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Abstract and Applied Analysis
Volume 2009 (2009), Article ID 523619, 11 pages
http://dx.doi.org/10.1155/2009/523619
Research Article

A Bayesian Abstract Economy with a Measure Space of Agents

Department of Probability, Statistics and Operations Research, Faculty of Mathematics and Computer Science, University of Bucharest, Street Academiei 14, 010014 Bucharest, Romania

Received 7 April 2009; Accepted 28 August 2009

Academic Editor: Nikolaos Papageorgiou

Copyright © 2009 Monica Patriche. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We define the model of an abstract economy with differential (asymmetric) information and a measure space of agents. We generalize N. C. Yannelis's result (2007), considering that each agent is characterised by a random preference correspondence instead of having a random utility function. We establish two different equilibrium existence results.

1. Introduction

We define the model of an abstract economy with asymmetric information and a measure set of agents, each of which is characterized by a private information set, an action (strategy) correspondence, a random constraint correspondence, and a random preference correspondence. The preference correspondences need not be representable by utility functions. The equilibrium concept is an extension of the deterministic equilibrium. We also present the model of Yannelis (see [1]) in which the agents maximize their expected utilities. Our model is a generalization of Yannelis's model.

A purpose of this paper is to prove the existence of equilibrium for an abstract economy with differential information and a measure space of agents. The assumptions on correspondences refer to upper semicontinuity and measurable graph. We use in this paper several results on the continuity and measurability of the set of integrable selections from a Banach-valued correspondences.

The model of an abstract economy with differential (asymmetric) information captures the meaning of trades under uncertainty. All economic activity in a society is made under conditions of uncertainty (incomplete information). The asymmetric information in the Arrow-Debreu model was introduced by Radner [2]. In his model each agent has his own private information set which is described by a partition of an exogenously given set of states of nature. The information partition of each agent generates a -algebra, and his net trades are measurable with respect to it (this -algebra). Thus, optimal choices reflect the private information of each agent.

The paper is organized as follows. In Section 2, some notational and terminological conventions are given. We also present, for the reader's convenience, some results on Bochner integration. In Section 3, Yannelis's expected utility model of differential information abstract economy and his main result in [1] are presented. Section 4 introduces our model, that is, the abstract economy with asymmetric information and a continuum of agents. Section 5 contains existence results for upper semicontinuous correspondences.

2. Mathematical Preliminaries

2.1. Notation and Definition

Throughout this paper, we will use the following notation:

(1) denotes the set of strictly positive reals,(2)co denotes the convex hull of the set ,(3) denotes the closed convex hull of the set ,(4) denotes the set of all nonempty subsets of the set ,(5)if , where is a topological space, cl denotes the closure of .

For the reader's convenience, we review a few basic definitions and results from continuity and measurability of correspondences, Bochner integrable functions, and the integral of a correspondence.

Let and be sets.

Definition 2.1. The graph of the correspondence is the set .
Let , be topological spaces and let be a correspondence.(1) is said to be upper semicontinuous if for each and each open set in with , there exists an open neighborhood of in such that for each .(2) is said to be lower semicontinuous if for each and each open set in with , there exists an open neighborhood of in such that for each .(3) is said to have open lower sections if is open in for each .

Lemma 2.2 (see [3]). Let and be two topological spaces and let be an open subset of . Suppose that , are upper semicontinuous such that for all . Then the correspondence defined by is also upper semicontinuous.

Let now be a complete, finite measure space, and let be a topological space.

Definition 2.3. (1) The correspondence is said to have a measurable graph if , where denotes the Borel -algebra on and denotes the product -algebra.
(2) The correspondence is said to be lower measurable if for every open subset of , the set is an element of .
Recall (see Debreu [4, page 359]) that if has a measurable graph, then is lower measurable. Furthermore, if is closed valued and lower measurable, then has a measurable graph.

Lemma 2.4 (see [5]). Let , be a sequence of correspondences with measurable graphs. Then the correspondences , and have measurable graphs.

Let , be a measure space and be a Banach space.

It is known (see [6, Theorem  2, page 45]) that if is a -measurable function, then is Bochner integrable if only if .

It is denoted by the space of equivalence classes of -valued Bochner integrable functions normed by .

Also it is known (see [6, page 50]) that is a Banach space.

We denote by the set of all selections of the correspondence that belong to the space , that is,

Definition 2.5 (see [7]). The integral of correspondence is the set .

We will denote the above set by or simply .

Definition 2.6. The correspondence is said to be integrably bounded if there exists a map such that .
Moreover, note that if is a complete measure space, is a separable Banach space and is an integrably bounded, nonempty valued correspondence having a measurable graph; then by the Aumann measurable selection theorem we can conclude that is nonempty and therefore is nonempty as well.
Let be a topological space and let be a nonempty valued correspondence.

Definition 2.7. A function is said to be a Carathéodory-type selection from if for all , is measurable for all and let is continuous for all

The results below have been used in the proof of our theorems. For more details and further references see the paper quoted.

Theorem 2.8 (Projection theorem). Let be a complete, finite measure space, and let be a complete separable metric space. If belongs to , its projection belongs to .

Theorem 2.9 (Aumann measurable selection theorem [2]). Let be a complete finite measure space, let be a complete, separable metric space, and let be a nonempty valued correspondence with a measurable graph, that is, . Then there is a measurable function such that

Theorem 2.10 (Diestel's theorem [2, Theorem  3.1]). Let be a complete finite measure space, let be a separable Banach space, and let be an integrably bounded, convex, weakly compact and nonempty valued correspondence. Then is weakly compact in .

Theorem 2.11 (Carathéodory-type selection theorem [5]). Let , be a complete measure space, let be a complete separable metric space, and let be a separable Banach space. Let be a correspondence with a measurable graph, that is, and let be a convex valued correspondence (possibly empty) with a meaurable graph, that is, where and are the Borel -algebras of and , respectively.
Suppose that(1)for each for all ,(2)for each , has open lower sections in ; that is, for each and , is open in ,(3)for each , if , then has a nonempty interior in .
Let and for each , and for each , . Then for each , is a measurable set in and there exists a Caratheodory-type selection from ; that is, there exists a function such that for all , for each , is measurable on and for each , is continuous on . Moreover, is jointly measurable.

Theorem 2.12 (u.s.c. lifting theorem [2]). Let be a separable space, let be a complete finite measure space, and let be an integrably bounded, nonempty, convex valued correspondence such that for all , is a weakly compact, convex subset of . Denote by the set . Let be a nonempty, closed, convex valued correspondence such that for all . Assume that for each fixed , has a measurable graph and that for each fixed , is u.s.c. in the sense that the set is weakly open in for every norm open subset of . Define the correspondence by
Then is weakly u.s.c.; that is, the set is weakly open in for every weakly open subset of .

Theorem 2.13 (Measurability lifting theorem [8]). Let and be separable Banach spaces, and let and be finite complete separable measure spaces. Let be a nonempty valued correspondence. Suppose that for each , has a measurable graph. Define the correspondence by
Then for each , has a measurable graph.

3. A Bayesian Social Equilibrium Existence Theorem

We present Yannelis's model [1] of an abstract economy with asymmetric information and a continuum of agents. In this model, there is assigned to each agent, in addition to his/her random utility function, a private information set, which is a measurable partition of the exogenously given probability measure space (which describes the states of nature of the world).

Let be a complete finite measure space, where denotes the set of states of nature of the world and the -algebra denotes the set of events. Let be a separable Banach space whose dual has the RNP, denoting the commodity or strategy space.

Definition 3.1. A Bayesian abstract economy (or social system) with differential information and a measure space of agents is a set , where(1) is the random action (strategy) correspondence, where is interpreted as the strategy set of agent of the state of nature (2)for each fixed , is the random utility function, where is interpreted as the utility function of agent , at the state of nature , using his/her strategy and all other players use the joint strategy ;(3) is the random constraint correspondence of agent , where for all , and is interpreted as the constraint of agent , when the state is and other agents use the joint strategy ;(4) is a sub- -algebra of which denotes the private information of agent ;(5) is the prior of agent , which is a Radon-Nikodym derivative such that .

Let be -measurable and . Notice that is the set of all Bocner integrable and -measurable selections from the random strategy of agent . In essence this is the set, out of which agent will pick his/her optimal choices. In particular, an element in is called a strategy for agent . The typical element of is denoted by and that of by . Let . An element of will be a joint strategy profile.

It will be convenient to assume that is a countable set and the -algebra is generated by a countable partition of . For each , let in denote the smallest set in containing and assume that, for each ,

Definition 3.2. For each , the interim expected utility of agent , is defined as where

Definition 3.3. A social equilibrium for is a strategy profile such that (i) ,(ii)

The following theorem is the main result of Yannelis in [1].

Theorem 3.4. Let be a social system with asymmetric information satisfying (A.1)–(A.4). Then there exists a social equilibrium for .
One has the following assumptions:(A.1)(a) is a nonempty, convex, compact valued, and integrably bounded correspondence,(b) for each , has an measurable graph, that is, for every open subset of , the set .(A.2)(a) for each , is continuous where is endowed with the weak topology and with the norm topology,(b) for each fixed , is a measurable function,(c) for each , is concave,(d) for each , is integrably bounded.(A.3)(a) has a measurable graph,(b) for each is a continuous correspondence with closed, convex, and nonempty values.(A.4) The correspondence has a measurable graph.

Remark 3.5. This theorem and its proof remain unchanged if the random constraint correspondence is defined as .

4. The Model

We will study the next model of the abstract economy.

Let be a complete finite measure space, where denotes the set of states of nature of the world and the -algebra denotes the set of events. Let be a separable Banach space whose dual has the RNP, denoting the commodity or strategy space.

Definition 4.1. A Bayesian abstract economy (or social system) with differential information and a measure space of agents is a set , where(1) is the random action (strategy) correspondence, where, is interpreted as the strategy set of agent of the state of nature ;(2) is a sub- -algebra of which denotes the private information of agent ;(3)for each , is the random constraint correspondence of agent , where for all , (4)for each , is the random preference correspondence of agent , where for all ,

Definition 4.2. A Bayesian equilibrium for is a strategy profile such that for -a.e.(i) -a.e.,(ii) -a.e.

Remark 4.3. This model of abstract economy is a generalization of Yannelis's model presented in Section 3, since for intern expected utilities we can define the correspondence by .

5. Bayesian Equilibrium Existence Theorems

Now we establish an equilibrium existence theorem for Bayesian abstract economies with a measure space of agents and with upper semicontinuous correspondences. Our theorem generalizes Theorem  1 in [1].

Theorem 5.1. Let be a measure space of agents and let be a Bayesian abstract economy satisfying (A.1)–(A.5). Then there exists a Bayesian equilibrium for .(A.1)(a) is a nonempty, convex, weakly compact-valued, and integrably bounded correspondence,(b) for each fixed , has an -measurable graph, that is, for every open subset of , the set .(A.2)(a) has a measurable graph,(b) for each , is an upper semicontinuous correspondence with closed, convex and nonempty values. (A.3)(a) has a measurable graph,(b) for each , is an upper semicontinuous correspondence with closed, convex, and nonempty values.(A.4) The correspondence has a measurable graph.(A.5)(a) for each , for each , ,(b) the set is weakly open in .

Proof. Define by . We will prove first that is nonempty, convex, weakly compact.
Since is a complete finite measure space, is a separable Banach space and has measurable graph, and by Aumann's selection theorem it follows that there exists a function such that -a.e. Since is integrably bounded, we have that , hence is nonempty, and is nonempty. is convex and is also convex. Since is integrably bounded and has convex weakly compact values, by Diestel's Theorem it follows that is a weakly compact subset of and so is . We have that is a metrizable set as being a weakly compact subset of the separable Banach space (Dunford-Schwartz [9, page 434]).
Since all the values of the correspondence are contained in the compact set and is closed and convex valued (hence weakly closed), it follows that is weakly compact valued.
Then is convex valued and for each , is upper semicontinuous. We have that has a measurable graph for each . Let . For each , let and for each , let . Define by
For each , the correspondence has a measurable graph.
By assumption (A5)(b), the set is weakly open in . For each , is upper semicontinuous.
Let be weakly open in and :
is an open set, because is open, is an upper semicontinuous map on and the set is open since is u.s.c. Moreover, is convex and nonempty valued.
Define the correspondence by .
By the measurability lifting theorem (Theorem 2.13), the correspondence has a measurable graph and so does by (A.4). Thus, for each fixed , has a measurable graph.
Since for each fixed , has a measurable graph and it is nonempty valued, then by the Aumann measurable selection theorem, it admits a measurable selection and we can conclude that is nonempty valued. It follows by the u.s.c. lifting theorem that for each fixed , is weakly u.s.c.
Define , by .
Another application of the u.s.c. lifting theorem enables us to conclude that is a weakly u.s.c. correspondence which is obviously convex valued (since is convex valued) and also nonempty valued (recall once more the Aumann measurable selection theorem and notice that the set is metrizable).
is an upper semicontinuous correspondence and has also nonempty convex closed values.
By Fan-Glicksberg's fixed-point theorem in [5], there exists such that . It follows that and -a.e. Thus, we have that and -a.e., -a.e.
By assumption (A.4)(a), it follows that , then we have that and .
Therefore, for -a.e.(1) -a.e.,(2) -a.e.

If there exists a selector for such that it has measurable graph and it is weakly upper semicontinuous in the third argument, we obtain the following theorem.

Theorem 5.2. Let be a measure space of agents and let be a Bayesian abstract economy satisfying (A.1)–(A.5). Then there exists a Bayesian equilibrium for .(A.1)(a) is a nonempty, convex, weakly compact-valued, and integrably bounded correspondence,(b) for each fixed , has an -measurable graph, that is, for every open subset of , the set .(A.2)(a) has a measurable graph,(b) for each , is an upper semicontinuous correspondence with closed convex and nonempty values. (A.3)(a) has a measurable graph,(b) for each , is a upper semicontinuous correspondence with closed, convex, and nonempty values.(A.4)(a) for each , the set is open in ,(b) for each , for each , .(A.5)(a) there exists a selector for such that for each has measurable graph and for each , is weakly upper semicontinuous with closed and convex values.

Proof. Define by
For each , the correspondence has a measurable graph.
By assumption (A4)(b), the set is weakly open in . For each , is upper semicontinuous.
Let be weakly open in and :
is an open set, because is open, is a upper semicontinuous map on , and the set is open since is u.s.c. Moreover, is convex and nonempty valued.
Define the correspondence by .
By the measurability lifting theorem (Theorem 2.13), the correspondence has a measurable graph and so does by (A.4). Thus, for each fixed , has a measurable graph.
Since for each fixed , has a measurable graph and it is nonempty valued, then by the Aumann measurable selection theorem, it admits a measurable selection and we can conclude that is nonempty valued. It follows by the u.s.c. lifting theorem that for each fixed , is weakly u.s.c.
Define , by .
Another application of the u.s.c. lifting Theorem enables us to conclude that is a weakly u.s.c. correspondence which is obviously convex valued (since is convex valued) and also nonempty valued (recall once more the Aumann measurable selection theorem and notice that the set is metrizable).
is an upper semicontinuous correspondence and has also nonempty convex closed values.
By Fan-Glicksberg's fixed-point theorem in [5], there exists such that . It follows that and -a.e. Thus, we have that and -a.e., -a.e.
By assumption (A.4)(a), it follows that , then we have that and .
Therefore, for -a.e.(1) -a.e.,(2) -a.e.

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