ki;e
turally consistent behavioral measure and a convex structurally consistent informational
t
measure 0 so that (B n
[e<j Ki;e
; n) =
i
R
0
R R
0
A1
At
1
IBn
i
t (
[e<j Ki;e
; a) ( ; da) 0 (
0; d 0)
0 (d
0)
0
1
I
for each B 2 Ii=0 ß( i ) ( tt0 =1
(Ati )) where 0 : Ii=1 ß( i ) ! [0; 1] is the marginal
i=1 ß
probability measure of obtained by marginalizing over 0 . Note that the variables 0 and
is convex structurally consistent behavioral. Thus, we have that
0 are independent and
0
0
0
0
0; ) = 0(
0,
0 and that ( 0 ;
0; ) = ( 0;
0(
0 ; ) for almost all
0 in
0 ; ) for
almost all ; 0 2 such that 0 = 0 0 . Moreover, the denominator of j;n ( ), which is (
t
A1
At 1 n i [e<j Ki;e
; n ), is constant and the measure j;n ( ) itself is de…ned
t
as a probability measure conditional on reaching
( i A1
At 1 n [e<j Ki;e
),
i
which includes the whole space 0 . Therefore, the conditional probability measure j;n ( )
can be represented by a convex structurally consistent behavioral measure 2n and a convex
structurally consistent informational measure 0 so that
j;n (B)
=
for each B 2 Ii=0 ß( i )
probability measure of
R
0
(
j;n
R R
0
t 1
t0 =1
A1
At
1
IB ( ; a)
0
I
(Ati ))
i=1 ß
2
n(
1
0 ; da) 0 (d 0 ) n (d
0)
where 1n : Ii=1 ß( i ) ! [0; 1] is the marginal
obtained by marginalizing j;n over 0 A1
At 1 .
31
Now, consider the measure j to which j;n converges weakly. Since
is topologically
29
complete, separable, and metric, according to Ash (1972, Theorems 6.6.5 and 6.6.630 ),
0
1
I
there are regular conditional probability measures 2 :
( tt0 =1
(Ati0 )) ! [0; 1] and
i0 =1 ß
~0 :
ß( 0 ) ! [0; 1] such that
0
j (B)
=
R
0
R R
0
A1
At
1
IB ( ; a) 2 ( ; da)~0 (
0; d 0)
1
(d
0)
0
1
I
(Ati )) where 1 : Ii=1 ß( i ) ! [0; 1] is the marginal
for each B 2 Ii=0 ß( i ) ( tt0 =1
i=1 ß
probability measure of j obtained by marginalizing j over 0 A1
At 1 . Note that
the product measure 1n 0 converges weakly to 1 ~0 according to Lemma 1 and, for every n,
1
0 ; ) = 0 ( ) for almost all
0
n 0 contains the same 0 in common. Thus, we have ~0 (
1 2
1 2
according to Lemma 1. Then, the product measure 0 ( n n ) converges weakly to 0 (
)
(= 1 0 2 ). In addition, Lemma 1 guarantees the existence of a probability measure
0
1
I
(Ati )) ! [0; 1] such that 0 = 0 ( 1 2 ). Again, since
: Ii=1 ß( i ) ( tt0 =1
0
i=1 ß
is topologically complete, separable, and metric, according to Ash (1972, Theorems 6.6.5
0
1
I
and 6.6.6), there is a regular conditional probability measure ti :
( tt0 =1
(Ati0 ))
0
i0 =1 ß
t
! [0; 1] such that
where : Ii=1 ß( i ) ! [0; 1] is the marginal probability
i =
measure31 of obtained by marginalizing over A1
At 1 . Therefore, we have
j (B)
=
R
0
R R
0
1
for each B 2 Ii=0 ß( i ) ( tt0 =1
turally consistent behavioral and
the proof.
A1
At
1
IB ( ; a) ti (
0 ; da) 0 (d 0 )
(d
0)
0
I
(Ati )).
i=1 ß
Finally, the observation that ti is convex struc0 is convex structurally consistent informational completes
29
Let Y : ( ; z) ! ( 0 ; z0 ) be a random object, and a sub -…eld of z. Suppose there is a map
1
: ( 0 ; z0 ) ! (R;ß
( R)) such that is one-to-one, E = ( 0 ) is a Borel subset of R, and
is measurable,
1
0
0
that is,
: (E;ß(E)) ! ( ; z ). Then there is a regular conditional probability for Y given .
30
Let 0 be a topologically complete separable metric space, and let z0 be the Borel subsets of 0 , that
is, the smallest -…eld containing open subsets of 0 . Then, 0 is Borel equivalent to a Borel subset E of
[0; 1].
31
Note that this marginal probability measure is not necessarily the same as the marginal probability
measure
over 0 . This is because
is originally derived from j as
0 obtained by marginalizing
1
its marginal probability measure. Here, j is a conditional probability measure on Ii=0 ß( i ) ( tt0 =1
I
t0
1
t 1
t
(Ai )) given
A
A
n i [e<j Ki;e . Thus, the measure j ignores probability assigned
i=1 ß
t
to the set
[
K
,
and
so
does
.
i
e<j i;e
32
References
1. Aliprantis, Charalambos D. and Border, Kim C. (1999): “In…nite Dimensional Analysis,”Springer, Second edition, New York.
2. Ash, Robert B. (1972): “Real Analysis and Probability,”Academic Press, New York.
3. Balder, Erik J. (1988): “Generalized Equilibrium Results for Games with Incomplete
Information,”Mathematics of Operations Research, 13, 265–276.
4. Billingsley, Patrick (1968): “Convergence of Probability Measures,”Wiley, New York.
5. Crawford, Vincent P. and Sobel, Joel (1982): “Strategic Information Transmission,”
Econometrica, 50, 1431–1451.
6. Fudenberg, Drew and Tirole, Jean (1991a): “Perfect Bayesian Equilibrium and Sequential Equilibrium,”Journal of Economic Theory, 53, 236–260.
7. Fudenberg, Drew and Tirole, Jean (1991b): “Game Theory,” MIT Press, Massachusetts.
8. Harsanyi, John C. (1967–68): “Games with Incomplete Information Played by Bayesian
Players,”Part I–III, Management Science, 14, 159–182, 320–334, and 486–502.
9. Kreps, David M. and Wilson, Robert (1982): “Sequential Equilibria,” Econometrica,
50, 863–894.
10. Kreps, David M. and Ramey, Garey (1987): “Structural Consistency, Consistency, and
Sequential Rationality,”Econometrica, 55, 1331–1348.
11. Kohlberg, Elon and Reny, Philip J. (1997): “Independence on Relative Probability
Spaces and Consistent Assessments in Game Trees,”Journal of Economic Theory, 75,
280–313.
12. Milgrom, Paul R. and Weber, Robert J. (1985): “Distributional Strategies for Games
with Incomplete Information,”Mathematics of Operations Research, 10, 619–632.
13. Nash, John (1951): “Non-Cooperative Games,”Annals of Mathematics, 54, 286–295.
14. Neveu, J. (1965): “Mathematical Foundations of the Calculus of Probability,”HoldenDay, San Francisco.
33
15. Osborne, Martin J. and Rubinstein, Ariel (1994): “A Course in Game Theory,” MIT
Press, Massachusetts.
16. Papoulis, Athanasios and Pillai, S. Unnikrishna (2002): “Probability, Random Variables, and Stochastic Processes,”McGraw-Hill, Fourth edition, Boston.
17. Reny, Philip J. (1999): “On the Existence of Pure and Mixed Strategy Nash Equilibria
in Discontinuous Games,”Econometrica, 67, 1029–1056.
18. Selten, R. (1975): “Reexamination of the Perfectness Concept for Equilibrium Points
in Extensive Games,”International Journal of Game Theory, 4, 25–55.
19. Uglanov, A. V. (1997): “Four Counterexamples to the Fubini Theorem,”Mathematical
Notes, 62, 104–107.
34