D Yannelis

temporary keynesian equilibria Rev

ral equilibrium theory Econometrica

ia Rev Econom Stud 45 (1978) 1shy

eneral equilibrium theory Reflections Microfoundations of JVlacroeconomics on 1978

ing the existence of an equilibrium in y of Interest Rates (eds F Hahn and don 1965

r Employment Interest and Mon ey

iployment Reconsidered Basil Blackshy

rices 2nd edition Harper and Row

Econometrica 45 (1977) 573- 590

ON THE EXISTENCE OF A B AYESIAN NASH EQUILIBRlUM

NICHOLAS C YANNELIS

Abstract We provide a new proof (based on a Caratheodory-type selection theorem) of the existence of a Bayesian Nash equilibrium

1 Introduction

The purpose of this note is to present an alternative proof of an existence reshysult for Bayesian Games (or games with differentia-l information) given in [8]1 The result we provide is identical to Theorem 41 in [8] However not only the present argument is different but it also has the advantage that it follows the footsteps of the argument given in [13J and therefore it can be used to generalize the Kim- Yannelis theorem to abstract Bayesian economies a la [3] Our argument combines several measure theoretic and fun ctional analytic reshysults In particular we employ a Caratheodory- type selection theorem a result on weak compactness (known as Diestels theorem) the Fatou Lemma in infinite-d imensional spaces and the Fan- Glicksberg fixed point theorem

2 Notation and definitions

Let 2x denote the set of all non-empty subsets of the set X If X and Yare sets the graph of the set- valued function (or correspondence) cent X -+ 2Y is Gcent = (x y) E X x Y y E cent(x) Let (0 F 1) be a complete finite measure space and Y be a separable Banach space The correspondence cent D --) 2Y is said to have a measurable graph if Gcent E F 0 B(Y) where B(Y) denotes the Borel (I-algebra on Y and 0 denotes the product (I-algebra The corresponshy

2Ydence cent deg--gt is said to be lower measurable if w ED cent(w) n V- q E F for every V open subset of Y We denote by Ll (IL Y) the space of equivashylence classes of Y - valued Bochner integrable functions Recall that Ll (11 Y) is a separable Banach space provided that (0 F 1) is separable

3 The game with differential informat ion

Let (0 F 1) be a complete finit e separable measure space where D denotes the set of states of the world and the (I- algebra F denote the set of events_ Let Y be a separable Banach space and T be a set of agents (either finit e or infinite )

IThe reader is referred to [I] or [8J for a discussion of the related literature

i

i

292 N C Yannelis

A Bayesian game (or a game with differential information) is a set G = (XtutFtgt) t E T where

2Yl X t D ---7 is the action set-valued function of agent t where Xt(w) is the set of actions available to t when the state is w

2 for each WED Ut(w) TIsET Xs(w) JR is the utility function of---7

agent t which can depend on the states

3 F t is a sub (J-algebra of F which denotes the private information of agent t

4 qt D ---7 lI4+ is the prior of agent t which is a Radon- Nikodym derivashytive such that Jqt(w) dfL(w) = l

Let LXt denote the set of all Bochner integrable and Ft- measurable selections from the action set- valued function X t D ---7 2Y of agent t ie

Lx = Xt E L1(fL Y) Xt is Ft-measurable and Xt(w) E Xt(w) fL- ae

The typical element of LXt is denoted as Xt while that of Xt(w) as Xt(w) (or Xt) Let Lx = TItET Lx and Lx _ = TIsft Lx Given a Bayesian game G a strategy for agent t is an element Xt in LXt

Throughout the paper we assume that for each t E T there exists a finite or countable partition 7ft of D Moreover the (J-algebra F t is generated by 7ft

For each wED let Et(w) E 7ft denote the smallest set in Ft containing w and we assume that for all t

1 qt(W) dfL(w) gt O wEEt(w)

For each wED the conditional (interim) expected utility function of agent t Vt(wmiddotmiddot) LX _ t x Xt( w) ---7 JR is defined as

Vt(w X-I Xt) = 1 Ut (W X_t(W) Xt)qt(wIEt(w)) dfL(w) w EEt (w)

where

if w ~ Et(w) qt(wIEt(w)) = 0 qt(w)

if w E Et(w) fwEEt(w) qt(w) dfL(w)

The function Vt(w X-t Xt) is interpreted as the conditional expected utility of agent t using the action Xt when the state is wand the other agents employ the strategy profile X-f where X-t is an element of Lx _t bull

A Bayesian Nash equilibrium for G is a strategy profile xmiddot E Lx such that for all t E T

On the existence of a Bayesian Nash eql

4 Statement of the theor

We state the assumptions needed fa

2Y(Al) X t D ---7 is a non-empty tegrably bounded correspond

GXt E Ft 0 B(Y)

(A2) l For each wED 1Lt(w ous where X s(w) (s i- t Xt(w) with the norm tor

2 For each x E llxET Yx wi able

3 For each wED and X- t

is concave

4 Ut is illtegrably bounded

41 T heorem Assume that T i t E T be a Bayesian game sat Bayesian Nash equilibrium for G

Proof It follows from a stand

or [2J or [8 Lemma AI]) that for e

is continuous where Lx (s i- t) Xt(w) with the norm topology I LX_t xX(t) Vt(-X-t Xt) D--JR the concavity of Ut in Xt that so is v

by

Pt(w X-t) = Yt E Xt(w) Ilt(W

By Lemma 213 in [14] for each fi) graph Since Vt is concave in Xt D X Lx _

t Pt(wX- t) is COllvex-v

that for each wE D Pt(wmiddot) h Lx (8 i- t) is endowed with the topology Therefore we can con open in X t (w) and that for each

(x-t) E Lx _t Yt E Pt(w X- t) is has open lower sections The lattE

Lx _ PtC Xt) has a measurable g a measurable graph For each t E

N C Yannelis

differential information) is a set

funciion of agent t where Xt(w) is the state is w

(w) --) lR is the utility junction of tes

tes the private information of agent

which is a Radon- Nikodym derivashy

rable and Ft- measurable selections 2--) 2Y of agent t ie

[able and 2(W) E Xt(w) p-ae

Xt while that of Xt(w) as Xt(w) Ost Lxbull Given a Bayesian game 1 Lxt

for each t E T there exists a finite he a-algebra F t is generated by 7ft

mallest set in Ft containing w and

(W) gt o

expected utility function of agent t

j)Xt)qt(wIEt(w)) dp(w)

if w ~ Et(w)

if w E Et(w)dp(w)

the conditional expected utility of is wand the other agents employ

ment of L x _t

strategy profile i E Lx such that

(w X_I Yt) p-ae

On the existence of a Bayesian Nash equilibrium 293

4 Statement of the t heorem and proof

We state the assumptions needed for our theorem

2Y(AI) X t 12 --gt is a non-empty convex weakly compact-valued and inshytegrably bounded correspondence having a Ft- measurable graph ie G Xt E F t 0 B(Y)

(A2) 1 For each wED Ut(wmiddot middot ) nsft Xs(w) X Xt(w) --gt lR is continushyous where Xs(w) (s =I- t) is endowed with the weak topology and Xt(w) with the norm topology

2 For each x E fLET Yx with Ys = Y Ut(- x) D --gt lR is F-measurshyable

3 For each wED and X-t E TIsttXs(w) Ut(wx-tmiddot) Xt(w) --gt lR is concave

4 Ut is integrably bounded

41 Theorem Assume that T is a countable set Let G = (Xt Ull Ft qt) t E T be a Bayesian game satisfying (AI)-(A2) Then there exists a Bayesian Nash equilibrium for G

Proof It follows from a standard argument (see for example [14 p 401 or [21 or [8 Lemma AI]) that for each wED Vt(wmiddotmiddot) L X _ t x Xt(w) -+ lR is continuous where Lx (s =I- t) is endowed with the weak topology and Xt(w) with the norm topology It is easy to see that for each (i_ t Xt) E

L X _ t xX(t) Vt(- i _t Xt) D -+ lR is Ft-measurable Moreover it follows from the concavity of Ut in Xt that so is Vt For each t E T define Pt 12 x L X _ t -+ 2Y

by

Pt(w X-t) = Yt E Xt(w) Vt(w i_t Yt) gt Vt(w i_Igt Xt) for all Xt E Xt(w)

By Lemma 213 in [14] for each fixed i_t E LX_tgt Pt(middot X-t) has a measurable graph Since Vt is concave in Xt we can conclude that for each (w i _t) E

12 x LX_tgt Pt(w X-t) is convex-valued It follows from the continuity of Vt that for each wED Pt(wmiddot) has an open graph in L X _ t x Xt(w) where Lx (s =I- t) is endowed with the weak topology and Xt(w) with the norm topology Therefore we can conclude that for each (w i_t) Pt(w L t ) is open in Xt(w) and that for each (w Yt) E D x Xt(w) the set Pt-1(w Yt) =

(x-t) E L X _ t Yt E Pt(wX-t) is open in L x _t ie for each wED Pt(wmiddot) has open lower sections The tatter together with the fact that for each X-t E

L x _ PtC Xt) has a measurable graph enables us to conclude that Pt(middotmiddot) has a measurable graph For each t E T let

Vt = (w i_t) E D X Lx- t Pt(wx-t) =I- 1gt

294 N C Yannelis

Restrict Pt to Ut and notice that it satisfies all conditions of the Caratheoshydory Selection Theorem in [11 Theorem 41] Hence there exists a function

ft Ut ---gt Y such that ft(w X-t) E Pt(w X-t) for all (w X_t) E Utmiddot Moreshyover for each w E 0 ft(wmiddot) is continuous on the set U( = x_t E Lx _ Pt(wX-t) -j cent and for each X-t E Lx _ ftCx-t) is measurable on the set U- = w EO Pt(wX-t) -j cent Also by Proposition 31 in [11] ftC) is

jointly measurable

2YFor each t E T define Ft 0 X L x _ ---gt by

if (w X-t) E Ut otherwise

By Lemma 212 in [14] for each X-t E Lx _ FtC X-t) is lower measurable Since FtC) is closed-valued we can conclude that for each X-t E Lx _ FtC X-t) has a measurable graph Clearly for each (w X-t) E 0 x Lx _ Ft(w X-t) is non-empty

By Lemma 61 in [13] for each wED Ft(wmiddot) is (weakly) usc2 For each t E T define Ft Lx_ ---gt 2Lx by

Since for each X-t E Lx _ FtC X-t) has a measurable graph by virtue of the Aumann measurable selection theorem there exists an Ft-measurable function

gt D ---gt Y such that gt(w) E Ft(w X-t) p-ae Since for each (w X-t) E

w x Lx- Ft(w X-t) C Xt(w) and X t(-) is integrably bounded it follows that gt E Lxmiddot Hence gt E FtCi_ t) ie Ft is non-empty-valued By Diestels

theorem3 Lx is a weakly compact subset of L J (p Y) Since the weak topology for a weakly compact subset of a separable Banach

space is metrizable [4 p 434] we can conclude that Lx is metrizable and

since T is countable so is Lx It follows from the Fatou Lemma in infiniteshy

dimensional spaces (see for example [12]) that for each t E T Ft (-) is (weakiy) upper semicontinuous and it is obviously convex and dosed-valued Define cent Lx ---gt 2Lx by

ltI(x) = II Ft(x_t) tET

Clearly Lx is compact convex non-empty and ltI is a (weakly) usc conshy

vex closed non-empty-valued correspondence from Lx to 2Lx By the Fan-shy

Glicksberg fixed point theorem there exists i E Lx such that x E ltI(xmiddot ) One

2i e for each wE 0 the set x - t E Lx_ Ft(wx_t) C V is open in Lx _ (where each Lx (8 i= t) is endowed with the weak topology) for every (norm) open subset V of Y

3For a proof using James theorem on weak compactness see [12 Theorem 31]

On the existence of a Bayesian Nas~

can easily check that x is by c

G

Remark One can introduce a (

[3] and [10]) and define an abstra in the above proof can be used t

References

1 E J Balder and A Rustichi

vate information and infinite

385-393

2 E J Balder and N C Yan

Economic Theory 3 (1993) e ~ 3 G Debreu A social equilibri

USA 38 (1952)886- 893

4 N Dunford and J T Schw(i

New York 1958

5 K Fan Fixed point and min

linear spaces Proc Nat Ace

6 A Fryszkowski CaratheodOl

variables Bull Acad Polan

7 1 L Glicksberg A further ge

orem with applications to J Soc 3 (1952) 170- 174

8 T Kim and N C Yannelis

with infinitely many players

9 J F Nash Noncooperative

10 W Schafer and H Sonnensd

out ordered preferences J jl

11 N C Yannelis Set- valued fu Equilib1illm Theory with Infi and N C Yannelis) Studies

12 N C Yannelis Integration

librium Theory with Infonite N C Yannelis) Studies in I

13 N C Yannelis and N D PI equilibria in linear topologic

N C Yannelis

isfies all conditions of the Caratheoshy141] Hence there exists a function u L t ) for all (w L t ) E Ut lVIoreshy

uous on the set U( = x-t E LX _ t

It (- X_ I) is measurable on the set

by Proposition 31 in [11] ftCmiddot) is

2Y--- by

) if (w L t ) E Ut

otherwise

X- tl FtC X-t) is lower measurable onclude that for each X-t E LX_tl

arl for each (w i_t ) E n x L x _ t

Ft(wmiddot) is (weakly) usc 2 For each

) E Ft(w X-t) It-ae

a measurable graph by virtue of the

ere exists an Fcmeasurable function

-t) It-ae Since for each (w x-d E

3 integrably bounded it follows that is non-empty-valued By Diestels of Ll (It Y)

)mpact subset of a separable Banach

melude that L Xt is metrizable and

from the Fatou Lemma in infinite-shy2]) that for each t E T C) isFt

lbviously convex and closed-valued

t(X-t)

ty and CP is a (weakly) USc conshyence from Lx to 2Lx _ By the Fanshy

x E Lx such that x E cp(x) One

i_ t ) C V is open in Lx _ (where each or every (norm) open subset V of y_ mpactness see [12 Theorem 31]

On the existence of a Bayesian Nash equilibrium 295

can easily check that x is by construction a Bayesian Nash equilibrium for

G bull

Remark One can introduce a constraint correspondence for each agent (a 10

[3] and [10]) and define an abstract Bayesian economy The argument adopted in the above proof can be used to cover abstract Bayesian economies as well

R eferenc es

1 E J Balder and A Rustichini An equilibrium result for games with pr ishy

vate information and infinitely many players J Econ Theory 62 (1994) 385-393

2 E J Balder and N C Yannelis On the continuity of expected utility Economic Theory 3 (1993) 625-643

3 3 G Debreu A social equilibrium existence theorem Proc Nail Acad Sci

USA 38 (1952)886-893

4 N Dunford and J T Schwartz Linear Operators Part I Interscience

New York 1958

5 K Fan Fixed point and minimax theorems in locally convex topological

linear spaces Proc Nat Acad Sci USA 38 (1952)131-136

6 A Fryszkowski Caratheodory type selectors of set-valued maps of two variables Bull Acad Polon Sci 25 (1977)41-46

7 1 L Glicksberg A further generalization of the Kakutani fixed poillt theshyorem with applications to Nash equilibrium points Proc Amer Math

Soc 3 (1952) 170-174

8 T Kim and N C Yannelis Existence of equilibrium in Bayesian games with infinitely many players J Econ Theory to appear

9 J F Nash Noncooperative games Ann Math 54 (1951)386-295

10 W Schafer and H Sonnenschein Equilibrium in abstract economies withshy

out ordered preferences J Math Econ 2 (1975) 345-348

w 11 N C Yannelis Set-valued function of two variables in economic theory in Equilibrium Theory with Infinitely Many Commodities (eds M A Khan

and N C Yannelis) Studies in Economic Theory Springer 1991 36-72

12 N C Yannelis Integration of Banach-valued correspondences in Equishy

librium Theory with Infinitely Many Commodities (eds lVI A Khan and N C Yannelis) Studies in Economic Theory Springer 19912-35

13 N C Yannelis and N D Prabhakar Existence of maximal elements and equilibria in linear topological spaces J Math Econ 12 (1983) 233-245

N C Yannelis296

11 N C Yannelis and A Rustichini Equilibrium points of non-cooperative random Bayesian games in Positive operators Riesz spaces and Economshy

ics (eds C D Aliprantis K Border and W A J Luxemburg) Springer

199123- 48

Nicholas C Yannelis

Department of Economics

Un iversity of Illinois

Champaign IL 61801

USA email nyanneli~uiucedu

i

i

292 N C Yannelis

A Bayesian game (or a game with differential information) is a set G = (XtutFtgt) t E T where

2Yl X t D ---7 is the action set-valued function of agent t where Xt(w) is the set of actions available to t when the state is w

2 for each WED Ut(w) TIsET Xs(w) JR is the utility function of---7

agent t which can depend on the states

3 F t is a sub (J-algebra of F which denotes the private information of agent t

4 qt D ---7 lI4+ is the prior of agent t which is a Radon- Nikodym derivashytive such that Jqt(w) dfL(w) = l

Let LXt denote the set of all Bochner integrable and Ft- measurable selections from the action set- valued function X t D ---7 2Y of agent t ie

Lx = Xt E L1(fL Y) Xt is Ft-measurable and Xt(w) E Xt(w) fL- ae

The typical element of LXt is denoted as Xt while that of Xt(w) as Xt(w) (or Xt) Let Lx = TItET Lx and Lx _ = TIsft Lx Given a Bayesian game G a strategy for agent t is an element Xt in LXt

Throughout the paper we assume that for each t E T there exists a finite or countable partition 7ft of D Moreover the (J-algebra F t is generated by 7ft

For each wED let Et(w) E 7ft denote the smallest set in Ft containing w and we assume that for all t

1 qt(W) dfL(w) gt O wEEt(w)

For each wED the conditional (interim) expected utility function of agent t Vt(wmiddotmiddot) LX _ t x Xt( w) ---7 JR is defined as

Vt(w X-I Xt) = 1 Ut (W X_t(W) Xt)qt(wIEt(w)) dfL(w) w EEt (w)

where

if w ~ Et(w) qt(wIEt(w)) = 0 qt(w)

if w E Et(w) fwEEt(w) qt(w) dfL(w)

The function Vt(w X-t Xt) is interpreted as the conditional expected utility of agent t using the action Xt when the state is wand the other agents employ the strategy profile X-f where X-t is an element of Lx _t bull

A Bayesian Nash equilibrium for G is a strategy profile xmiddot E Lx such that for all t E T

On the existence of a Bayesian Nash eql

4 Statement of the theor

We state the assumptions needed fa

2Y(Al) X t D ---7 is a non-empty tegrably bounded correspond

GXt E Ft 0 B(Y)

(A2) l For each wED 1Lt(w ous where X s(w) (s i- t Xt(w) with the norm tor

2 For each x E llxET Yx wi able

3 For each wED and X- t

is concave

4 Ut is illtegrably bounded

41 T heorem Assume that T i t E T be a Bayesian game sat Bayesian Nash equilibrium for G

Proof It follows from a stand

or [2J or [8 Lemma AI]) that for e

is continuous where Lx (s i- t) Xt(w) with the norm topology I LX_t xX(t) Vt(-X-t Xt) D--JR the concavity of Ut in Xt that so is v

by

Pt(w X-t) = Yt E Xt(w) Ilt(W

By Lemma 213 in [14] for each fi) graph Since Vt is concave in Xt D X Lx _

t Pt(wX- t) is COllvex-v

that for each wE D Pt(wmiddot) h Lx (8 i- t) is endowed with the topology Therefore we can con open in X t (w) and that for each

(x-t) E Lx _t Yt E Pt(w X- t) is has open lower sections The lattE

Lx _ PtC Xt) has a measurable g a measurable graph For each t E

N C Yannelis

differential information) is a set

funciion of agent t where Xt(w) is the state is w

(w) --) lR is the utility junction of tes

tes the private information of agent

which is a Radon- Nikodym derivashy

rable and Ft- measurable selections 2--) 2Y of agent t ie

[able and 2(W) E Xt(w) p-ae

Xt while that of Xt(w) as Xt(w) Ost Lxbull Given a Bayesian game 1 Lxt

for each t E T there exists a finite he a-algebra F t is generated by 7ft

mallest set in Ft containing w and

(W) gt o

expected utility function of agent t

j)Xt)qt(wIEt(w)) dp(w)

if w ~ Et(w)

if w E Et(w)dp(w)

the conditional expected utility of is wand the other agents employ

ment of L x _t

strategy profile i E Lx such that

(w X_I Yt) p-ae

On the existence of a Bayesian Nash equilibrium 293

4 Statement of the t heorem and proof

We state the assumptions needed for our theorem

2Y(AI) X t 12 --gt is a non-empty convex weakly compact-valued and inshytegrably bounded correspondence having a Ft- measurable graph ie G Xt E F t 0 B(Y)

(A2) 1 For each wED Ut(wmiddot middot ) nsft Xs(w) X Xt(w) --gt lR is continushyous where Xs(w) (s =I- t) is endowed with the weak topology and Xt(w) with the norm topology

2 For each x E fLET Yx with Ys = Y Ut(- x) D --gt lR is F-measurshyable

3 For each wED and X-t E TIsttXs(w) Ut(wx-tmiddot) Xt(w) --gt lR is concave

4 Ut is integrably bounded

41 Theorem Assume that T is a countable set Let G = (Xt Ull Ft qt) t E T be a Bayesian game satisfying (AI)-(A2) Then there exists a Bayesian Nash equilibrium for G

Proof It follows from a standard argument (see for example [14 p 401 or [21 or [8 Lemma AI]) that for each wED Vt(wmiddotmiddot) L X _ t x Xt(w) -+ lR is continuous where Lx (s =I- t) is endowed with the weak topology and Xt(w) with the norm topology It is easy to see that for each (i_ t Xt) E

L X _ t xX(t) Vt(- i _t Xt) D -+ lR is Ft-measurable Moreover it follows from the concavity of Ut in Xt that so is Vt For each t E T define Pt 12 x L X _ t -+ 2Y

by

Pt(w X-t) = Yt E Xt(w) Vt(w i_t Yt) gt Vt(w i_Igt Xt) for all Xt E Xt(w)

By Lemma 213 in [14] for each fixed i_t E LX_tgt Pt(middot X-t) has a measurable graph Since Vt is concave in Xt we can conclude that for each (w i _t) E

12 x LX_tgt Pt(w X-t) is convex-valued It follows from the continuity of Vt that for each wED Pt(wmiddot) has an open graph in L X _ t x Xt(w) where Lx (s =I- t) is endowed with the weak topology and Xt(w) with the norm topology Therefore we can conclude that for each (w i_t) Pt(w L t ) is open in Xt(w) and that for each (w Yt) E D x Xt(w) the set Pt-1(w Yt) =

(x-t) E L X _ t Yt E Pt(wX-t) is open in L x _t ie for each wED Pt(wmiddot) has open lower sections The tatter together with the fact that for each X-t E

L x _ PtC Xt) has a measurable graph enables us to conclude that Pt(middotmiddot) has a measurable graph For each t E T let

Vt = (w i_t) E D X Lx- t Pt(wx-t) =I- 1gt

294 N C Yannelis

Restrict Pt to Ut and notice that it satisfies all conditions of the Caratheoshydory Selection Theorem in [11 Theorem 41] Hence there exists a function

ft Ut ---gt Y such that ft(w X-t) E Pt(w X-t) for all (w X_t) E Utmiddot Moreshyover for each w E 0 ft(wmiddot) is continuous on the set U( = x_t E Lx _ Pt(wX-t) -j cent and for each X-t E Lx _ ftCx-t) is measurable on the set U- = w EO Pt(wX-t) -j cent Also by Proposition 31 in [11] ftC) is

jointly measurable

2YFor each t E T define Ft 0 X L x _ ---gt by

if (w X-t) E Ut otherwise

By Lemma 212 in [14] for each X-t E Lx _ FtC X-t) is lower measurable Since FtC) is closed-valued we can conclude that for each X-t E Lx _ FtC X-t) has a measurable graph Clearly for each (w X-t) E 0 x Lx _ Ft(w X-t) is non-empty

By Lemma 61 in [13] for each wED Ft(wmiddot) is (weakly) usc2 For each t E T define Ft Lx_ ---gt 2Lx by

Since for each X-t E Lx _ FtC X-t) has a measurable graph by virtue of the Aumann measurable selection theorem there exists an Ft-measurable function

gt D ---gt Y such that gt(w) E Ft(w X-t) p-ae Since for each (w X-t) E

w x Lx- Ft(w X-t) C Xt(w) and X t(-) is integrably bounded it follows that gt E Lxmiddot Hence gt E FtCi_ t) ie Ft is non-empty-valued By Diestels

theorem3 Lx is a weakly compact subset of L J (p Y) Since the weak topology for a weakly compact subset of a separable Banach

space is metrizable [4 p 434] we can conclude that Lx is metrizable and

since T is countable so is Lx It follows from the Fatou Lemma in infiniteshy

dimensional spaces (see for example [12]) that for each t E T Ft (-) is (weakiy) upper semicontinuous and it is obviously convex and dosed-valued Define cent Lx ---gt 2Lx by

ltI(x) = II Ft(x_t) tET

Clearly Lx is compact convex non-empty and ltI is a (weakly) usc conshy

vex closed non-empty-valued correspondence from Lx to 2Lx By the Fan-shy

Glicksberg fixed point theorem there exists i E Lx such that x E ltI(xmiddot ) One

2i e for each wE 0 the set x - t E Lx_ Ft(wx_t) C V is open in Lx _ (where each Lx (8 i= t) is endowed with the weak topology) for every (norm) open subset V of Y

3For a proof using James theorem on weak compactness see [12 Theorem 31]

On the existence of a Bayesian Nas~

can easily check that x is by c

G

Remark One can introduce a (

[3] and [10]) and define an abstra in the above proof can be used t

References

1 E J Balder and A Rustichi

vate information and infinite

385-393

2 E J Balder and N C Yan

Economic Theory 3 (1993) e ~ 3 G Debreu A social equilibri

USA 38 (1952)886- 893

4 N Dunford and J T Schw(i

New York 1958

5 K Fan Fixed point and min

linear spaces Proc Nat Ace

6 A Fryszkowski CaratheodOl

variables Bull Acad Polan

7 1 L Glicksberg A further ge

orem with applications to J Soc 3 (1952) 170- 174

8 T Kim and N C Yannelis

with infinitely many players

9 J F Nash Noncooperative

10 W Schafer and H Sonnensd

out ordered preferences J jl

11 N C Yannelis Set- valued fu Equilib1illm Theory with Infi and N C Yannelis) Studies

12 N C Yannelis Integration

librium Theory with Infonite N C Yannelis) Studies in I

13 N C Yannelis and N D PI equilibria in linear topologic

N C Yannelis

isfies all conditions of the Caratheoshy141] Hence there exists a function u L t ) for all (w L t ) E Ut lVIoreshy

uous on the set U( = x-t E LX _ t

It (- X_ I) is measurable on the set

by Proposition 31 in [11] ftCmiddot) is

2Y--- by

) if (w L t ) E Ut

otherwise

X- tl FtC X-t) is lower measurable onclude that for each X-t E LX_tl

arl for each (w i_t ) E n x L x _ t

Ft(wmiddot) is (weakly) usc 2 For each

) E Ft(w X-t) It-ae

a measurable graph by virtue of the

ere exists an Fcmeasurable function

-t) It-ae Since for each (w x-d E

3 integrably bounded it follows that is non-empty-valued By Diestels of Ll (It Y)

)mpact subset of a separable Banach

melude that L Xt is metrizable and

from the Fatou Lemma in infinite-shy2]) that for each t E T C) isFt

lbviously convex and closed-valued

t(X-t)

ty and CP is a (weakly) USc conshyence from Lx to 2Lx _ By the Fanshy

x E Lx such that x E cp(x) One

i_ t ) C V is open in Lx _ (where each or every (norm) open subset V of y_ mpactness see [12 Theorem 31]

On the existence of a Bayesian Nash equilibrium 295

can easily check that x is by construction a Bayesian Nash equilibrium for

G bull

Remark One can introduce a constraint correspondence for each agent (a 10

[3] and [10]) and define an abstract Bayesian economy The argument adopted in the above proof can be used to cover abstract Bayesian economies as well

R eferenc es

1 E J Balder and A Rustichini An equilibrium result for games with pr ishy

vate information and infinitely many players J Econ Theory 62 (1994) 385-393

2 E J Balder and N C Yannelis On the continuity of expected utility Economic Theory 3 (1993) 625-643

3 3 G Debreu A social equilibrium existence theorem Proc Nail Acad Sci

USA 38 (1952)886-893

4 N Dunford and J T Schwartz Linear Operators Part I Interscience

New York 1958

5 K Fan Fixed point and minimax theorems in locally convex topological

linear spaces Proc Nat Acad Sci USA 38 (1952)131-136

6 A Fryszkowski Caratheodory type selectors of set-valued maps of two variables Bull Acad Polon Sci 25 (1977)41-46

7 1 L Glicksberg A further generalization of the Kakutani fixed poillt theshyorem with applications to Nash equilibrium points Proc Amer Math

Soc 3 (1952) 170-174

8 T Kim and N C Yannelis Existence of equilibrium in Bayesian games with infinitely many players J Econ Theory to appear

9 J F Nash Noncooperative games Ann Math 54 (1951)386-295

10 W Schafer and H Sonnenschein Equilibrium in abstract economies withshy

out ordered preferences J Math Econ 2 (1975) 345-348

w 11 N C Yannelis Set-valued function of two variables in economic theory in Equilibrium Theory with Infinitely Many Commodities (eds M A Khan

and N C Yannelis) Studies in Economic Theory Springer 1991 36-72

12 N C Yannelis Integration of Banach-valued correspondences in Equishy

librium Theory with Infinitely Many Commodities (eds lVI A Khan and N C Yannelis) Studies in Economic Theory Springer 19912-35

13 N C Yannelis and N D Prabhakar Existence of maximal elements and equilibria in linear topological spaces J Math Econ 12 (1983) 233-245

N C Yannelis296

11 N C Yannelis and A Rustichini Equilibrium points of non-cooperative random Bayesian games in Positive operators Riesz spaces and Economshy

ics (eds C D Aliprantis K Border and W A J Luxemburg) Springer

199123- 48

Nicholas C Yannelis

Department of Economics

Un iversity of Illinois

Champaign IL 61801

USA email nyanneli~uiucedu

N C Yannelis

differential information) is a set

funciion of agent t where Xt(w) is the state is w

(w) --) lR is the utility junction of tes

tes the private information of agent

which is a Radon- Nikodym derivashy

rable and Ft- measurable selections 2--) 2Y of agent t ie

[able and 2(W) E Xt(w) p-ae

Xt while that of Xt(w) as Xt(w) Ost Lxbull Given a Bayesian game 1 Lxt

for each t E T there exists a finite he a-algebra F t is generated by 7ft

mallest set in Ft containing w and

(W) gt o

expected utility function of agent t

j)Xt)qt(wIEt(w)) dp(w)

if w ~ Et(w)

if w E Et(w)dp(w)

the conditional expected utility of is wand the other agents employ

ment of L x _t

strategy profile i E Lx such that

(w X_I Yt) p-ae

On the existence of a Bayesian Nash equilibrium 293

4 Statement of the t heorem and proof

We state the assumptions needed for our theorem

2Y(AI) X t 12 --gt is a non-empty convex weakly compact-valued and inshytegrably bounded correspondence having a Ft- measurable graph ie G Xt E F t 0 B(Y)

(A2) 1 For each wED Ut(wmiddot middot ) nsft Xs(w) X Xt(w) --gt lR is continushyous where Xs(w) (s =I- t) is endowed with the weak topology and Xt(w) with the norm topology

2 For each x E fLET Yx with Ys = Y Ut(- x) D --gt lR is F-measurshyable

3 For each wED and X-t E TIsttXs(w) Ut(wx-tmiddot) Xt(w) --gt lR is concave

4 Ut is integrably bounded

41 Theorem Assume that T is a countable set Let G = (Xt Ull Ft qt) t E T be a Bayesian game satisfying (AI)-(A2) Then there exists a Bayesian Nash equilibrium for G

Proof It follows from a standard argument (see for example [14 p 401 or [21 or [8 Lemma AI]) that for each wED Vt(wmiddotmiddot) L X _ t x Xt(w) -+ lR is continuous where Lx (s =I- t) is endowed with the weak topology and Xt(w) with the norm topology It is easy to see that for each (i_ t Xt) E

L X _ t xX(t) Vt(- i _t Xt) D -+ lR is Ft-measurable Moreover it follows from the concavity of Ut in Xt that so is Vt For each t E T define Pt 12 x L X _ t -+ 2Y

by

Pt(w X-t) = Yt E Xt(w) Vt(w i_t Yt) gt Vt(w i_Igt Xt) for all Xt E Xt(w)

By Lemma 213 in [14] for each fixed i_t E LX_tgt Pt(middot X-t) has a measurable graph Since Vt is concave in Xt we can conclude that for each (w i _t) E

12 x LX_tgt Pt(w X-t) is convex-valued It follows from the continuity of Vt that for each wED Pt(wmiddot) has an open graph in L X _ t x Xt(w) where Lx (s =I- t) is endowed with the weak topology and Xt(w) with the norm topology Therefore we can conclude that for each (w i_t) Pt(w L t ) is open in Xt(w) and that for each (w Yt) E D x Xt(w) the set Pt-1(w Yt) =

(x-t) E L X _ t Yt E Pt(wX-t) is open in L x _t ie for each wED Pt(wmiddot) has open lower sections The tatter together with the fact that for each X-t E

L x _ PtC Xt) has a measurable graph enables us to conclude that Pt(middotmiddot) has a measurable graph For each t E T let

Vt = (w i_t) E D X Lx- t Pt(wx-t) =I- 1gt

294 N C Yannelis

Restrict Pt to Ut and notice that it satisfies all conditions of the Caratheoshydory Selection Theorem in [11 Theorem 41] Hence there exists a function

ft Ut ---gt Y such that ft(w X-t) E Pt(w X-t) for all (w X_t) E Utmiddot Moreshyover for each w E 0 ft(wmiddot) is continuous on the set U( = x_t E Lx _ Pt(wX-t) -j cent and for each X-t E Lx _ ftCx-t) is measurable on the set U- = w EO Pt(wX-t) -j cent Also by Proposition 31 in [11] ftC) is

jointly measurable

2YFor each t E T define Ft 0 X L x _ ---gt by

if (w X-t) E Ut otherwise

By Lemma 212 in [14] for each X-t E Lx _ FtC X-t) is lower measurable Since FtC) is closed-valued we can conclude that for each X-t E Lx _ FtC X-t) has a measurable graph Clearly for each (w X-t) E 0 x Lx _ Ft(w X-t) is non-empty

By Lemma 61 in [13] for each wED Ft(wmiddot) is (weakly) usc2 For each t E T define Ft Lx_ ---gt 2Lx by

Since for each X-t E Lx _ FtC X-t) has a measurable graph by virtue of the Aumann measurable selection theorem there exists an Ft-measurable function

gt D ---gt Y such that gt(w) E Ft(w X-t) p-ae Since for each (w X-t) E

w x Lx- Ft(w X-t) C Xt(w) and X t(-) is integrably bounded it follows that gt E Lxmiddot Hence gt E FtCi_ t) ie Ft is non-empty-valued By Diestels

theorem3 Lx is a weakly compact subset of L J (p Y) Since the weak topology for a weakly compact subset of a separable Banach

space is metrizable [4 p 434] we can conclude that Lx is metrizable and

since T is countable so is Lx It follows from the Fatou Lemma in infiniteshy

dimensional spaces (see for example [12]) that for each t E T Ft (-) is (weakiy) upper semicontinuous and it is obviously convex and dosed-valued Define cent Lx ---gt 2Lx by

ltI(x) = II Ft(x_t) tET

Clearly Lx is compact convex non-empty and ltI is a (weakly) usc conshy

vex closed non-empty-valued correspondence from Lx to 2Lx By the Fan-shy

Glicksberg fixed point theorem there exists i E Lx such that x E ltI(xmiddot ) One

2i e for each wE 0 the set x - t E Lx_ Ft(wx_t) C V is open in Lx _ (where each Lx (8 i= t) is endowed with the weak topology) for every (norm) open subset V of Y

3For a proof using James theorem on weak compactness see [12 Theorem 31]

On the existence of a Bayesian Nas~

can easily check that x is by c

G

Remark One can introduce a (

[3] and [10]) and define an abstra in the above proof can be used t

References

1 E J Balder and A Rustichi

vate information and infinite

385-393

2 E J Balder and N C Yan

Economic Theory 3 (1993) e ~ 3 G Debreu A social equilibri

USA 38 (1952)886- 893

4 N Dunford and J T Schw(i

New York 1958

5 K Fan Fixed point and min

linear spaces Proc Nat Ace

6 A Fryszkowski CaratheodOl

variables Bull Acad Polan

7 1 L Glicksberg A further ge

orem with applications to J Soc 3 (1952) 170- 174

8 T Kim and N C Yannelis

with infinitely many players

9 J F Nash Noncooperative

10 W Schafer and H Sonnensd

out ordered preferences J jl

11 N C Yannelis Set- valued fu Equilib1illm Theory with Infi and N C Yannelis) Studies

12 N C Yannelis Integration

librium Theory with Infonite N C Yannelis) Studies in I

13 N C Yannelis and N D PI equilibria in linear topologic

N C Yannelis

isfies all conditions of the Caratheoshy141] Hence there exists a function u L t ) for all (w L t ) E Ut lVIoreshy

uous on the set U( = x-t E LX _ t

It (- X_ I) is measurable on the set

by Proposition 31 in [11] ftCmiddot) is

2Y--- by

) if (w L t ) E Ut

otherwise

X- tl FtC X-t) is lower measurable onclude that for each X-t E LX_tl

arl for each (w i_t ) E n x L x _ t

Ft(wmiddot) is (weakly) usc 2 For each

) E Ft(w X-t) It-ae

a measurable graph by virtue of the

ere exists an Fcmeasurable function

-t) It-ae Since for each (w x-d E

3 integrably bounded it follows that is non-empty-valued By Diestels of Ll (It Y)

)mpact subset of a separable Banach

melude that L Xt is metrizable and

from the Fatou Lemma in infinite-shy2]) that for each t E T C) isFt

lbviously convex and closed-valued

t(X-t)

ty and CP is a (weakly) USc conshyence from Lx to 2Lx _ By the Fanshy

x E Lx such that x E cp(x) One

i_ t ) C V is open in Lx _ (where each or every (norm) open subset V of y_ mpactness see [12 Theorem 31]

On the existence of a Bayesian Nash equilibrium 295

can easily check that x is by construction a Bayesian Nash equilibrium for

G bull

Remark One can introduce a constraint correspondence for each agent (a 10

[3] and [10]) and define an abstract Bayesian economy The argument adopted in the above proof can be used to cover abstract Bayesian economies as well

R eferenc es

1 E J Balder and A Rustichini An equilibrium result for games with pr ishy

vate information and infinitely many players J Econ Theory 62 (1994) 385-393

2 E J Balder and N C Yannelis On the continuity of expected utility Economic Theory 3 (1993) 625-643

3 3 G Debreu A social equilibrium existence theorem Proc Nail Acad Sci

USA 38 (1952)886-893

4 N Dunford and J T Schwartz Linear Operators Part I Interscience

New York 1958

5 K Fan Fixed point and minimax theorems in locally convex topological

linear spaces Proc Nat Acad Sci USA 38 (1952)131-136

6 A Fryszkowski Caratheodory type selectors of set-valued maps of two variables Bull Acad Polon Sci 25 (1977)41-46

7 1 L Glicksberg A further generalization of the Kakutani fixed poillt theshyorem with applications to Nash equilibrium points Proc Amer Math

Soc 3 (1952) 170-174

8 T Kim and N C Yannelis Existence of equilibrium in Bayesian games with infinitely many players J Econ Theory to appear

9 J F Nash Noncooperative games Ann Math 54 (1951)386-295

10 W Schafer and H Sonnenschein Equilibrium in abstract economies withshy

out ordered preferences J Math Econ 2 (1975) 345-348

w 11 N C Yannelis Set-valued function of two variables in economic theory in Equilibrium Theory with Infinitely Many Commodities (eds M A Khan

and N C Yannelis) Studies in Economic Theory Springer 1991 36-72

12 N C Yannelis Integration of Banach-valued correspondences in Equishy

librium Theory with Infinitely Many Commodities (eds lVI A Khan and N C Yannelis) Studies in Economic Theory Springer 19912-35

13 N C Yannelis and N D Prabhakar Existence of maximal elements and equilibria in linear topological spaces J Math Econ 12 (1983) 233-245

N C Yannelis296

11 N C Yannelis and A Rustichini Equilibrium points of non-cooperative random Bayesian games in Positive operators Riesz spaces and Economshy

ics (eds C D Aliprantis K Border and W A J Luxemburg) Springer

199123- 48

Nicholas C Yannelis

Department of Economics

Un iversity of Illinois

Champaign IL 61801

USA email nyanneli~uiucedu

294 N C Yannelis

Restrict Pt to Ut and notice that it satisfies all conditions of the Caratheoshydory Selection Theorem in [11 Theorem 41] Hence there exists a function

ft Ut ---gt Y such that ft(w X-t) E Pt(w X-t) for all (w X_t) E Utmiddot Moreshyover for each w E 0 ft(wmiddot) is continuous on the set U( = x_t E Lx _ Pt(wX-t) -j cent and for each X-t E Lx _ ftCx-t) is measurable on the set U- = w EO Pt(wX-t) -j cent Also by Proposition 31 in [11] ftC) is

jointly measurable

2YFor each t E T define Ft 0 X L x _ ---gt by

if (w X-t) E Ut otherwise

By Lemma 212 in [14] for each X-t E Lx _ FtC X-t) is lower measurable Since FtC) is closed-valued we can conclude that for each X-t E Lx _ FtC X-t) has a measurable graph Clearly for each (w X-t) E 0 x Lx _ Ft(w X-t) is non-empty

By Lemma 61 in [13] for each wED Ft(wmiddot) is (weakly) usc2 For each t E T define Ft Lx_ ---gt 2Lx by

Since for each X-t E Lx _ FtC X-t) has a measurable graph by virtue of the Aumann measurable selection theorem there exists an Ft-measurable function

gt D ---gt Y such that gt(w) E Ft(w X-t) p-ae Since for each (w X-t) E

w x Lx- Ft(w X-t) C Xt(w) and X t(-) is integrably bounded it follows that gt E Lxmiddot Hence gt E FtCi_ t) ie Ft is non-empty-valued By Diestels

theorem3 Lx is a weakly compact subset of L J (p Y) Since the weak topology for a weakly compact subset of a separable Banach

space is metrizable [4 p 434] we can conclude that Lx is metrizable and

since T is countable so is Lx It follows from the Fatou Lemma in infiniteshy

dimensional spaces (see for example [12]) that for each t E T Ft (-) is (weakiy) upper semicontinuous and it is obviously convex and dosed-valued Define cent Lx ---gt 2Lx by

ltI(x) = II Ft(x_t) tET

Clearly Lx is compact convex non-empty and ltI is a (weakly) usc conshy

vex closed non-empty-valued correspondence from Lx to 2Lx By the Fan-shy

Glicksberg fixed point theorem there exists i E Lx such that x E ltI(xmiddot ) One

2i e for each wE 0 the set x - t E Lx_ Ft(wx_t) C V is open in Lx _ (where each Lx (8 i= t) is endowed with the weak topology) for every (norm) open subset V of Y

3For a proof using James theorem on weak compactness see [12 Theorem 31]

On the existence of a Bayesian Nas~

can easily check that x is by c

G

Remark One can introduce a (

[3] and [10]) and define an abstra in the above proof can be used t

References

1 E J Balder and A Rustichi

vate information and infinite

385-393

2 E J Balder and N C Yan

Economic Theory 3 (1993) e ~ 3 G Debreu A social equilibri

USA 38 (1952)886- 893

4 N Dunford and J T Schw(i

New York 1958

5 K Fan Fixed point and min

linear spaces Proc Nat Ace

6 A Fryszkowski CaratheodOl

variables Bull Acad Polan

7 1 L Glicksberg A further ge

orem with applications to J Soc 3 (1952) 170- 174

8 T Kim and N C Yannelis

with infinitely many players

9 J F Nash Noncooperative

10 W Schafer and H Sonnensd

out ordered preferences J jl

11 N C Yannelis Set- valued fu Equilib1illm Theory with Infi and N C Yannelis) Studies

12 N C Yannelis Integration

librium Theory with Infonite N C Yannelis) Studies in I

13 N C Yannelis and N D PI equilibria in linear topologic

N C Yannelis

isfies all conditions of the Caratheoshy141] Hence there exists a function u L t ) for all (w L t ) E Ut lVIoreshy

uous on the set U( = x-t E LX _ t

It (- X_ I) is measurable on the set

by Proposition 31 in [11] ftCmiddot) is

2Y--- by

) if (w L t ) E Ut

otherwise

X- tl FtC X-t) is lower measurable onclude that for each X-t E LX_tl

arl for each (w i_t ) E n x L x _ t

Ft(wmiddot) is (weakly) usc 2 For each

) E Ft(w X-t) It-ae

a measurable graph by virtue of the

ere exists an Fcmeasurable function

-t) It-ae Since for each (w x-d E

3 integrably bounded it follows that is non-empty-valued By Diestels of Ll (It Y)

)mpact subset of a separable Banach

melude that L Xt is metrizable and

from the Fatou Lemma in infinite-shy2]) that for each t E T C) isFt

lbviously convex and closed-valued

t(X-t)

ty and CP is a (weakly) USc conshyence from Lx to 2Lx _ By the Fanshy

x E Lx such that x E cp(x) One

i_ t ) C V is open in Lx _ (where each or every (norm) open subset V of y_ mpactness see [12 Theorem 31]

On the existence of a Bayesian Nash equilibrium 295

can easily check that x is by construction a Bayesian Nash equilibrium for

G bull

Remark One can introduce a constraint correspondence for each agent (a 10

[3] and [10]) and define an abstract Bayesian economy The argument adopted in the above proof can be used to cover abstract Bayesian economies as well

R eferenc es

1 E J Balder and A Rustichini An equilibrium result for games with pr ishy

vate information and infinitely many players J Econ Theory 62 (1994) 385-393

2 E J Balder and N C Yannelis On the continuity of expected utility Economic Theory 3 (1993) 625-643

3 3 G Debreu A social equilibrium existence theorem Proc Nail Acad Sci

USA 38 (1952)886-893

4 N Dunford and J T Schwartz Linear Operators Part I Interscience

New York 1958

5 K Fan Fixed point and minimax theorems in locally convex topological

linear spaces Proc Nat Acad Sci USA 38 (1952)131-136

6 A Fryszkowski Caratheodory type selectors of set-valued maps of two variables Bull Acad Polon Sci 25 (1977)41-46

7 1 L Glicksberg A further generalization of the Kakutani fixed poillt theshyorem with applications to Nash equilibrium points Proc Amer Math

Soc 3 (1952) 170-174

8 T Kim and N C Yannelis Existence of equilibrium in Bayesian games with infinitely many players J Econ Theory to appear

9 J F Nash Noncooperative games Ann Math 54 (1951)386-295

10 W Schafer and H Sonnenschein Equilibrium in abstract economies withshy

out ordered preferences J Math Econ 2 (1975) 345-348

w 11 N C Yannelis Set-valued function of two variables in economic theory in Equilibrium Theory with Infinitely Many Commodities (eds M A Khan

and N C Yannelis) Studies in Economic Theory Springer 1991 36-72

12 N C Yannelis Integration of Banach-valued correspondences in Equishy

librium Theory with Infinitely Many Commodities (eds lVI A Khan and N C Yannelis) Studies in Economic Theory Springer 19912-35

13 N C Yannelis and N D Prabhakar Existence of maximal elements and equilibria in linear topological spaces J Math Econ 12 (1983) 233-245

N C Yannelis296

11 N C Yannelis and A Rustichini Equilibrium points of non-cooperative random Bayesian games in Positive operators Riesz spaces and Economshy

ics (eds C D Aliprantis K Border and W A J Luxemburg) Springer

199123- 48

Nicholas C Yannelis

Department of Economics

Un iversity of Illinois

Champaign IL 61801

USA email nyanneli~uiucedu

N C Yannelis

isfies all conditions of the Caratheoshy141] Hence there exists a function u L t ) for all (w L t ) E Ut lVIoreshy

uous on the set U( = x-t E LX _ t

It (- X_ I) is measurable on the set

by Proposition 31 in [11] ftCmiddot) is

2Y--- by

) if (w L t ) E Ut

otherwise

X- tl FtC X-t) is lower measurable onclude that for each X-t E LX_tl

arl for each (w i_t ) E n x L x _ t

Ft(wmiddot) is (weakly) usc 2 For each

) E Ft(w X-t) It-ae

a measurable graph by virtue of the

ere exists an Fcmeasurable function

-t) It-ae Since for each (w x-d E

3 integrably bounded it follows that is non-empty-valued By Diestels of Ll (It Y)

)mpact subset of a separable Banach

melude that L Xt is metrizable and

from the Fatou Lemma in infinite-shy2]) that for each t E T C) isFt

lbviously convex and closed-valued

t(X-t)

ty and CP is a (weakly) USc conshyence from Lx to 2Lx _ By the Fanshy

x E Lx such that x E cp(x) One

i_ t ) C V is open in Lx _ (where each or every (norm) open subset V of y_ mpactness see [12 Theorem 31]

On the existence of a Bayesian Nash equilibrium 295

can easily check that x is by construction a Bayesian Nash equilibrium for

G bull

Remark One can introduce a constraint correspondence for each agent (a 10

[3] and [10]) and define an abstract Bayesian economy The argument adopted in the above proof can be used to cover abstract Bayesian economies as well

R eferenc es

1 E J Balder and A Rustichini An equilibrium result for games with pr ishy

vate information and infinitely many players J Econ Theory 62 (1994) 385-393

2 E J Balder and N C Yannelis On the continuity of expected utility Economic Theory 3 (1993) 625-643

3 3 G Debreu A social equilibrium existence theorem Proc Nail Acad Sci

USA 38 (1952)886-893

4 N Dunford and J T Schwartz Linear Operators Part I Interscience

New York 1958

5 K Fan Fixed point and minimax theorems in locally convex topological

linear spaces Proc Nat Acad Sci USA 38 (1952)131-136

6 A Fryszkowski Caratheodory type selectors of set-valued maps of two variables Bull Acad Polon Sci 25 (1977)41-46

7 1 L Glicksberg A further generalization of the Kakutani fixed poillt theshyorem with applications to Nash equilibrium points Proc Amer Math

Soc 3 (1952) 170-174

8 T Kim and N C Yannelis Existence of equilibrium in Bayesian games with infinitely many players J Econ Theory to appear

9 J F Nash Noncooperative games Ann Math 54 (1951)386-295

10 W Schafer and H Sonnenschein Equilibrium in abstract economies withshy

out ordered preferences J Math Econ 2 (1975) 345-348

w 11 N C Yannelis Set-valued function of two variables in economic theory in Equilibrium Theory with Infinitely Many Commodities (eds M A Khan

and N C Yannelis) Studies in Economic Theory Springer 1991 36-72

12 N C Yannelis Integration of Banach-valued correspondences in Equishy

librium Theory with Infinitely Many Commodities (eds lVI A Khan and N C Yannelis) Studies in Economic Theory Springer 19912-35

13 N C Yannelis and N D Prabhakar Existence of maximal elements and equilibria in linear topological spaces J Math Econ 12 (1983) 233-245

N C Yannelis296

11 N C Yannelis and A Rustichini Equilibrium points of non-cooperative random Bayesian games in Positive operators Riesz spaces and Economshy

ics (eds C D Aliprantis K Border and W A J Luxemburg) Springer

199123- 48

Nicholas C Yannelis

Department of Economics

Un iversity of Illinois

Champaign IL 61801

USA email nyanneli~uiucedu

N C Yannelis296

11 N C Yannelis and A Rustichini Equilibrium points of non-cooperative random Bayesian games in Positive operators Riesz spaces and Economshy

ics (eds C D Aliprantis K Border and W A J Luxemburg) Springer

199123- 48

Nicholas C Yannelis

Department of Economics

Un iversity of Illinois

Champaign IL 61801

USA email nyanneli~uiucedu