Equilibria of large games and bayesian games with private and public information

EQUILIBRIA OF LARGE GAMES AND
BAYESIAN GAMES WITH PRIVATE AND
PUBLIC INFORMATION
FU HAIFENG
(B.S., Fudan Univ. and M.A., East China Normal Univ.)
A THESIS SUBMITTED
FOR THE DEGREE OF DOCTOR OF PHILOSOPHY
DEPARTMENT OF STATISTICS AND APPLIED
PROBABILITY
NATIONAL UNIVERSITY OF SINGAPORE
2008
Acknowledgements
At the outset, I would like to express my heartfelt gratitude to my advisor Professor
Sun Yeneng for his great guidance and assistance during my doctoral research
endeavor. I thank him for leading me into this wonderful area of game theory and
providing me with the opportunity to work with him and other talented researchers
in this area. Without his help, this thesis could not have been completed.
I am indebted to my co-advisor Professor Bai Zhidong who is very knowl-
edgable, kind and helpful. Whenever I have a question of which I think he may
know the answer, I always go to his office without hesitation, knock on his door
and he is always there for me.
I thank my co-authors, Professor Nicholas C. Yannelis, Dr Zhang Zhixiang, Ms
Xu Ying and Ms Zhang Luyi for their help and collaboration. Prof. Yannelis has
ii
Acknowledgements iii
helped me in many ways and provided me with helpful feedbacks. So does Dr
Zhang. Xu Ying and Luyi are my junior classmates and they have brought a lot
of fun to my life in NUS.
I am very grateful to Professor M. Ali Khan for his very helpful comments and
suggestions on several of my research papers on which this thesis is based. His
very warm encouragement also inspired me greatly.

I thank my classmates, Zhao Yudong, Xu Yuhong, Yang Jialiang, Wu lei and
Zhang Yongchao for their help and support at all the times, and I also thank Want-
ing, Jingyuan, Rongli and Hao Ying for sharing with me a quite and harmonious
studying environment in our small PhD students’ room.
I thank Jolene, Ziyi and other friends in NUSBS for their company and friend-
ship, which makes my life in this doctoral research period more colorful.
I am grateful to my landlord Madam Huang whose help excused me from clean-
ing my room and doing my laundry.
Last, but not least, I would like to dedicate this thesis to my parents and my
sister for their life-long love, support and understanding.
Contents
Acknowledgements ii
Summary viii
1 Introduction 1
1.1 Some backgrounds . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Motivations and contributions . . . . . . . . . . . . . . . . . . . . . 4
1.3 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2 Pure-strategy equilibria in games with private and public infor-
mation 8
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
iv
Contents v
2.2 Games with private and public information . . . . . . . . . . . . . . 9
2.3 Distribution of correspondences via vector measures . . . . . . . . . 13
2.4 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.5 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.5.1 Proof of Theorem 1 . . . . . . . . . . . . . . . . . . . . . . . 16
2.5.2 Proof of Proposition 1 . . . . . . . . . . . . . . . . . . . . . 19
3 Mixed-strategy equilibria and strong purification 25
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.2 Games with private and public information . . . . . . . . . . . . . . 29
3.3 The existence of mixed-strategy equilibria . . . . . . . . . . . . . . 31
3.4 Strong purification and pure-strategy equilibria . . . . . . . . . . . 32
3.5 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.5.1 Proof of Theorem 2 . . . . . . . . . . . . . . . . . . . . . . . 37
3.5.2 Proof of Theorem 3 . . . . . . . . . . . . . . . . . . . . . . . 39
3.5.3 Proof of Corollary 1 . . . . . . . . . . . . . . . . . . . . . . 43
4 Characterizing pure-strategy equilibria in large games 44
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
4.2 The model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
4.3 The results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
4.4 A counterexample . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
Contents vi
4.5 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
4.6 Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
4.6.1 Proof of Theorem 4 . . . . . . . . . . . . . . . . . . . . . . . 53
4.6.2 Proof of Theorem 5 . . . . . . . . . . . . . . . . . . . . . . . 55
4.6.3 Proof of Theorem 6 . . . . . . . . . . . . . . . . . . . . . . . 57
5 From large games to Bayesian games: connection and generaliza-
tion 59
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
5.2 Large Games . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
5.2.1 Game model . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
5.2.2 Pure-strategy equilibrium . . . . . . . . . . . . . . . . . . . 66
5.3 Bayesian Games with countable players . . . . . . . . . . . . . . . . 69
5.3.1 Game model . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
5.3.2 Connecting Bayesian games with large games . . . . . . . . 72
5.3.3 Pure-strategy equilibria for Bayesian games . . . . . . . . . 75
5.4 Bayesian games with private and public information . . . . . . . . . 76
5.5 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

5.6 Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
5.6.1 Proof of Theorem 7 . . . . . . . . . . . . . . . . . . . . . . . 81
5.6.2 Proof of Theorem 9 . . . . . . . . . . . . . . . . . . . . . . . 84
Contents vii
5.6.3 Proof of Theorem 11 . . . . . . . . . . . . . . . . . . . . . . 86
5.6.4 Proof of Theorem 12 . . . . . . . . . . . . . . . . . . . . . . 88
Summary
This thesis studies the equilibria in large games and Bayesian games and it consists
of four parts.
In the first part, we generalize the traditional Bayesian games by introduc-
ing a new game form, the so-called games with private and public information.
This new game model allows the players’ strategies to depend on their strategy-
relevant private information as well as on some publicly announced information.
The players’ payoffs dep end on their own payoff-relevant private information and
some payoff-relevant common information. Under the assumption that the play-
ers’ strategy-relevant private information is diffuse and their private information is
conditionally independent given the public and payoff-relevant common informa-
tion, we directly prove the existence of pure strategy equilibrium for such a game
viii
Summary ix
by developing a distribution theory of correspondences via vector measures.
In the second part, we further explore this new game model by showing the
existence of mixed-strategy equilibria under general conditions. Moreover, under
the additional assumptions of finiteness of action spaces and diffuseness and condi-
tional independence of private information, a strong purification result is obtained
for the mixed strategies in such games. As a corollary, the existence of pure-
strategy equilibria follows. This corollary generalizes the main result in our first
part.
In the third part, we consider a generalized large game model where the agent
space is divided into countable subgroups and each players payoff depends on her

own action and the action distribution in each of the subgroups. Focusing on
the interaction between Nash equilibria and the best response correspondence of
the players, we characterize the pure-strategy equilibrium distributions in large
games endowed with countable actions, countable homogeneous groups of players,
or atomless Loeb agent spaces by showing that a given distribution is an equilibrium
distribution if and only if for any (Borel) subset of actions the proportion of players
in each group playing this subset of actions is no larger than the proportion of
players in that group having a best response in this subset. Furthermore, we also
present a counterexample showing that this characterization result does not hold
for a more general setting.
In the fourth part, we firstly present a unified proof for the existence of pure
Summary x
strategy equilibria in the three settings of large games mentioned above by showing
the existence of their common characterizing counterpart. Then we show that each
Bayesian game with countable players can induce a large game and the Bayesian
game has a pure strategy equilibria if and only if the induced large game has one.
This result enables us to apply the existence results in large games to Bayesian
games and obtain existence of pure strategy equilibria in four different settings of
Bayesian games. Finally, we also establish a connection between the generalized
Bayesian games with private and public information and large games. Based on
this connection and the existence results in large games, we obtain more general-
ized existence results of pure strategy equilibria in both Bayesian games and the
generalized Bayesian games with private and public information. These results
cover and improve the main results in Parts 1 and 2.
Chapter 1
Introduction
1.1 Some backgrounds
As a field of modern science, game theory was founded by John von Neumann and
Oskar Morgenstern in 1944 in their classic, Theory of games and economic behavior.
In 1950, John F. Nash showed that finite games with complete information always

have an equilibrium point, at which all players choose actions that are best for
them given other players’ choices. After Nash’s work, game theory has gradually
become a central part of the modern economics. Moreover, game theory also
finds applications in numerous other fields including biology, political science and
computer science.
In order to facilitate analysis, games are often classified into different types.
Depending on whether or not the players are allowed to form binding commitments,
1
1.1 Some backgrounds 2
games are classified into cooperative games or noncooperative games. Depending
on whether or not the game is played simultaneously by all the players, games are
classified into static games and dynamic games. The games discussed in this paper,
i.e., Bayesian games and large games, all belong to noncooperative static games.
Bayesian games, also called games of incomplete information, are games in
which at least one player is uncertain about another player’s payoff function. While
players may not know other player’s exact payoff function, we assume they have
certain ‘belief’ about other player’s payoff function, that is, they know the ex ante
probability distribution of other player’s payoff function. Or equivalently, we can
view Bayesian games as games where each player’s payoff function is determined
by the realization of a random variable. The random variable’s actual realization
is observed only by the player but its ex ante probability distribution is known by
all the players. (See Harsanyi (1967-68).) The probability space underlying that
random variable can be regarded as the private information space pertaining to
the player.
The idea of diffuse information
1
was introduced by Dvoretsky, Wald and Wol-
fowitz (See Dvoretsky et al. (1950, 1951)) and was used as an tool to eliminate the
randomization in decision rules and to ensure the existence of a pure strategy equi-
librium in two-person zero-sum games. Following Dvoretsky et al.’s idea of diffuse

information and Harsanyi (1967-68, 1973)’s framework, Milgrom and Weber (1981,
1
The information space is said to be diffuse if it is an atomless probability space.
1.1 Some backgrounds 3
1985) and Radner and Rosenthal (1982) gave a comprehensive theory of Bayesian
games and proved the existence of pure strategy equilibria in Bayesian games with
a finite number of players and a finite number of actions. Khan and Sun (1995) pre-
sented a generalized existence result of pure strategy equilibria, which allows play-
ers to have countably many (finite or countably infinite) actions. Khan and Sun
(1999) models the set of players as a Loeb space and shows the existence of pure
strategy equilibria in Bayesian games with uncountable actions.
In contrast, a large game is a game where the set of players is endowed with
an atomless measure. Thus the number of the players in a large game is at least
uncountable. Here, the atomless assumption formalizes the “negligible” influence
of each individual player and hence large games “enable us to analyze a conflict
situation where the single player has no influence on the situation but the ag-
gregative behavior of ‘large’ sets of players can change the payoffs.”
2
Therefore,
large games are good models for large economies. Examples of large games are nu-
merous, include elections, markets, exchanges, corporations (from the shareholders
viewpoint) and so on.
The idea of modeling the set of players as an atomless measure space was intro-
duced in 1961 by Milnor and Shapley. Aumann (1964) made important contribu-
tions to the justification and distribution of this idea. Using Aumann’s methods,
Schmeidler (1973) shows the existence of a pure strategy equilibrium in a large
2
Quoted from Schmeidler (1973).
1.2 Motivations and contributions 4
game where each player is endowed with finite actions. Khan and Sun (1995) gen-

eralized the result of Schmeidler (1973) to allow a countable set of pure strategies.
The usage of hyperfinite Loeb spaces in modeling large games was systematically
studied in Khan and Sun (1996, 1999). By modeling the set of players as a Loeb
space, Khan and Sun (1999) shows the existence of Nash equilibria in large games
without any countability assumption on action or payoff space, which is false when
the agent space is modeled by Lebesgue unit interval (see Khan et al., 1997). This
major success, among others, led them to argue Loeb spaces as the ‘right’ tool for
modeling games with a large number of players.
3
1.2 Motivations and contributions
In this paper, we first notice that in some situations, the players in a Bayesian game
may encounter another type of information which is to be publicly announced to
them and may influence their strategies. To study such a situation, we introduce
a new game form which incorporates this new type of information, the so-called
“public information”. Our game model thus generalizes the game models consid-
ered in Milgrom and Weber (1985) and Radner and Rosenthal (1982).
Our next two chapters focus on this generalized Bayesian game model. The
next chapter gives a direct proof of the existence of a pure-strategy equilibrium
3
For a recent survey of large games, see Khan and Sun (2002).
1.2 Motivations and contributions 5
without using mixed-strategies. The proof itself has its conceptual advantage (if
the players play a pure-strategy equilibrium, they will search among the pure
strategies to reach an equilibrium) and is shorter than the indirect approach of
using mixed-strategies and then purification. The mathematical method for the
direct proof also has independent interest.
While the second chapter is solely focusing on pure strategy equilibrium, we
notice that the existence result of a pure strategy equilibrium relies on some strong
assumptions including the finiteness of the action spaces, finiteness of the public
and the common information spaces and diffuseness and conditional independence

of the private information spaces. However, those assumptions may not be always
satisfied in realistic situations. Thus, the pure-strategy equilibria may not always
exist and it is worth examining the existence of mixed-strategy equilibria under
more general conditions, which becomes the main objective of the third chapter.
In the third chapter, we first show the existence existence of mixed-strategy
equilibria for such a game without those strong assumptions. Moreover, by using a
similar technique as in Khan et al. (2006), a strong purification result is obtained
for all mixed strategies in such a game under similar conditions as in the first
chapter.
Thus the strong purification result, together with the existence result of mixed-
strategy equilibria, also shows the existence of pure-strategy equilibria for such a
game. This existence result of pure-strategy equilibria also covers and improves
1.2 Motivations and contributions 6
the corresponding result in the first chapter. Therefore, all the existence results
of pure-strategy equilibria in Radner and Rosenthal (1982), Milgrom and Weber
(1985) and Fu et al. (2007a) can be regarded as special cases of this result.
Chapter 4 is for characterizing large games. We notice that in the past few
decades, there have been a lot of famous existence or nonexistence results for pure-
strategy Nash equilibria in different settings of large games (see, for example, the
survey Chapter in Khan and Sun (2002)). However, very few studies focus on char-
acterizing the pure-strategy Nash equilibria or equilibrium distributions. Clearly,
good characterization results are also valuable since they can help us better un-
derstand the Nash equilibria and also provide alternative ideas for proving the
existence of Nash equilibria. It is the aim of this chapter to make some contribu-
tions in filling this gap. In particular, this chapter presents three characterization
results and a counterexample for the equilibrium distributions in large games.
Chapter 5 is for connecting large games and Bayesian games. It has long been
noted that there is a close relationship between large games and Bayesian games.
(see eg, Mas-Colell (1984), Khan and Sun (1995, 1999)). But no formal connection
was established between the two types of games. They are still regarded as two

separate types of games without any direct links. In this chapter, we shall establish
a formal connection among them, which shows that any Bayesian game can induce
a generic large game and the Bayesian game has a pure strategy (Bayesian Nash)
equilibrium iff the induced large game has a pure strategy (Nash) equilibrium.
1.3 Acknowledgements 7
Based on the above connection, we aim to unify the existence results of pure
strategy equilibria for large games and Bayesian games. We also provide a unified
approach for showing the existence of the pure strategy equilibria such that the
proofs are greatly simplified and new results are discovered.
4
1.3 Acknowledgements
During my doctoral research period, I have collaborated with a number of scholars
within the university and outside. The work resulted in a joint publication and a
joint working paper from which two of the chapters are based. More specifically,
chapter 2 is based on the following joint publication:
• Fu, H.F., Sun, Y.N., Yannelis, N.C., Zhang, Z.X.: Pure-strategy equilibria in
games with private and public information. J Math Econ 43, 523-531 (2007)
and Chapter 4 is based on the following joint working paper:
• H. Fu, Y. Xu, L. Zhang, Characterizing pure-strategy equilibria in large
games, Working Paper, 2008.
4
As this thesis is based on four essays each of which is completed in itself, there will be some
repetitions of definitions among different chapters.
Chapter 2
Pure-strategy equilibria in games with
private and public information
1
2.1 Introduction
We introduce a generalized Bayesian game model which allows the players’ strate-
gies to depend on their strategy-relevant private information as well as on some

publicly announced information. The players’ payoffs depend on their own payoff-
relevant private information and some payoff-relevant common information. The
purpose of this chapter is to show that pure strategy equilibrium exists for such
game if the players’ strategy-relevant private information is diffuse and their pri-
vate information is conditionally independent given the public and payoff-relevant
common information.
1
This chapter is based on the joint publication of Fu, Sun, Yannlis and Zhang in 2007.
8
2.2 Games with private and public information 9
The proof of the existence of pure strategy equilibrium in our setting is far from
trivial and requires the use of some new mathematical techniques. In particular, we
develop a distribution theory of correspondences via vector measures that involves
convexity, compactness and preservation of upper semi-continuity. This type of
results allows us to apply Kakutani’s fixed point theorem to prove the existence
result based only on pure strategies. As noted in (Khan and Sun, 1995, p. 637),
such a direct proof on the existence of pure strategy equilibrium using only pure
strategies does have some advantages from a game-theoretic point of view. In
particular, one does not need to go through mixed (or behavioral) strategies that
are considered to have limited appeal in many practical situations.
The chapter is organized as follows. In section 2, we introduce the game with
private and public information and state the existence of pure strategy equilibrium
for such a game. Section 3 contains the main mathematical tool that is needed for
our existence proof. Section 4 contains some concluding remarks. All the proofs
are given in the appendix.
2.2 Games with private and public information
Consider a game Γ with private and public information formulated as follows. The
game has finitely many players i = 1, . . . , l. Each player i is endowed with a
finite action set A
i

, a measurable space (T
i
, T
i
) representing her strategy-relevant
2.2 Games with private and public information 10
private information, and another measurable space (S
i
, S
i
) representing her payoff-
relevant private information. A finite set T
0
= {t
01
, . . . , t
0m
} represents those
states that are to be publicly announced to all the players; let T
0
be the power
set on T
0
. Another finite set S
0
= {s
01
, . . . , s
0n
} represents the payoff-relevant

common states that affect the payoffs of all the players with S
0
the power set on
S
0
. Thus, the product measurable space (Ω, F) = (Π
l
j=0
(S
j
× T
j
), Π
l
j=0
(S
j
× T
j
))
equipped with a probability measure η constitutes the information space of the
game. For each player i, her payoff function is a mapping from A × S
0
× S
i
to R,
i.e. u
i
: A × S
0

× S
i
−→ R. Here A = Π
l
j=1
A
j
is the set of the players’ action
profiles; and assume that for any a ∈ A, u
i
(a, s
0
, s
i
) is integrable on (Ω, F, η).
For each player i, she can use her private information as well as the publicly
announced information. Thus, a pure strategy for player i is a measurable mapping
from T
0
×T
i
to A
i
; and let Meas(T
0
×T
i
, A
i
) be the space of all measurable mappings

from T
0
× T
i
to A
i
. A pure strategy profile is a collection g = (g
1
, . . . , g
l
) of pure
strategies that specify a pure strategy for each player. For a player i = 1, . . . , l, we
shall use the following (conventional) notation: A
−i
= Π
1≤j≤l,j=i
A
j
, a = (a
i
, a
−i
)
for a ∈ A, and g = (g
i
, g
−i
) for a strategy profile g.
2
To sum up, our game is of the form Γ = {A

1
, . . . , A
l
; T
0
; S
0
; T
1
, . . . , T
l
;
S
1
, . . . , S
l
; u
1
, . . . , u
l
}, where A
1
, . . . , A
l
are the player’s action spaces, T
0
is their
public information space, S
0
is their payoff-relevant common information space,

2
From now on, without any ambiguity, we shall abbreviate Π
1≤j≤l,j=i
to Π
j=i
.
2.2 Games with private and public information 11
T
1
, . . . , T
l
are their strategy-relevant private information spaces, S
1
, . . . , S
l
are their
payoff-relevant private information spaces and u
1
, . . . , u
l
are their payoff functions.
If the players play a pure strategy profile g = (g
1
, . . . , g
l
), the resulting expected
payoff for player i can be written as
U
i
(g) = U

i
(g
1
, . . . , g
l
) =

Ω
u
i
(g
1
(t
0
, t
1
), . . . , g
l
(t
0
, t
l
), s
0
, s
i
)dη. (2.1)
A pure strategy equilibrium for Γ is a pure strategy profile g
∗
= (g

∗
1
, . . . , g
∗
l
), such
that for each i = 1, . . . , l, g
∗
i
maximizes U
i
(g
i
, g
∗
−i
) for g
i
∈ Meas(T
0
× T
i
, A
i
).
The marginal measure of η on (T
0
×S
0
, T

0
×S
0
) is denoted by η
0
. For simplicity,
we denote η
0
({t
0k
, s
0q
}) by α
kq
. For each given t
0k
∈ T
0
and s
0q
∈ S
0
, let η
kq
denote
the conditional probability measure of η on the space (Π
l
j=1
(T
j

×S
j
), Π
l
j=1
(T
j
×S
j
)).
For each player i = 1, . . . , l, let τ
i
be the marginal measure of η on the space (T
i
, T
i
),
ρ
kq
i
the marginal measure of η
kq
on the space ((T
i
×S
i
)×Π
j=i
T
j

, (S
i
×T
i
)×Π
j=i
T
j
),
ν
kq
i
the marginal measure of η
kq
on the space (T
i
× S
i
, T
i
× S
i
), and µ
kq
i
be the
marginal measure of η
kq
on the space (T
i

, T
i
).
Definition 1. (1) The players’ strategy-relevant private information is said to be
diffuse if the marginal measure τ
i
of η on the space (T
i
, T
i
) is atomless for each
player i = 1, . . . , l.
(2) The players’ private information is said to be conditionally independent
given the public and payoff-relevant common information if for each player i =
1, . . . , l, her strategy and payoff-relevant information is conditionally independent
2.2 Games with private and public information 12
of all other players’ strategy-relevant information, given t
0
∈ T
0
and s
0
∈ S
0
. That
is, ρ
kq
i
= ν
kq

i
×

j=i
µ
kq
j
for k = 1, . . . , m and q = 1, . . . , n.
The following result shows the existence of pure strategy equilibrium for the
game Γ under the assumption of diffuse and conditionally independent information.
Theorem 1. If the players’ strategy-relevant private information is diffuse and
their private information is conditionally independent given the public and payoff-
relevant common information, then there exists a pure strategy equilibrium for the
game Γ.
Independent payoff-relevant and strategy-relevant private information is used
in the game studied in Radner and Rosenthal (1982). Milgrom and Weber (1985)
considers games with payoff-relevant common information and private information
that influences players’ strategies and payoffs.
3
Our model introduces the new
concept of public information that influences all players’ strategies, in addition
to payoff-relevant and strategy-relevant private information and payoff-relevant
common information. It is obvious that the existence results of pure strategy
equilibrium in Milgrom and Weber (1985) and Radner and Rosenthal (1982) are
special cases of our Theorem 1.
4
3
See Khan et al. (2006) for a unified approach to the purification of mixed strategies by using
a consequence of the Dvoretzky-Wald-Wolfowitz Theorem in Dvoretsky et al. (1951).
4

The existence result of pure strategy equilibrium in Milgrom and Weber (1985) is stated as
a consequence of purification. However, the purification result in Milgrom and Weber (1985)
does not follow directly from the original result in Dvoretsky et al. (1951) as claimed therein,
but from a new corollary of the Dvoretzky-Wald-Wolfowitz Theorem formulated in Khan et al.
(2006), where a stronger result on purification is also proved.
2.3 Distribution of correspondences via vector measures 13
2.3 Distribution of correspondences via vector
measures
In this section we present some properties of the distribution of correspondences
induced by vector measures, which will be used to prove Theorem 1. We recall
some basic notions first.
Let Ω and X be nonempty sets, and P(X) the power set of X. A mapping
from Ω to P(X) \ {∅} is called a corresp ondence from Ω to X.
Let F be a correspondence from a measurable space (Ω, F) to a complete
separable metric space X with its Borel σ-algebra B(X), where F is a σ-algebra
on Ω. The correspondence F is said to be measurable if for each closed subset C of
X, the set {ω ∈ Ω : F(ω) ∩ C = ∅} is measurable in F. The correspondence F is
said to be closed valued if F (ω) is a closed subset of X for each ω ∈ Ω. A function
f from (Ω, F) to X is said to be a measurable selection of F if f is measurable and
f(ω) ∈ F (ω) for all ω ∈ Ω. When F is measurable and closed valued, the classical
Kuratowski-Ryll-Nardzewski Theorem (see, for example, (Aliprantis and Border,
1994, p.505)) says that F has a measurable selection.
Let M(X) be the space of Borel probability measures on X endowed with
the topology of weak convergence of measures. Let ν be a probability measure
and µ = (µ
1
, . . . , µ
m
) a vector measure on (Ω, F), where each µ
k

is a probability
measure for k = 1, . . . , m. (Ω, F, µ) is called a vector probability measure space.
2.3 Distribution of correspondences via vector measures 14
For a measurable mapping ϕ from a probability space (Ω, F, ν) to X, we use νϕ
−1
to denote the Borel probability measure on X induced by ϕ, which is often called
the distribution of ϕ. We also use µϕ
−1
to denote (µ
1
ϕ
−1
, . . . , µ
m
ϕ
−1
), which
belongs to (M(X))
m
. When X is a finite set {x
1
, . . . , x
d
}, M(X) can be identified
with the simplex ∆ = {(x
1
, . . . , x
d
) : x
i

≥ 0,

d
i=1
x
i
= 1} under the Euclidean
metric.
Next, let G be a correspondence from a topological space Y to another topo-
logical space Z. Let y
0
be a p oint in Y . Then G is said to be upper semicontinuous
at y
0
if for any open set U which contains G(y
0
), there exists a neighborhood V of
y
0
such that y ∈ V implies that G(y) ⊆ U. G is said to be upper semicontinuous
on Y if it is upper semicontinuous at every point y ∈ Y .
Now we state our main result about the distribution of correspondences induced
by a vector measure when the target space is a finite set.
Proposition 1. Let A be a finite set, Y a metric space, (Ω, F, µ) an atomless
vector probability measure space,
5
and F a correspondence from Ω × Y to A. For
each fixed y ∈ Y , let F
y
denote the correspondence F (·, y) from Ω to A, which is

assumed to be measurable. Let G be a correspondence from Y to (M(A))
m
such
that for each y ∈ Y ,
G(y) = {µϕ
−1
: ϕ(·) is a measurable selection from F
y
(·)}. (2.2)
5
It means that µ
k
is atomless for each 1 ≤ k ≤ m.
2.4 Concluding remarks 15
Then, (1) G is convex and compact valued; (2) if, in addition, the correspondence
F (ω, ·) is upper semicontinuous on Y for each fixed ω ∈ Ω, then G is upper semi-
continuous on Y .
Consider the simple case that µ is a scalar probability measure (i.e., m =
1). All the three properties of convexity, compactness, and preservation of upper
semicontinuity in the above theorem on the distribution of correspondences may
fail when A is not assumed to be finite (see Examples 1, 2 and 3 in Sun (1996) for
the case that A = [−1, 1]).
2.4 Concluding remarks
The game introduced in this chapter can be easily extended to a social system
by including constraint correspondences where action sets depend on the informa-
tion of individual players . Such a framework may be useful to applications for
economies with private information and also public information (see, for exam-
ple Glycopantis and Yannelis (2005)). Thus, the standard Walrasian expectation
equilibrium notions may be generalized by including the public information aspect
as used in this chapter.