A Bayesian Abstract Economy with a Measure Space of Agents

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 𝜎-algebra, and his net trades are measurable with respect to it (this 𝜎-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)ℝ₊₊ denotes the set of strictly positive reals,(2)co𝐴 denotes the convex hull of the set 𝐴,(3)co𝐴 denotes the closed convex hull of the set 𝐴,(4)2^𝐴 denotes the set of all nonempty subsets of the set 𝐴,(5)if 𝐴⊂𝑋, where 𝑋 is a topological space, cl𝐴 denotes the closure of 𝐴.

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 𝑋 and 𝑌 be sets.

Definition 2.1. The graph of the correspondence Φ∶𝑋→2^𝑌 is the set 𝐺_Φ={(𝑥,𝑦)∈𝑋×𝑌∶𝑦∈Φ(𝑥)}.
Let 𝑋, 𝑌 be topological spaces and let Φ∶𝑋→2^𝑌 be a correspondence.(1)Φ is said to be upper semicontinuous if for each 𝑥∈𝑋 and each open set 𝑉 in 𝑌 with Φ(𝑥)⊂𝑉, there exists an open neighborhood 𝑈 of 𝑥 in 𝑋 such that Φ(𝑦)⊂𝑉 for each 𝑦∈𝑈.(2)Φ is said to be lower semicontinuous if for each 𝑥∈𝑋 and each open set 𝑉 in 𝑌 with Φ(𝑥)∩𝑉≠∅, there exists an open neighborhood 𝑈 of 𝑥 in 𝑋 such that Φ(𝑦)∩𝑉≠∅ for each 𝑦∈𝑈.(3)Φ is said to have open lower sections if Φ⁻¹(𝑦)∶={𝑥∈𝑋∶𝑦∈Φ(𝑥)} is open in 𝑋 for each 𝑦∈𝑌.

Lemma 2.2 (see [3]). Let 𝑋 and 𝑌 be two topological spaces and let 𝐴 be an open subset of 𝑋. Suppose that Φ₁∶𝑋→2^𝑌, Φ₂∶𝑋→2^𝑌 are upper semicontinuous such that Φ₂(𝑥)⊂Φ₁(𝑥) for all 𝑥∈𝐴. Then the correspondence Φ∶𝑋→2^𝑌 defined by ΦΦ(𝑥)=¹Φ(𝑥),if𝑥∉𝐴,₂(𝑥),if𝑥∈𝐴(2.1) is also upper semicontinuous.

Let now (Ω,ℱ,𝜇) be a complete, finite measure space, and let 𝑌 be a topological space.

Definition 2.3. (1) The correspondence Φ∶Ω→2^𝑌 is said to have a measurable graph if 𝐺_Φ∈ℱ⊗𝛽(𝑌), where 𝛽(𝑌) denotes the Borel 𝜎-algebra on 𝑌 and ⊗ denotes the product 𝜎-algebra.
(2) The correspondence Φ∶Ω→2^𝑌 is said to be lower measurable if for every open subset 𝑉 of 𝑌, the set {𝜔∈Ω∶Φ(𝜔)∩𝑉≠∅} is an element of ℱ.
Recall (see Debreu [4, page 359]) that if Φ∶Ω→2^𝑌 has a measurable graph, then Φ is lower measurable. Furthermore, if Φ(⋅) is closed valued and lower measurable, then Φ∶Ω→2^𝑌 has a measurable graph.

Lemma 2.4 (see [5]). Let Φ_𝑛∶Ω→2^𝑌, 𝑛=1,2,… be a sequence of correspondences with measurable graphs. Then the correspondences ⋃_𝑛Φ_𝑛, ⋂_𝑛Φ_𝑛 and 𝑌⧵Φ_𝑛 have measurable graphs.

Let (Ω,ℱ, 𝜇) be a measure space and 𝑌 be a Banach space.

It is known (see [6, Theorem  2, page 45]) that if 𝑥∶Ω→𝑌 is a 𝜇-measurable function, then 𝑥 is Bochner integrable if only if ∫_Ω‖𝑥(𝜔)‖𝑑𝜇(𝜔)<∞.

It is denoted by 𝐿₁(𝜇,𝑌) the space of equivalence classes of 𝑌-valued Bochner integrable functions 𝑥∶Ω→𝑌 normed by ‖𝑥‖=∫_Ω‖𝑥(𝜔)‖𝑑𝜇(𝜔).

Also it is known (see [6, page 50]) that 𝐿₁(𝜇,𝑌) is a Banach space.

We denote by 𝑆_Φ the set of all selections of the correspondence Φ∶Ω→2^𝑌 that belong to the space 𝐿₁(𝜇,𝑌), that is,

𝑆_Φ={𝑥∈𝐿₁(𝜇,𝑌)∶𝑥(𝜔)∈Φ(𝜔)𝜇-a.e.}.(2.2)

Definition 2.5 (see [7]). The integral of correspondence Φ∶Ω→2^𝑌 is the set {∫_Ω𝑥(𝜔)𝑑𝜇(𝜔)∶𝑥∈𝑆_Φ}.

We will denote the above set by ∫Φ(𝜔)𝑑𝜔 or simply ∫Φ.

Definition 2.6. The correspondence Φ∶Ω→2^𝑌 is said to be integrably bounded if there exists a map ℎ∈𝐿₁(𝜇,𝑅) such that sup{‖𝑥‖∶𝑥∈Φ(𝜔)}≤ℎ(𝜔)𝜇-a.e.
Moreover, note that if Ω is a complete measure space, 𝑌 is a separable Banach space and Φ∶Ω→2^𝑌 is an integrably bounded, nonempty valued correspondence having a measurable graph; then by the Aumann measurable selection theorem we can conclude that 𝑆_Φ is nonempty and therefore ∫_ΩΦ(𝜔)𝑑𝜇(𝜔) is nonempty as well.
Let 𝑋 be a topological space and let Φ∶Ω×𝑋→2^𝑌 be a nonempty valued correspondence.

Definition 2.7. A function 𝑓∶Ω×𝑋→𝑌 is said to be a Carathéodory-type selection from Φ if 𝑓(𝜔,𝑥)∈Φ(𝜔,𝑥) for all (𝜔,𝑥)∈Ω×𝑋, 𝑓(⋅,𝑥) is measurable for all 𝑥∈𝑋 and let 𝑓(𝜔,⋅) is continuous for all 𝜔∈Ω.

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 (Ω,ℱ,𝜇) be a complete, finite measure space, and let 𝑌 be a complete separable metric space. If 𝐻 belongs to ℱ⊗𝛽(𝑌), its projection Proj_Ω(𝐻) belongs to ℱ.

Theorem 2.9 (Aumann measurable selection theorem [2]). Let (Ω,ℱ,𝜇) be a complete finite measure space, let 𝑌 be a complete, separable metric space, and let Φ∶Ω→2^𝑌 be a nonempty valued correspondence with a measurable graph, that is, 𝐺_Φ∈ℱ⊗𝛽(𝑌). Then there is a measurable function 𝑓∶Ω→𝑌 such that 𝑓(𝜔)∈Φ(𝜔)𝜇-a.e.

Theorem 2.10 (Diestel's theorem [2, Theorem  3.1]). Let (Ω,ℱ,𝜇) be a complete finite measure space, let 𝑋 be a separable Banach space, and let Φ∶Ω→2^𝑌 be an integrably bounded, convex, weakly compact and nonempty valued correspondence. Then 𝑆_Φ={𝑥∈𝐿₁(𝜇,𝑌)∶𝑥(𝜔)∈Φ(𝜔)𝜇-a.e.} is weakly compact in 𝐿₁(𝜇,𝑌).

Theorem 2.11 (Carathéodory-type selection theorem [5]). Let (Ω,ℱ,𝜇) be a complete measure space, let 𝑍 be a complete separable metric space, and let 𝑌 be a separable Banach space. Let 𝑋∶Ω→2^𝑌 be a correspondence with a measurable graph, that is, 𝐺_𝑋∈ℱ⊗𝛽(𝑌) and let Φ∶Ω×𝑍→2^𝑌 be a convex valued correspondence (possibly empty) with a meaurable graph, that is, 𝐺_Φ∈ℱ⊗𝛽(𝑍)⊗𝛽(𝑌) where 𝛽(𝑌) and 𝛽(𝑍) are the Borel 𝜎-algebras of 𝑌 and 𝑍, respectively.
Suppose that(1)for each 𝜔∈Ω,Φ(𝜔,𝑥)⊂𝑋(𝜔) for all 𝑥∈𝑍,(2)for each 𝜔∈Ω, Φ(𝜔,⋅) has open lower sections in 𝑍; that is, for each 𝜔∈Ω and 𝑦∈𝑌, Φ⁻¹(𝜔,𝑦)={𝑥∈𝑍∶𝑦∈Φ(𝜔,𝑥)} is open in 𝑍,(3)for each (𝜔,𝑥)∈Ω×𝑍, if Φ(𝜔,𝑥)≠∅, then Φ(𝜔,𝑥) has a nonempty interior in 𝑋(𝜔).
Let 𝑈={(𝜔,𝑥)∈Ω×𝑍∶Φ(𝜔,𝑥)≠∅} and for each 𝑥∈𝑍, 𝑈^𝑥={𝜔∈Ω∶(𝜔,𝑥)∈𝑈} and for each 𝜔∈Ω, 𝑈^𝜔={𝑥∈𝑍∶(𝜔,𝑥)∈𝑈}. Then for each 𝑥∈𝑍, 𝑈^𝑥 is a measurable set in Ω and there exists a Caratheodory-type selection from Φ_|𝑈; that is, there exists a function 𝑓∶𝑈→𝑌 such that 𝑓(𝜔,𝑥)∈Φ(𝜔,𝑥) for all (𝜔,𝑥)∈𝑈, for each 𝑥∈𝑍, 𝑓(⋅,𝑥) is measurable on 𝑈^𝑥, and for each 𝜔∈Ω, 𝑓(𝜔,⋅) is continuous on 𝑈^𝜔. Moreover, 𝑓(⋅,⋅) is jointly measurable.

Theorem 2.12 (u.s.c. lifting theorem [2]). Let 𝑌 be a separable space, let (Ω,ℱ,𝜇) be a complete finite measure space, and let 𝑋∶Ω→2^𝑌 be an integrably bounded, nonempty, convex valued correspondence such that for all 𝜔∈Ω, 𝑋(𝜔) is a weakly compact, convex subset of 𝑌. Denote by 𝑆_𝑋 the set {𝑥∈𝐿₁(𝜇,𝑌)∶𝑥(𝜔)∈𝑋(𝜔)𝜇-a.e.}. Let Φ∶Ω×𝑆_𝑋→2^𝑌 be a nonempty, closed, convex valued correspondence such that Φ(𝜔,𝑥)⊂𝑋(𝜔) for all (𝜔,𝑥)∈Ω×𝑆¹_𝑋. Assume that for each fixed 𝑥∈𝑆_𝑋, Φ(⋅,𝑥) has a measurable graph and that for each fixed 𝜔∈Ω, Φ(𝜔,⋅)∶𝑆_𝑋→2^𝑌 is u.s.c. in the sense that the set {𝑥∈𝑆_𝑋∶Φ(𝜔,𝑥)⊂𝑉} is weakly open in 𝑆_𝑋 for every norm open subset 𝑉 of 𝑌. Define the correspondence Ψ∶𝑆_𝑋→2^𝑆𝑋 by Ψ(𝑥)={𝑦∈𝑆_𝑋∶𝑦(𝜔)∈Φ(𝜔,𝑥)𝜇-a.e.}.(2.3)
Then Ψ is weakly u.s.c.; that is, the set {𝑥∈𝑆_𝑋∶Ψ(𝑥)⊂𝑉} is weakly open in 𝑆_𝑋 for every weakly open subset 𝑉 of 𝑆_𝑋.

Theorem 2.13 (Measurability lifting theorem [8]). Let 𝑌 and 𝐸 be separable Banach spaces, and let (𝑇,𝜏,𝜐) and (Ω,ℱ,𝜇) be finite complete separable measure spaces. Let 𝛾∶𝑇×Ω×𝐸→2^𝑌 be a nonempty valued correspondence. Suppose that for each 𝑦∈𝐸, 𝛾(⋅,⋅,𝑦) has a measurable graph. Define the correspondence 𝜓∶Ω×𝐸→2^{𝐿1(𝜇,𝑌)} by 𝜓(𝑡,𝑦)={𝑥(𝑡)∈𝐿₁(𝜇,𝑌)∶𝑥(𝑡,𝜔)∈𝛾(𝑡,𝜔,𝑦)𝜇-a.e.}.(2.4)
Then for each 𝑦∈𝐸, 𝜓(⋅,𝑦) 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 (Ω,ℱ,𝜇) be a complete finite measure space, where Ω denotes the set of states of nature of the world and the 𝜎-algebra ℱ denotes the set of events. Let 𝑌 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 (𝑇,𝜏,𝜐) is a set 𝐺={(𝑋,𝑢,𝐴,ℱ_𝑡,𝑞_𝑡),𝑡∈𝑇}, where(1)𝑋∶𝑇×Ω→2^𝑌 is the random action (strategy) correspondence, where 𝑋(𝑡,𝜔)⊂𝑌 is interpreted as the strategy set of agent 𝑡 of the state of nature 𝜔;(2)for each fixed (𝑡,𝜔)∈𝑇×Ω, 𝑢(𝑡,𝜔,⋅,⋅)∶𝐿₁(𝜈,𝑌)×𝑋(𝑡,𝜔)→ℝ is the random utility function, where 𝑢(𝑡,𝜔,𝑥,𝑥_𝑡) is interpreted as the utility function of agent 𝑡, at the state of nature 𝜔, using his/her strategy 𝑥_𝑡 and all other players use the joint strategy 𝑥;(3)𝐴∶𝑇×Ω×𝐿₁(𝜐,𝑌)→2^𝑌 is the random constraint correspondence of agent 𝑡, where for all (𝑡,𝜔,𝑥)∈𝑇×Ω×𝐿₁(𝜐,𝑌), 𝐴(𝑡,𝜔,𝑥)⊂𝑋(𝑡,𝜔), and 𝐴(𝑡,𝜔,𝑥) is interpreted as the constraint of agent 𝑡, when the state is 𝜔 and other agents use the joint strategy 𝑥;(4)ℱ_𝑡 is a sub-𝜎-algebra of ℱ which denotes the private information of agent 𝑡;(5)𝑞_𝑡∶Ω→ℝ₊₊ is the prior of agent 𝑡, which is a Radon-Nikodym derivative such that ∫_Ω𝑞_𝑡(𝜔)𝑑𝜇(𝜔)=1.

Let 𝑆¹_𝑋𝑡={𝑦(𝑡)∈𝐿₁(𝜇,𝑋)∶𝑦(𝑡,⋅)∶Ω→𝑌 be ℱ_𝑡-measurable and 𝑦(𝑡,𝜔)∈𝑋(𝑡,𝜔)𝜇-a.e.}. Notice that 𝑆¹_𝑋𝑡 is the set of all Bocner integrable and ℱ_𝑡-measurable selections from the random strategy of agent 𝑡. In essence this is the set, out of which agent 𝑡 will pick his/her optimal choices. In particular, an element 𝑥_𝑡 in 𝑆¹_𝑋𝑡 is called a strategy for agent 𝑡. The typical element of 𝑆¹_𝑋𝑡 is denoted by 𝑥_𝑡 and that of 𝑋(𝑡,𝜔) by 𝑥_𝑡(𝜔). Let 𝑆¹_𝑋={𝑥∈𝐿₁(𝜐,𝐿₁(𝜇,𝑌))∶𝑥(𝑡,⋅)∈𝑆¹_𝑋𝑡𝜐-a.e.}. An element of 𝑆¹_𝑋 will be a joint strategy profile.

It will be convenient to assume that Ω is a countable set and the 𝜎-algebra ℱ_𝑡 is generated by a countable partition Λ of Ω. For each 𝜔∈Ω, let 𝐸_𝑡(𝜔) in Λ denote the smallest set in ℱ_𝑡 containing 𝜔 and assume that, for each 𝑡,

_{𝜔′∈𝐸𝑡(𝜔)}𝑞_𝑡(𝜔^′)𝑑𝜇(𝜔^′)>0.(3.1)

Definition 3.2. For each (𝑡,𝜔)∈𝑇×Ω, the interim expected utility of agent 𝑡, 𝑈(𝑡,𝜔,⋅,⋅)∶𝑆¹_𝑋×𝑋(𝑡,𝜔)→ℝ is defined as 𝑈(𝑡,𝜔,𝑥,𝑥_𝑡)=_{𝜔′∈𝐸𝑡(𝜔)}𝑢(𝑡,𝜔^′,𝑥(𝜔^′),𝑥_𝑡(𝜔^′))𝑞_𝑡(𝜔^′∣𝐸_𝑡(𝜔))𝑑𝜇(𝜔^′),(3.2) where 𝑞_𝑡(𝜔^′∣𝐸_𝑡(𝜔))=0,if𝜔^′∉𝐸_𝑡𝑞(𝜔)_𝑡(𝜔^′)∫_{𝜔∈𝐸𝑡(𝜔)}𝑞_𝑡(𝜔)𝑑𝜇(𝜔),if𝜔^′∈𝐸_𝑡(𝜔).(3.3)

Definition 3.3. A social equilibrium for 𝐺 is a strategy profile 𝑥^∗∈𝑆¹_𝑋 such that 𝜐-a.e.(i)𝑥^∗(𝑡,𝜔)∈𝐴(𝑡,𝜔,𝑥^∗)𝜇-a.e.,(ii)𝑈(𝑡,𝜔,𝑥,𝑥^∗(𝑡,𝜔))=max_{𝑦∈𝐴(𝑡,𝜔,}𝑥^∗₎𝑈(𝑡,𝜔,𝑥,𝑦)𝜇-a.e.

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

Theorem 3.4. Let 𝐺 be a social system with asymmetric information satisfying (A.1)–(A.4). Then there exists a social equilibrium for 𝐺.
One has the following assumptions:(A.1) (a)𝑋∶𝑇×Ω→2^𝑌 is a nonempty, convex, compact valued, and integrably bounded correspondence,(b) for each 𝑡∈𝑇, 𝑋(𝑡,⋅)∶Ω→2^𝑌 has an ℱ_𝑡 measurable graph, that is, for every open subset 𝑉 of 𝑌, the set 𝐺_𝑋(𝑡,⋅)∈ℱ_𝑡×𝛽(𝑌).(A.2) (a) for each (𝑡,𝜔)∈𝑇×Ω, 𝑢(𝑡,𝜔,⋅,⋅)∶𝐿₁(𝜐,𝑌)×𝑋(𝑡,𝜔)→ℝ is continuous where 𝐿₁(𝜐,𝑌) is endowed with the weak topology and 𝑋(𝑡,𝜔) with the norm topology,(b) for each fixed (𝑥,𝑦)∈𝐿₁(𝜐,𝑌)×𝑌, 𝑢(⋅,⋅,𝑥,𝑦)∶𝑇×Ω→ℝ is a measurable function,(c) for each (𝑡,𝜔,𝑥)∈𝑇×Ω×𝐿₁(𝜐,𝑌), 𝑢(𝑡,𝜔,𝑥,⋅)∶𝑋(𝑡,𝜔)→ℝ is concave,(d) for each 𝑡∈𝑇, 𝑢(𝑡,⋅,⋅,⋅) is integrably bounded.(A.3) (a)𝐴∶𝑇×Ω×𝐿₁(𝜐,𝑌)→2^𝑌 has a measurable graph,(b) for each (𝑡,𝜔)∈𝑇×Ω,𝐴(𝑡,𝜔,⋅)∶𝐿₁(𝜐,𝑌)×→2^𝑌 is a continuous correspondence with closed, convex, and nonempty values.(A.4) The correspondence 𝑡→𝑆¹_𝑋𝑡 has a measurable graph.

Remark 3.5. This theorem and its proof remain unchanged if the random constraint correspondence is defined as 𝐴_𝑖∶Ω×𝐿_𝑋→2^𝑌.

4. The Model

We will study the next model of the abstract economy.

Let (Ω,ℱ,𝜇) be a complete finite measure space, where Ω denotes the set of states of nature of the world and the 𝜎-algebra ℱ denotes the set of events. Let 𝑌 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 (𝑇,𝜏,𝜐) is a set 𝐺={(𝑋,ℱ_𝑡,𝐴,𝑃),𝑡∈𝑇}, where(1)𝑋∶𝑇×Ω→2^𝑌 is the random action (strategy) correspondence, where, 𝑋(𝑡,𝜔)⊂𝑌 is interpreted as the strategy set of agent 𝑡 of the state of nature 𝜔;(2)ℱ_𝑡 is a sub-𝜎-algebra of ℱ which denotes the private information of agent 𝑡;(3)for each 𝑡∈𝑇, 𝐴(𝑡,⋅,⋅)∶Ω×𝑆¹_𝑋→2^𝑌 is the random constraint correspondence of agent 𝑡, where for all (𝑡,𝜔,𝑥)∈𝑇×Ω×𝑆¹_𝑋, 𝐴(𝑡,𝜔,𝑥)⊂𝑋(𝑡,𝜔);(4)for each 𝑡∈𝑇, 𝑃(𝑡,⋅,⋅)∶Ω×𝑆¹_𝑋→2^𝑌 is the random preference correspondence of agent 𝑡, where for all (𝑡,𝜔,𝑥)∈𝑇×Ω×𝑆¹_𝑋, 𝑃(𝑡,𝜔,𝑥)⊂𝑋(𝑡,𝜔).

Definition 4.2. A Bayesian equilibrium for 𝐺 is a strategy profile 𝑥^∗∈𝑆¹_𝑋 such that for 𝜐-a.e.(i)𝑥^∗(𝑡,𝜔)∈𝐴(𝑡,𝜔,𝑥^∗)𝜇-a.e.,(ii)𝐴(𝑡,𝜔,𝑥^∗)∩𝑃(𝑡,𝜔,𝑥^∗)=∅𝜇-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 𝑈 we can define the correspondence 𝑃∶𝑇×Ω×𝑆¹_𝑋→2^𝑌 by 𝑃(𝑡,𝜔,𝑥)={𝑦∈𝑌∶𝑈(𝑡,𝜔,𝑥(𝑡,𝜔),𝑦)>𝑈(𝑡,𝜔,𝑥(𝑡,𝜔)𝜇-a.e.}.

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 (𝑇,𝜏,𝜐) be a measure space of agents and let 𝐺={(𝑋,ℱ_𝑡,𝐴,𝑃),𝑡∈𝑇} be a Bayesian abstract economy satisfying (A.1)–(A.5). Then there exists a Bayesian equilibrium for 𝐺.(A.1) (a)𝑋∶𝑇×Ω→2^𝑌 is a nonempty, convex, weakly compact-valued, and integrably bounded correspondence,(b) for each fixed 𝑡∈𝑇, 𝑋(𝑡,⋅) has an ℱ_𝑡-measurable graph, that is, for every open subset 𝑉 of 𝑌, the set 𝐺_{𝑋(𝑡,⋅)}∈ℱ_𝑡×𝛽(𝑌).(A.2) (a)𝐴∶𝑇×Ω×𝑆¹_𝑋→2^𝑌 has a measurable graph,(b) for each (𝑡,𝜔)∈𝑇×Ω, 𝐴(𝑡,𝜔,⋅)∶𝑆¹_𝑋→2^𝑌 is an upper semicontinuous correspondence with closed, convex and nonempty values. (A.3) (a)𝑃∶𝑇×Ω×𝑆¹_𝑋→2^𝑌 has a measurable graph,(b) for each (𝑡,𝜔)∈𝑇×Ω, 𝑃(𝑡,𝜔,⋅)∶𝑆¹_𝑋→2^𝑌 is an upper semicontinuous correspondence with closed, convex, and nonempty values.(A.4) The correspondence 𝑡→𝑆¹_𝑋𝑡 has a measurable graph.(A.5) (a) for each 𝜔∈Ω, for each 𝑥(𝑡)∈𝑆¹_𝑋, 𝑥(𝑡,𝜔)∉𝐴(𝑡,𝜔,𝑥)∩𝑃(𝑡,𝜔,𝑥),(b) the set 𝑈^(𝑡,𝜔)={𝑥∈𝑆¹_𝑋∶ 𝐴(𝑡,𝜔,𝑥)∩𝑃(𝑡,𝜔,𝑥)≠∅} is weakly open in 𝑆¹_𝑋.

Proof. Define Φ∶𝑇×Ω×𝑆¹_𝑋→2^𝑌 by Φ(𝑡,𝜔,𝑥)=𝐴(𝑡,𝜔,𝑥)∩𝑃(𝑡,𝜔,𝑥). We will prove first that 𝑆¹_𝑋 is nonempty, convex, weakly compact.
Since (Ω,ℱ,𝜇) is a complete finite measure space, 𝑌 is a separable Banach space and 𝑋(𝑡,⋅)∶Ω→2^𝑌 has measurable graph, and by Aumann's selection theorem it follows that there exists a function 𝑓(𝑡,⋅)∶Ω→𝑌 such that 𝑓(𝑡,𝜔)∈𝑋(𝑡,𝜔)𝜇-a.e. Since 𝑋(𝑡,⋅) is integrably bounded, we have that 𝑓(𝑡,⋅)∈𝐿₁(𝜇,𝑌), hence 𝑆_𝑋𝑡 is nonempty, and 𝑆_𝑋=∏_𝑡∈𝑇𝑆_𝑋𝑡 is nonempty. 𝑆¹_𝑋𝑡 is convex and 𝑆¹_𝑋 is also convex. Since 𝑋(𝑡,⋅)∶Ω→2^𝑌 is integrably bounded and has convex weakly compact values, by Diestel's Theorem it follows that 𝑆¹_𝑋𝑡 is a weakly compact subset of 𝐿₁(𝜇,𝑌) and so is 𝑆¹_𝑋. We have that 𝑆¹_𝑋 is a metrizable set as being a weakly compact subset of the separable Banach space 𝐿₁(𝜐,𝐿₁(𝜇,𝑌)) (Dunford-Schwartz [9, page 434]).
Since all the values of the correspondence 𝐴 are contained in the compact set 𝑋(⋅,⋅) and 𝐴 is closed and convex valued (hence weakly closed), it follows that 𝐴 is weakly compact valued.
Then Φ is convex valued and for each (𝑡,𝜔)∈𝑇×Ω, Φ(𝑡,𝜔,⋅) is upper semicontinuous. We have that Φ(⋅,⋅,𝑥) has a measurable graph for each 𝑥∈𝑆¹_𝑋. Let 𝑈={(𝑡,𝜔,𝑥)∈Ω×𝑆¹_𝑋∶Φ(𝑡,𝜔,𝑥)≠∅}. For each 𝑥∈𝑆¹_𝑋, let 𝑈^𝑥={(𝑡,𝜔)∈𝑇×Ω∶Φ(𝑡,𝜔,𝑥)≠∅} and for each 𝜔∈Ω, let 𝑈^(𝑡,𝜔)={𝑥∈𝑆¹_𝑋∶Φ(𝑡,𝜔,𝑥)≠∅}. Define 𝐺∶𝑇×Ω×𝑆¹_𝑋→2^𝑌 by 𝐺(𝑡,𝜔,𝑥)=Φ(𝑡,𝜔,𝑥),if(𝑡,𝜔,𝑥)∈𝑈,𝐴(𝑡,𝜔,𝑥),if(𝑡,𝜔,𝑥)∉𝑈.(5.1)
For each 𝑥∈𝑆¹_𝑋, the correspondence 𝐺(⋅,⋅,𝑥) has a measurable graph.
By assumption (A5)(b), the set 𝑈^(𝑡,𝜔)={𝑥∈𝑆¹_𝑋∶𝐴(𝑡,𝜔,𝑥)∩𝑃(𝑡,𝜔,𝑥)≠∅} is weakly open in 𝑆¹_𝑋. For each (𝑡,𝜔)∈𝑇×Ω, 𝐺(𝑡,𝜔,⋅)∶𝑆¹_𝑋→2^𝑌 is upper semicontinuous.
Let 𝑉 be weakly open in 𝑆¹_𝑋 and 𝑥∈𝑆¹_𝑋: 𝑊={𝑥∈𝑆¹_𝑋∶𝐺(𝑡,𝜔,𝑥)⊂𝑉}={𝑥∈𝑈^(𝑡,𝜔)∶𝐺(𝑡,𝜔,𝑥)⊂𝑉}∪{𝑥∈𝑆¹_𝑋⧵𝑈^(𝑡,𝜔)∶𝐺(𝑡,𝜔,𝑥)⊂𝑉}={𝑥∈𝑈^(𝑡,𝜔)∶Φ(𝑡,𝜔,𝑥)⊂𝑉}∪{𝑥∈𝑆¹_𝑋∶𝐺(𝑡,𝜔,𝑥)⊂𝑉}.(5.2)
𝑊 is an open set, because 𝑈^(𝑡,𝜔) is open, Φ(𝑡,𝜔,⋅) is an upper semicontinuous map on 𝑈^(𝑡,𝜔), and the set {𝑥∈𝑆¹_𝑋∶𝐺(𝑡,𝜔,𝑥)⊂𝑉} is open since 𝐺(𝑡,𝜔,⋅) is u.s.c. Moreover, 𝐺 is convex and nonempty valued.
Define the correspondence 𝜑∶𝑇×𝑆¹_𝑋→2^{𝐿1(𝜇,𝑌)} by 𝜑(𝑡,𝑥)={𝑦(𝑡)∈𝐿₁(𝜇,𝑌)∶𝑦(𝑡,𝜔)∈𝐺(𝑡,𝜔,𝑥)𝜇-a.e.}∩𝑆¹_𝑋𝑡.
By the measurability lifting theorem (Theorem 2.13), the correspondence 𝑡→{𝑦(𝑡)∈𝐿₁(𝜇,𝑌)∶𝑦(𝑡,𝜔)∈𝐺(𝑡,𝜔,𝑥)𝜇-a.e.} has a measurable graph and so does 𝑡→𝑆¹_𝑋𝑡 by (A.4). Thus, for each fixed 𝑥∈𝑆¹_𝑋, 𝜑(⋅,𝑥) has a measurable graph.
Since for each fixed 𝑥∈𝑆¹_𝑋, 𝐺(⋅,⋅,𝑥) 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 𝜑 is nonempty valued. It follows by the u.s.c. lifting theorem that for each fixed 𝑡, 𝜑(𝑡,⋅) is weakly u.s.c.
Define 𝐺^′∶𝑆¹_𝑋→2^𝑆1𝑋, by 𝐺^′(𝑥)={𝑦∈𝑆¹_𝑋∶𝑦(𝑡)∈𝜑(𝑡,𝑥)𝜈-a.e.}.
Another application of the u.s.c. lifting theorem enables us to conclude that 𝐺^′ is a weakly u.s.c. correspondence which is obviously convex valued (since 𝜑 is convex valued) and also nonempty valued (recall once more the Aumann measurable selection theorem and notice that the set 𝑆¹_𝑋 is metrizable).
𝐺^′ is an upper semicontinuous correspondence and has also nonempty convex closed values.
By Fan-Glicksberg's fixed-point theorem in [5], there exists 𝑥^∗∈𝑆¹_𝑋 such that 𝑥^∗∈𝐺^′(𝑥^∗). It follows that 𝑥^∗∈𝑆¹_𝑋 and 𝑥^∗(𝑡)∈𝜑(𝑡,𝑥^∗)𝜈-a.e. Thus, we have that 𝑥^∗∈𝑆¹_𝑋 and 𝑥^∗(𝑡,𝜔)∈𝐺(𝑡,𝜔,𝑥^∗)𝜇-a.e., 𝜈-a.e.
By assumption (A.4)(a), it follows that 𝑥^∗(𝑡,𝜔)∉(𝐴∩𝑃)(𝑡,𝜔,𝑥^∗), then we have that 𝑥^∗∉𝑈 and 𝑥^∗(𝑡,𝜔)∈𝐴(𝑡,𝜔,𝑥^∗).
Therefore, for 𝜐-a.e.(1)𝑥^∗(𝑡,𝜔)∈𝐴(𝑡,𝜔,𝑥^∗)𝜇-a.e.,(2)𝐴(𝑡,𝜔,𝑥^∗)∩𝑃(𝑡,𝜔,𝑥^∗)=∅𝜇-a.e.

If there exists a selector 𝐹 for 𝐴∩𝑃 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 (𝑇,𝜏,𝜐) be a measure space of agents and let 𝐺={(𝑋,ℱ_𝑡,𝐴,𝑃),𝑡∈𝑇} be a Bayesian abstract economy satisfying (A.1)–(A.5). Then there exists a Bayesian equilibrium for 𝐺.(A.1) (a)𝑋∶𝑇×Ω→2^𝑌 is a nonempty, convex, weakly compact-valued, and integrably bounded correspondence,(b) for each fixed 𝑡∈𝑇, 𝑋(𝑡,⋅) has an ℱ_𝑡-measurable graph, that is, for every open subset 𝑉 of 𝑌, the set 𝐺_𝑋(𝑡,⋅)∈ℱ_𝑡×𝛽(𝑌).(A.2) (a)𝐴∶𝑇×Ω×𝑆¹_𝑋→2^𝑌 has a measurable graph,(b) for each (𝑡,𝜔)∈𝑇×Ω, 𝐴(𝑡,𝜔,⋅)∶𝑆¹_𝑋→2^𝑌 is an upper semicontinuous correspondence with closed convex and nonempty values. (A.3) (a)𝑃∶𝑇×Ω×𝑆¹_𝑋→2^𝑌 has a measurable graph,(b) for each (𝑡,𝜔)∈𝑇×Ω, 𝑃(𝑡,𝜔,⋅)∶𝑆¹_𝑋→2^𝑌 is a upper semicontinuous correspondence with closed, convex, and nonempty values.(A.4) (a) for each (𝑡,𝜔)∈𝑇×Ω, the set 𝑈^(𝑡,𝜔)={𝑥∈𝑆¹_𝑋∶𝐴(𝑡,𝜔,𝑥)∩𝑃(𝜔,𝑥)=∅𝜇-a.e.} is open in 𝑆¹_𝑋,(b) for each 𝜔∈Ω, for each 𝑥∈𝑈, 𝑥(𝑡,𝜔)∉𝐴(𝑡,𝜔,𝑥)∩𝑃(𝑡,𝜔,𝑥).(A.5) (a) there exists a selector 𝐹∶𝑆¹_𝑋→2^𝑌 for 𝐴∩𝑃∶𝑆¹_𝑋→2^𝑌 such that for each 𝑥∈𝑆¹_𝑋,𝐹(⋅,⋅,𝑥) has measurable graph and for each (𝑡,𝜔)∈𝑇×Ω, 𝐹(𝑡,𝜔,⋅)∶𝑈^(𝑡,𝜔)→2^𝑌 is weakly upper semicontinuous with closed and convex values.

Proof. Define 𝐺∶𝑇×Ω×𝑆¹_𝑋→2^𝑌 by 𝐺(𝑡,𝜔,𝑥)=𝐹(𝑡,𝜔,𝑥)if(𝑡,𝜔,𝑥)∈𝑈,𝐴(𝑡,𝜔,𝑥)if(𝑡,𝜔,𝑥)∉𝑈.(5.3)
For each 𝑥∈𝑆¹_𝑋, the correspondence 𝐺(⋅,⋅,𝑥) has a measurable graph.
By assumption (A4)(b), the set 𝑈^(𝑡,𝜔)={𝑥∈𝑆¹_𝑋∶𝐴(𝑡,𝜔,𝑥)∩𝑃(𝑡,𝜔,𝑥)≠∅} is weakly open in 𝑆¹_𝑋. For each (𝑡,𝜔)∈𝑇×Ω, 𝐺(𝑡,𝜔,⋅)∶𝑆¹_𝑋→2^𝑌 is upper semicontinuous.
Let 𝑉 be weakly open in 𝑆¹_𝑋 and 𝑥∈𝑆¹_𝑋: 𝑊={𝑥∈𝑆¹_𝑋∶𝐺(𝑡,𝜔,𝑥)⊂𝑉}={𝑥∈𝑈^(𝑡,𝜔)∶𝐺(𝑡,𝜔,𝑥)⊂𝑉}∪{𝑥∈𝑆¹_𝑋⧵𝑈^(𝑡,𝜔)∶𝐺(𝑡,𝜔,𝑥)⊂𝑉}={𝑥∈𝑈^(𝑡,𝜔)∶𝐹(𝑡,𝜔,𝑥)⊂𝑉}∪{𝑥∈𝑆¹_𝑋∶𝐺(𝑡,𝜔,𝑥)⊂𝑉}.(5.4)
𝑊 is an open set, because 𝑈^(𝑡,𝜔) is open, Φ(𝑡,𝜔,⋅) is a upper semicontinuous map on 𝑈^(𝑡,𝜔), and the set {𝑥∈𝑆¹_𝑋∶𝐺(𝑡,𝜔,𝑥)⊂𝑉} is open since 𝐺(𝑡,𝜔,⋅) is u.s.c. Moreover, 𝐺 is convex and nonempty valued.
Define the correspondence 𝜑∶𝑇×𝑆¹_𝑋→2^{𝐿1(𝜇,𝑌)} by 𝜑(𝑡,𝑥)={𝑦(𝑡)∈𝐿₁(𝜇,𝑌)∶𝑦(𝑡,𝜔)∈𝐺(𝑡,𝜔,𝑥)𝜇-a.e.}∩𝑆¹_𝑋𝑡.
By the measurability lifting theorem (Theorem 2.13), the correspondence 𝑡→{𝑦(𝑡)∈𝐿₁(𝜇,𝑌)∶𝑦(𝑡,𝜔)∈𝐺(𝑡,𝜔,𝑥)𝜇-a.e.} has a measurable graph and so does 𝑡→𝑆¹_𝑋𝑡 by (A.4). Thus, for each fixed 𝑥∈𝑆¹_𝑋, 𝜑(⋅,𝑥) has a measurable graph.
Since for each fixed 𝑥∈𝑆¹_𝑋, 𝐺(⋅,⋅,𝑥) 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 𝜑 is nonempty valued. It follows by the u.s.c. lifting theorem that for each fixed 𝑡, 𝜑(𝑡,⋅) is weakly u.s.c.
Define 𝐺^′∶𝑆¹_𝑋→2^𝑆1𝑋, by 𝐺^′(𝑥)={𝑦∈𝑆¹_𝑋∶𝑦(𝑡)∈𝜑(𝑡,𝑥)𝜈-a.e.}.
Another application of the u.s.c. lifting Theorem enables us to conclude that 𝐺^′ is a weakly u.s.c. correspondence which is obviously convex valued (since 𝜑 is convex valued) and also nonempty valued (recall once more the Aumann measurable selection theorem and notice that the set 𝑆¹_𝑋 is metrizable).
𝐺^′ is an upper semicontinuous correspondence and has also nonempty convex closed values.
By Fan-Glicksberg's fixed-point theorem in [5], there exists 𝑥^∗∈𝑆¹_𝑋 such that 𝑥^∗∈𝐺^′(𝑥^∗). It follows that 𝑥^∗∈𝑆¹_𝑋 and 𝑥^∗(𝑡)∈𝜑(𝑡,𝑥^∗)𝜈-a.e. Thus, we have that 𝑥^∗∈𝑆¹_𝑋 and 𝑥^∗(𝑡,𝜔)∈𝐺(𝑡,𝜔,𝑥^∗)𝜇-a.e., 𝜈-a.e.
By assumption (A.4)(a), it follows that 𝑥^∗(𝑡,𝜔)∉(𝐴∩𝑃)(𝑡,𝜔,𝑥^∗), then we have that 𝑥^∗∉𝑈 and 𝑥^∗(𝑡,𝜔)∈𝐴(𝑡,𝜔,𝑥^∗).
Therefore, for 𝜐-a.e.(1)𝑥^∗(𝑡,𝜔)∈𝐴(𝑡,𝜔,𝑥^∗)𝜇-a.e.,(2)𝐴(𝑡,𝜔,𝑥^∗)∩𝑃(𝑡,𝜔,𝑥^∗)=∅𝜇-a.e.