The necessity of nowhere equivalence

Abstract

We prove some regularity properties (convexity, closedness, compactness and preservation of upper hemicontinuity) for distribution and regular conditional distribution of correspondences under the nowhere equivalence condition. We show the necessity of such a condition for any of these properties to hold. As an application, we demonstrate that the nowhere equivalence condition is satisfied on the underlying agent space if and only if pure-strategy Nash equilibria exist in general large games with any fixed uncountable compact action space. 1 See [22] and [26] for the construction of Loeb spaces. 2 As noted in [15], atomless Loeb probability spaces are saturated. For some other applications of Loeb and saturated probability spaces, see, for example, [5], [7], [8], [10], [16], [19], [24], [34], [35], and [36]. 3 When the target space is a Banach space, one can also consider Bochner and Gelfand integration of correspondences. The same kind of regularity properties were shown to hold under Loeb/saturated probability spaces in [30], [38] and [39], while the necessity of saturation for these properties was indicated in [30] and [39]. A related issue on the purification of measure-valued maps on Loeb/saturated probability spaces was considered in [23], [25] and [31] with the necessity of saturation in [25] and [31]. 4 See Definition 1 in [13] and Definition 1 in Section 2 below.

Conditional Expectation of Banach Valued Correspondences

“…To prove the necessity result, He, Sun and Sun [14] constructed a sequence of large games that have the same player space but different action spaces. This result was improved in He and Sun [15], which could work with large games with any specific uncountable action space. In all these papers, the societal aggregate is formalized as the distribution induced by the strategy profile.…”

A bang-bang principle for the conditional expectation vector measure

“…The set IN * refers to IN \ {0}. The following notion is introduced in [13] as a "nowhere equivalence" between σ-algebra. Even if "nowhere equivalence" reflects intuitively the introduced notion, we rename it in order to reflect the influence of the underlying measure µ.…”

“…We will see below that this is the case for the Lyapunov convexity property for the conditional expectation vector measure. An equivalent concept, as proved in [13] (Lemma 1, p 616), is the following notion: Definition 2 ( [33,8,16]). A is said to be atomless over a sub-σ-algebra C ⊂ A iff for every E ∈ A + , there exists E 0 ∈ A, E 0 ⊂ E such that: on some set of…”

“…This notion is reformulated as "setwise coarser subσ-algebra" in He and Sun [12] where it is successfully used to establish existence of pure strategy equilibria of incomplete information games and some purification process. It is renamed as "nowhere equivalent" σ-algebras and used in He and Sun [13] to handle basic properties such as convexity, closure, and compactness for the distribution of correspondences that are necessary for applied work. He and Sun [13] show their condition to be necessary and sufficient for the existence of pure strategy equilibria in some formulations of large games.…”

“…It is renamed as "nowhere equivalent" σ-algebras and used in He and Sun [13] to handle basic properties such as convexity, closure, and compactness for the distribution of correspondences that are necessary for applied work. He and Sun [13] show their condition to be necessary and sufficient for the existence of pure strategy equilibria in some formulations of large games.…”