The equivalence of Bell's inequality and the Nash inequality in a quantum game-theoretic setting

TL;DR

This paper explores the connection between Bell's inequality and Nash equilibrium in quantum games, showing that in certain cases, the existence of a Nash equilibrium is tied to Bell's inequality violation, revealing a fundamental link between quantum nonlocality and game theory.

Contribution

It demonstrates the equivalence of Bell's inequality and Nash inequality in specific quantum games using an EPR setting, highlighting a fundamental link between quantum nonlocality and strategic stability.

Findings

Identifies games where Nash equilibrium exists only when Bell's inequality is violated.

Shows the equivalence of Bell's inequality and Nash inequality in certain quantum strategies.

Provides a framework connecting quantum nonlocality with game-theoretic concepts.

Abstract

The interaction of competing agents is described by classical game theory. It is now well known that this can be extended to the quantum domain, where agents obey the rules of quantum mechanics. This is of emerging interest for exploring quantum foundations, quantum protocols, quantum auctions, quantum cryptography, and the dynamics of quantum cryptocurrency, for example. In this paper, we investigate two-player games in which a strategy pair can exist as a Nash equilibrium when the games obey the rules of quantum mechanics. Using a generalized Einstein-Podolsky-Rosen (EPR) setting for two-player quantum games, and considering a particular strategy pair, we identify sets of games for which the pair can exist as a Nash equilibrium only when Bell's inequality is violated. We thus determine specific games for which the Nash inequality becomes equivalent to Bell's inequality for the considered strategy pair.