Conditional Expectation of Banach Valued Correspondences

TL;DR

This paper investigates the regularity properties of Banach valued correspondences under conditional expectations, establishing conditions for convexity, compactness, and upper hemicontinuity, with applications to game theory.

Contribution

It introduces the nowhere equivalence condition as necessary for regularity properties of Banach valued correspondences and links it to the existence of Nash equilibria in large games.

Findings

Regularity properties hold under the nowhere equivalence condition.

Necessity of the nowhere equivalence condition is proven.

Application to existence of pure-strategy Nash equilibria in large games.

Abstract

We present some regularity properties (convexity, weak/weak* compactness and preservation of weak/weak* upper hemicontinuity) for Bochner/Gelfand conditional expectation of Banach valued correspondences under the nowhere equivalence condition. These regularity properties for Bochner/Gelfand integral of Banach valued correspondences are obtained as corollaries. Similar properties for regular conditional distributions are also covered by the corresponding results for Gelfand conditional expectation of correspondences. We prove the necessity of the nowhere equivalence condition for any of these properties to hold. As an application, we show that the nowhere equivalence condition is satisfied on the space of players if and only if a pure-strategy Nash equilibrium exists in a general class of large games.