The Necessity of Nowhere Equivalence

TL;DR

This paper establishes the importance of the nowhere equivalence condition for regularity properties of distributions and demonstrates its equivalence to the existence of pure-strategy Nash equilibria in large games with uncountable action spaces.

Contribution

It proves regularity properties under the nowhere equivalence condition and shows this condition is necessary for these properties and for the existence of Nash equilibria in large games.

Findings

Regularity properties hold under nowhere equivalence.

Necessity of nowhere equivalence for these properties.

Equivalence between nowhere equivalence and Nash equilibria existence.

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.