Quantum Strategic Game Theory

TL;DR

This paper extends classical game theory to the quantum setting, exploring how quantum strategies can significantly increase players' payoffs and introducing correlation complexity as a new measure of quantum advantage.

Contribution

It introduces a quantum extension of Nash and correlated equilibria, analyzes payoff improvements, and proposes correlation complexity as a novel measure with implications in physics and computational complexity.

Findings

Quantum strategies can dramatically increase payoffs in certain games.

Some correlated equilibria can be generated with minimal quantum resources but require more classical communication.

Correlation complexity relates to Bell's inequality and has implications in physics and complexity theory.

Abstract

We propose a simple yet rich model to extend the notions of Nash equilibria and correlated equilibria of strategic games to the quantum setting, in which we then study the relations between classical and quantum equilibria. Unlike the previous work that focus on qualitative questions on specific games of small sizes, we address the following fundamental and quantitative question for general games: How much "advantage" can playing quantum strategies provide, if any? Two measures of the advantage are studied, summarized as follows. 1. A natural measure is the increase of payoff. We consider natural mappings between classical and quantum states, and study how well those mappings preserve the equilibrium properties. Among other results, we exhibit correlated equilibrium $p$ whose quantum superposition counterpart $\sum_s \sqrt{p(s)}\ket{s}$ is far from being a quantum correlated equilibrium; actually a player can increase her payoff from almost 0 to almost 1 in a [0,1]-normalized game. We achieve this by a tensor product construction on carefully designed base cases. 2. For studying the hardness of generating correlated equilibria, we propose to examine \emph{correlation complexity}, a new complexity measure for correlation generation. We show that there are $n$-bit correlated equilibria which can be generated by only one EPR pair followed by local operation (without communication), but need at least $\log(n)$ classical shared random bits plus communication. The randomized lower bound can be improved to $n$, the best possible, assuming (even a much weaker version of) a recent conjecture in linear algebra. We believe that the correlation complexity, as a complexity-theoretical counterpart of the celebrated Bell's inequality, has independent interest in both physics and computational complexity theory and deserves more explorations.