EXISTENCE OF EQUILIBRIUM FOR AN ABSTRACT ECONOMY WITH PRIVATE INFORMATION AND A COUNTABLE SPACE OF ACTIONS

(1)

WITH PRIVATE INFORMATION AND A COUNTABLE SPACE OF ACTIONS

MONICA PATRICHE

Communicated by the former editorial board

We define the model of an abstract economy with private information and a countable set of actions. We generalize the H. Yu and Z. Zhang’s model (2007), considering that each agent is characterised by a preference correspondence in- stead of having an utility function. We establish two different equilibrium existence results.

AMS 2010 Subject Classification: 47H10, 55M20, 91B50.

Key words:private information, upper semicontinuous correspondences, abstract economy, equilibrium.

1. INTRODUCTION

We define the model of an abstract economy with private information and a countable set of actions. The preference correspondences need not to be represented by utility functions. The equilibrium concept is an extension of the deterministic equilibrium. We present the H. Yu and Z. Zhang’s model in [16], in which the agents maximize their expected utilities. Our model is a generalization of H. Yu and Z. Zhang’s one.

A purpose in this paper is to prove the existence of equilibrium for an abstract economy with private information and a countable set of actions. The assumptions on correspondences refer to upper semicontinuity and measura- bility.

The existence of pure strategy equilibrium for a game with finitely many players, finite action space and diffuse and independent private information was first proved by Radner and Rosenthal [12]. This result was extended by Khan and Sun [7] to the case of a finite game with diffuse and independent private information and with countable compact metric spaces as their action spaces.

We quote the papers of M.A. Khan, Y. Sun [8, 9], concerning this subject of research. In [16], H. Yu and Z. Zhang showed the existence of pure strategy equilibrium for games with countable complete metric spaces and worked with

MATH. REPORTS15(65),3(2013), 233–242

(2)

compact-valued correspondences. He relied on the Bollob´as and Varopulos’s [2] extension of the marriage lemma to construct a theory of the distribution of a correspondence from an atomless probability space to a countable complete metric space. They also studied the case of the game with a continuum of players.

The classical model of Nash [11] was generalized by many authors. Models were proposed in his pioneering works by Debreu [3] or later by A. Borglin and H. Keiding [2], Shafer and Sonnenschein [13], Yannelis and Prahbakar [15]. Yannelis and Prahbakar developed new tehniques of work for showing the existence of equilibrium. That is the reason we defined a new model that can be integrated in this direction of development of the games theory. We use the fixed point method of finding the equilibrium, precisely, we use the Ky Fan fixed point Theorem for upper semicontinuous correspondences.

The paper is organised as follows. In Section 2, some notation and ter- minological convention are given. In Section 3, H. Yu and Z. Zhang’s expected utility model with a finite number of agents and private information and their main result in [16] are presented. Section 4 introduces our model, that is, an abstract economy with private information and a countable space of actions.

Section 5 contains existence results for upper semicontinuous correspondences.

2.PRELIMINARIES AND NOTATION

Throughout this paper, we shall use the following notations and defini- tions:

Let Abe a subset of a topological space X.

1. F(A) denotes the family of all non-empty finite subset of A.

2. 2^A denotes the family of all subsets ofA.

3. clA denotes the closure of Ain X.

4. IfA is a subset of a vector space, coAdenotes the convex hull of A.

5. IfF,G:X →2^Y are correspondences, then coG, clG,G∩F :X→2^Y are correspondences defined by (coG)(x) =coG(x), (clG)(x) =clG(x) and (G∩F)(x) =G(x)∩F(x) for each x∈X, respectively.

Definition 1.Each correspondenceF :X →2^Y has two natural inverses:

1. the upper inverse F^u (also called the strong inverse) of a subset A of Y is defined by F^u(A) ={x∈A:F(x)⊂A}.

2. the lower inverse F^l (also called theweak inverse) of a subset A ofY is defined byF^l(A) ={x∈A:F(x)∩A6=∅}.

(3)

Definition 2. Let X, Y be topological spaces and F : X → 2^Y be a correspondence. F is said to be upper semicontinuous if for each x ∈ X and each open set V inY with F(x)⊂V, there exists an open neighborhood U of x inX such that F(y)⊂V for each y∈U.

Lemma 1 ([17]). Let X and Y be two topological spaces and let A be a closed (resp. open) subset of X. Suppose F₁ :X →2^Y, F₂ :X →2^Yare lower semicontinuous (resp. upper semicontinuous) such that F2(x) ⊂F1(x) for all x∈A. Then the correspondence F :X→2^Y defined by

F(x) =

F₁(x), if x /∈A, F₂(x), if x∈A

is also lower semicontinuous (resp. upper semicontinuous).

Definition 3. Let (T, T) be a measurable space, Y a topological space and F :T →2^Y a corespondence.

1. F is weakly measurable ifF^l(A)∈ T for each open subsetA of Y; 2. F is measurable ifF^l(A)∈ T for each closed subset Aof Y.

Remark. Let (T, T) be a measurable space, Y a countable set and F : T → 2^Y a corespondence. ThenF is measurable if for each y ∈Y, F⁻¹(y) = {t∈T :z∈F(t)} is T −measurable.

Lemma 2 ([1]). For a correspondence. F : (T, T) →2^Y from a measurable space into a metrizable space we have the following:

1. If F is measurable, then it is also weakly measurable;

2. If F is compact valued and weakly measurable, it is measurable.

Definition 4 ([16]). LetY be a countable complete metric space, (T,T,λ) an atomless probability space and F : T → 2^Ya measurable correspondence.

The function f :T → Y is said to be a selection of F if f(t) ∈ F(t) for λ−

almost t∈T. DenoteDF =

λf⁻¹:f is a measurable selection ofF . Lemma 3 ([16]). Let Y be a countable complete metric space, (T,T,λ) an atomless probability space and F : T → 2^Y a measurable correspondence.

ThenD_F is nonempty and convex in the spaceM(Y) - the space of probability measure on Y, equipped with the topology of weak convergence.

Lemma4 ([16]). LetY be a countable complete metric space, and(T,T,λ) be an atomless probability space and F : T → 2^Y be a measurable correspondence. If F is compact valued, then D_Fis compact in M(Y).

(4)

Lemma 5 ([16]). Let X be a metric space, (T,T,λ) be an atomless probability space, Y be a countable complete metric space and F : T ×X → 2^Y a correspondence. Assume that for any fixed x in X, F(·, x) (also denoted by F_x) is a compact-valued measurable correspondence, and for each fixed t∈ T, F(t,·)is upper semicontinuous on X. Also, assume that there exists a compact valued correspondence H :T ×X →2^Y such that F(t, x)⊂H(t)for all t and x. Then D_F_x is upper semicontinuous on X.

Theorem 1 (Kuratowski-Ryll-Nardzewski Selection Theorem [1]). A weakly measurable correspondence with nonempty closed values from a measurable space into a Polish space admits a measurable selector.

3. A NASH EQUILIBRIUM EXISTENCE THEOREM

We present Yu and Zhang’s model of a finite game with private information. In this model it is assigned to each agent a private information related to his action and payoff described by the random mappings τ_i and χ_i,mappings defined on (Ω,F) = (Q

i∈I

(Z_i, X_i),Q

i∈I

(Z_i,X_i)), where (X_i, X_i) and (Z_i,Z_i) are measurable spaces. For a point ω = (z₁, x₁, ...z_n, x_n) ∈ Ω, τ_i and χ_i are the coordinate projections τi(ω) =zi,χi(ω) =xi.Each player iinI first observes the realization, sayz_i ∈Z_i,of the random element τ_i(ω),then chooses his own action from a nonempty compact subsetD_i(z_i) of a countable complete metric spaceAi. Di(·) is measurable. The payoff of each playeriis given by the utility functionu_i :A×X_i →R,where A= Q

j∈I

A_j is the set of of all combination of all players’ moves. It is assumed the following uniform integrability condition (UI):

(UI) For every i ∈ I, there is a real-valued integrable function hi on (Ω,F, µ) such that µ−almost all ω ∈ Ω, | ui(a, χi(ω)) |≤ hi(ω) holds for all a∈A.

Definition 5 ([16]).A finite game with private information is a family Γ = (I,((Z_i,Z_i),(X_i,X_i),(A_i, D_i), u_i)i∈I, µ).

For each i ∈ I, let meas(Z_i, D_i) be the set of measurable mappings f from (Z_i,Z_i) toA_i such that f(z_i)∈D_i(z_i) for eachz_i ∈Z_i.An element g_i of meas(Zi, Ai) is calleda pure strategy for playeri.A pure strategy profile gis an n-vector function (g1, g2, ..., gn) that specifies a pure strategy for each player.

Definition 6 ([16]). For a pure strategy profile g = (g1, g2, ..., gn), the expected payoff for playeriis

U_i(g) = R

ω∈Ω

u_i(g₁(τ₁(ω)), ..., g_n(τ_n(ω)), χ_i(ω))µd(ω).

(5)

Definition 7 ([16]). A Nash equilibrium in pure strategies is defined as a pure strategy profile (g₁^∗, g^∗₂, ..., g_n^∗) such that for each player i∈I

Ui(g^∗)≥Ui(gi, g_−i^∗ ) for all gi∈ Meas(Zi, Di).

The following theorem is the main result of Zhang in [16].

Theorem 2. Suppose that for every player i, the compact valuedD_i correspondence is measurable, and

a) the distribution µτ_i⁻¹ of τi is an atomless measure;

b) the random elements {τ_j :j6=i} together with the random element ξ_i ≡(τ_i, χ_i) form a mutually independent set;

c) for any fixed x_i ∈ X_i, u_i(·, x_i)is a continuous function on A; for any fixed a∈A,u_i(a,·) is a measurable function on (X_i,X_i);

d) the uniform integrability condition (UI) holds.

Then the game Γ has a Nash equilibrium in pure strategies.

4. THE MODEL OF AN ABSTRACT ECONOMY WITH PRIVATE INFORMATION

In this section, we define a model of abstract economy with private information and a countable set of actions. We prove the existence of equilibrium of abstract economies in four cases.

Let I be a nonempty and finite set (the set of agents). For each i∈ I, the space of actions, A_i is a countable complete metric space and (Z_i,Z_i) is measurable space. Let (Ω,F) be the product measurable space,(Q

i∈I

Zi,Q

i∈I

Z_i), and µ a probability measure on (Ω,F).For a point ω= (z1, ...zn)∈Ω,define the coordinate projections

τ_i(ω) =z_i.

The random mappingτi(ω) is interpreted as playeri’s private information related to his action.

We also denote, for each i∈I, meas(Z_i, A_i) the set of measurable map- pingsf from (Zi,Z_i) toAi.An elementgiof meas(Zi, Ai) is called apure strategy for player i.A pure strategy profile gis an n-vector function (g1, g2, ..., gn) that specifies a pure strategy for each player.

We suppose that there exists a correspondence D_i :Z_i →2^Aⁱ such that each agent ican choose an action from Di(zi)⊂Ai for each zi∈Zi.

Let DDi be the set

(µτ_i⁻¹)g_i⁻¹ :gi is a measurable selection of Di . For each i∈I, let the constraint correspondence beαi :Zi×Q

i∈I

D_D_i → 2^Aⁱ, such that α_i(z_i,(µτ₁⁻¹)g₁⁻¹,(µτ₂⁻¹)g⁻¹₂ , ...,(µτ_n⁻¹)g⁻¹_n ) ∈ A_i and the pref-

(6)

erence correspondence is P_i :Z_i× Q

i∈I

D_D_i →2^Aⁱ, such that P_i(z_i,(µτ₁⁻¹)g₁⁻¹, (µτ₂⁻¹)g⁻¹₂ , ...,(µτ_n⁻¹)g⁻¹_n )∈Ai.

Definition 8. An abstract economy (or a generalized game) with private information and a countable space of actions is defined as Γ = (I,((Z_i,Z_i), (A_i, α_i, P_i))i∈I, µ).

Definition 9. An equilibrium for Γ is defined as a strategy profile (g₁^∗, g^∗₂, ..., g_n^∗)∈ Q

i∈I

Meas(Zi, Ai) such that for eachi∈I :

1) g^∗_i(z_i) ∈ α_i(z_i,(µτ₁⁻¹)(g^∗₁)⁻¹,(µτ₂⁻¹)(g₂^∗)⁻¹, ...,(µτ_n⁻¹)(g^∗_n)⁻¹) for each zi∈Zi;

2)αi(zi,(µτ₁⁻¹)(g₁^∗)⁻¹,(µτ₂⁻¹)(g₂^∗)⁻¹, ...,(µτ_n⁻¹)(g_n^∗)⁻¹)∩P_i(zi,(µτ₁⁻¹)(g^∗₁)⁻¹, (µτ₂⁻¹)(g^∗₂)⁻¹, ...,(µτ_n⁻¹)(g^∗_n)⁻¹) =φfor each z_i ∈Z_i.

5. EXISTENCE OF EQUILIBRIUM FOR ABSTRACT ECONOMIES WITH PRIVATE INFORMATION

We state some new equilibrium existence theorems for abstract economies.

Theorem 3 is an existence theorem of equilibrium for an abstract economy with upper semicontinuous correspondences αi and Pi.

Theorem 3. Let Γ = (I,((Zi,Z_i),(Ai, αi, Pi))i∈I, µ) be an abstract economy with private information and a countable space of action, where I is a finite index set such that for each i∈I :

a) Ai is a countable complete metric space and (Zi,Z_i) is a measurable space; (Ω, F)is the product measurable space (Q

i∈I

(Z_i,Z_i)) and µ an atomless probability measure on (Ω, F);

b) the correspondenceDi :Zi→2^Aⁱ is measurable with compact values;

c)the correspondenceαi:Zi×Q

i∈I

D_D_i →2^Aⁱ is measurable with respect to z_i and, for all z_i ∈Z_i, α_i(z_i,·,·, ...,·) is upper semicontinuous with nonempty, compact values;

d)the correspondencePi :Zi×Q

i∈I

D_D_i →2^Aⁱ is measurable with respect to zi and, for all zi ∈Zi, Pi(zi,·,·, ...,·) is upper semicontinuous with nonempty, compact values;

e) for each z_i ∈Z_i and each(g₁, g₂, ..., g_n)∈ Q

i∈I

Meas(Z_i, A_i), gi(zi)6∈Pi(zi,(µτ₁⁻¹)g₁⁻¹,(µτ₂⁻¹)g₂⁻¹, ...,(µτ_n⁻¹)g_n⁻¹);

f) the set U_i :={(z_i, λ₁, λ₂, ..., λ_n) ∈Z_i× Q

i∈I

D_D_i : (α_i∩P_i)(z_i, λ₁, λ₂, ..., λ_n) =∅} is open for each z_i ∈Z_i.

(7)

Then there exists (g^∗₁, g^∗₂, ..., g_n^∗)∈ Q

i∈I

Meas(Z_i, A_i) an equilibrium for Γ.

Proof. By Lemma 3,D_D_i is nonempty and convex. By Lemma 4,D_D_i is compact.For eachi∈I the set

Ui:=

(zi, λ1, λ2, ..., λn)∈Zi×Q

i∈I

D_D_i : (αi∩Pi)(zi, λ1, λ2, ..., λn) =∅

is open and we defineFi:Zi× Q

i∈I

D_D_i →2^Aⁱ by

F_i(z_i, λ₁, λ₂, ..., λ_n) =

(αi∩Pi)(zi, λ1, λ2, ..., λn) if (zi, λ1, λ2, ..., λn)6∈Ui, α_i(z_i, λ₁, λ₂, ..., λ_n) if (z_i, λ₁, λ₂, ..., λ_n)∈U_i. Then, the correspondence F_i has nonempty compact values. It is also measurable with respect to z_i and upper semicontinuous with respect to (λ₁, λ2, ..., λn)∈ Q

i∈I

D_D_i.

We denoteD_F_i(λ₁, λ₂, ..., λ_n) =

(µτ_i⁻¹)g_i⁻¹ :g_i is a measurable selection of F_i(·, λ₁, λ₂, ..., λ_n)}.Then:

i) D_F_i(λ1, λ2, ..., λn) is nonempty because there exists a measurable selection from the correspondence F_i by Kuratowski-Ryll-Nardewski Selection Theorem.

ii) D_F_i(λ1, λ2, ..., λn) is convex and compact by Lemma 3 and Lemma 4.

We define Φ : Q

i∈I

D_D_i →2

Q

i∈I

D_Di

,Φ(λ1, λ2, ..., λn) = Q

i∈I

D_F_i(λ1, λ2, ..., λn).

The set Q

i∈I

D_D_i is nonempty, compact, convex. By Lemma 5 the correspondence D_F_i is upper semicontinuous. Then, the correspondence Φ is upper semicontinuous and has nonempty compact and convex values. By Ky Fan fixed point Theorem, we know that there exists a fixed point (λ^∗₁, λ^∗₂, ..., λ^∗_n)∈ Φ(λ^∗₁, λ^∗₂, ..., λ^∗_n).In particular, for each playeri, λ^∗_i ∈ D_F_i(λ^∗₁, λ^∗₂, ..., λ^∗_n).There- fore, for each playeri,there existsg_i^∗∈Meas(Z_i, A_i) such thatg_i^∗ is a selection of Fi(·, λ^∗₁, λ^∗₂, ..., λ^∗_n) and (µτ_i⁻¹)(g^∗_i)⁻¹=λ^∗_i.

We prove that (g₁^∗, g₂^∗, ..., g^∗_n) is an equilibrium for Γ. For each i ∈ I, because g_i^∗ is a selection of F_i(·, λ^∗₁, λ^∗₂, ..., λ^∗_n), it follows that g^∗_i(z_i) ∈ (α_i∩ Pi)(zi, λ^∗₁, λ^∗₂, ..., λ^∗_n) if (zi, λ^∗₁, λ^∗₂, ..., λ^∗_n)6∈Ui org_i^∗(zi)∈α⁰_i(zi, λ^∗₁, λ^∗₂, ..., λ^∗_n) if (zi, λ^∗₁, λ^∗₂, ..., λ^∗_n)∈Ui.

By the assumption d) it follows that g_i^∗(z_i)6∈P_i(z_i, λ^∗₁, λ^∗₂, ..., λ^∗_n) for each z_i ∈ Z_i. Then g_i^∗(z_i) ∈ α_i(z_i, λ^∗₁, λ^∗₂, ..., λ^∗_n) and (z_i, λ^∗₁, λ^∗₂, ..., λ^∗_n) ∈ U_i. This is equivalent with the fact that g^∗_i(zi) ∈ αi(zi,(µτ₁⁻¹)(g^∗₁)⁻¹,(µτ₂⁻¹)(g^∗₂)⁻¹, ..., (µτ_n⁻¹)(g^∗_n)⁻¹) and (α_i∩P_i)(z_i,(µτ₁⁻¹)(g^∗₁)⁻¹,(µτ₂⁻¹)(g^∗₂)⁻¹, ...,(µτ_n⁻¹)(g^∗_n)⁻¹) =

∅ for each z_i ∈Z_i.Consequently, (g₁^∗, g₂^∗, ..., g^∗_n) is an equilibrium for Γ.

(8)

Theorem 4. Let Γ = (I,((Z_i,Z_i),(A_i, α_i, P_i))i∈I, µ) be an abstract economy with private information and a countable space of action, whereI is afinite index set such that for each i∈I,

a) A_i is a countable complete metric space and (Z_i, Z_i) is a measurable space; (Ω, F) is the product measurable space Q

i∈I

(Z_i,Z_i) and µ an atomless probability measure on (Ω, F);

b) the correspondence D_i :Z_i → 2^Aⁱ is measurable with compact values;

let D_D_i be the set

(µτ_i⁻¹)g_i⁻¹:gi is a measurable selection of Di

c)the correspondenceα_i:Z_i×Q

i∈I

D_D_i →2^Aⁱ is measurable with respect to z_i and, for all z_i ∈Z_i, α_i(z_i,·,·, ...,·) is upper semicontinuous with nonempty, compact values;

d) there exists a selector G_i : Z_i × Q

i∈I

D_D_i → 2^Aⁱ for (α_i ∩P_i) : Z_i× Q

i∈I

D_D_i → 2^Aⁱ such that Gi(zi,(µτ₁⁻¹)g⁻¹₁ ,(µτ₂⁻¹)g⁻¹₂ , ...,(µτ_n⁻¹)g⁻¹_n ) is measurable with respect to zi and, for all zi ∈Zi, Gi(zi,·,·, ...,·) is upper semicontinuous with nonempty, compact values;

e) for each z_i ∈Z_i and each(g₁, g₂, ..., g_n)∈ Q

i∈I

Meas(Z_i, A_i), gi(zi)6∈Gi(zi,(µτ₁⁻¹)g₁⁻¹,(µτ₂⁻¹)g⁻¹₂ , ...,(µτ_n⁻¹)g⁻¹_n )

f) the set U_i :={(z_i, λ₁, λ₂, ..., λ_n) ∈Z_i× Q

i∈I

D_D_i : (α_i∩P_i)(z_i, λ₁, λ₂, ..., λ_n) =∅} is open for each z_i ∈Z_i.

Then there exists (g^∗₁, g^∗₂, ..., g_n^∗)∈ Q

i∈I

Meas(Zi, Ai) an equilibrium for Γ.

Proof. By Lemma 3,D_D_i is nonempty and convex. By Lemma 4,D_D_i is compact.For eachi∈I the set

U_i:=

(z_i, λ₁, λ₂, ..., λ_n)∈Z_i×Q

i∈I

D_D_i : (α_i∩P_i)(z_i, λ₁, λ₂, ..., λ_n) =∅

is open and we defineF_i:Z_i× Q

i∈I

D_D_i →2^Aⁱ by

Fi(zi, λ1, λ2, ..., λn) =

G_i(z_i, λ₁, λ₂, ..., λ_n) if (z_i, λ₁, λ₂, ..., λ_n)6∈U_i, αi(zi, λ1, λ2, ..., λn) if (zi, λ1, λ2, ..., λn)∈Ui. Then, the correspondence Fi has nonempty compact values. It is also measurable with respect to z_i and upper semicontinuous with respect to (λ₁, λ₂, ..., λ_n)∈ Q

i∈I

D_D_i. We denote

D_F_i(λ₁, λ₂, ..., λ_n) ={(µτ_i⁻¹)g⁻¹_i :g_i is a measurable selection ofF_i(·, λ₁, λ₂, ..., λ_n)}.Then:

(9)

i) D_F_i(λ₁, λ₂, ..., λ_n) is nonempty because there exists a measurable selection from the correspondence F_i by Kuratowski-Ryll-Nardewski Selection Theorem.

ii) D_F_i(λ₁, λ₂, ..., λ_n) is convex and compact by Lemma 3 and Lemma 4.

We define Φ : Q

i∈I

D_D_i →2

Q

i∈I

D_Di

,Φ(λ₁, λ₂, ..., λ_n) = Q

i∈I

D_F_i(λ₁, λ₂, ..., λ_n).

The set Q

i∈I

D_D_i is nonempty, compact, convex. By Lemma 5 the correspondence D_F_i is upper semicontinuous. Then, the correspondence Φ is upper semicontinuous and has nonempty, compact and convex values. By Ky Fan fixed point Theorem, we know that there exists a fixed point (λ^∗₁, λ^∗₂, ..., λ^∗_n)∈ Φ(λ^∗₁, λ^∗₂, ..., λ^∗_n).In particular, for each playeri, λ^∗_i ∈ D_F_i(λ^∗₁, λ^∗₂, ..., λ^∗_n).There- fore, for each playeri,there existsg_i^∗∈Meas(Zi, Ai) such thatg_i^∗ is a selection of F_i(·, λ^∗₁, λ^∗₂, ..., λ^∗_n) and (µτ_i⁻¹)(g^∗_i)⁻¹=λ^∗_i.

We prove that (g₁^∗, g₂^∗, ..., g^∗_n) is an equilibrium for Γ. For each i ∈ I, because g_i^∗ is a selection of Fi(·, λ^∗₁, λ^∗₂, ..., λ^∗_n), it follows that g^∗_i(zi) ∈ (αi∩ P_i)(z_i, λ^∗₁, λ^∗₂, ..., λ^∗_n) if (z_i, λ^∗₁, λ^∗₂, ..., λ^∗_n)6∈U_i org_i^∗(z_i)∈α_i(z_i, λ^∗₁, λ^∗₂, ..., λ^∗_n) if (z_i, λ^∗₁, λ^∗₂, ..., λ^∗_n)∈U_i.

By the hypothesis d) it follows that g_i^∗(zi)6∈Gi(zi, λ^∗₁, λ^∗₂, ..., λ^∗_n) for each z_i ∈ Z_i. Then g_i^∗(z_i) ∈ α_i(z_i, λ^∗₁, λ^∗₂, ..., λ^∗_n) and (z_i, λ^∗₁, λ^∗₂, ..., λ^∗_n) ∈ U_i. This is equivalent with the fact that g^∗_i(z_i) ∈ α_i(z_i,(µτ₁⁻¹)(g^∗₁)⁻¹,(µτ₂⁻¹)(g^∗₂)⁻¹, ..., (µτ_n⁻¹)(g^∗_n)⁻¹) and (αi∩P_i)(zi,(µτ₁⁻¹)(g^∗₁)⁻¹,(µτ₂⁻¹)(g^∗₂)⁻¹, ...,(µτ_n⁻¹)(g^∗_n)⁻¹) =

∅ for each z_i ∈Z_i.Consequently, (g₁^∗, g₂^∗, ..., g^∗_n) is an equilibrium for Γ.

Acknowledgment. This work was supported by the strategic grant POSDRU/89/

1.5/S/58852, Project “Postdoctoral programme for training scientific researchers” co- financed by the European Social Found within the Sectorial Operational Program Human Resources Development 2007–2013.

REFERENCES

[1] C.D. Aliprantis and K.C. Border,Infinite Dimensional Analysis: A Hitchhiker’s Guide.

Springer-Verlag, Berlin, 1994.

[2] B. Bollobas and N.T. Varopouloos,Representation of systems of measurable sets. Math.

Proc. Cambridge Philos. Soc. 78(1975), 323–325.

[3] A. Borglin and H. Keiding,Existence of equilibrium actions and of equilibrium: A note on the ’new’ existence theorem,J. Math. Econom. 3(1976), 313–316.

[4] G. Debreu, A social equilibrium existence theorem. Proc. Natl. Acad. Sci. USA 38 (1952), 886–893.

[5] K. Fan, Fixed-point and minimax theorems in locally convex topological linear spaces.

Proc. Natl. Acad. Sci. USA38(1952), 121–126.

[6] H. Fu, Y. Sun, N.C. Yannelis and N.C. Zhang, Pure strategy equilibria in games with private and public information. J. Math. Econom. 43(2007), 523–531.

(10)

[7] M.A. Khan and Y. Sun,Pure strategies in games with private information. J. Math.

Econom. 24(1995), 633–653.

[8] M.A. Khan and Y. Sun,Integrals of set valued functions with a countable range. Math.

Oper. Res. 21(1996), 946–954.

[9] M.A. Khan and Y. Sun, Non-cooperative games on hyperfinite Loeb spaces. J. Math.

Econom. 31(1999), 455–492.

[10] E. Klein and A.C. Thompson,Theory of correspondences. Canadian Mathematical So- ciety Series of Monographs and Advanced Texts, John Wiley&Sons, Inc., New York, 1984.

[11] J.F. Nash,Non-cooperative games. Ann. of Math. (2)54(1951), 286–295.

[12] R. Radner and R. W. Rosenthal,Private information and pure-strategy equilibria. Math.

Oper. Res. 7(1982), 401–409.

[13] W. Shafer and H. Sonnenschein,Equilibrium in abstract economies without ordered pref- erences. J. Math. Econom.2(1975), 345–348.

[14] Y. Sun, Distributional properties of correspondences on Loeb Spaces. J. Funct. Anal.

139(1996), 68–93.

[15] N.C. Yannelis and N.D. Prabhakar, Existence of maximal elements and equilibrium in linear topological spaces. J. Math. Econom.12(1983), 233–245.

[16] H.Yu and Z. Zhang,Pure strategy equilibria in games with countable actions. J. Math.

Econom. 43(2007), 192–200.

[17] X.Z. Yuan,The Study of Minimax Inequalities and Applications to Economies and Vari- ational Inequalities. Mem. Amer. Math. Soc. 132(1988).

Received 13 September 2011 University of Bucharest,

Faculty of Mathematics and Computer Science, Department of Mathematics

14 Academiei St., 010014 Bucharest, Romania monica.patriche@yahoo.com