A Quantum Protocol for Sampling Correlated Equilibria Unconditionally and without a Mediator

Abstract

A correlated equilibrium is a fundamental solution concept in game theory that enjoys many desirable mathematical and algorithmic properties: it can achieve more fair and higher payoffs than a Nash equilibrium and it can be efficiently computed for a vast class of games. However, it requires a trusted mediator to assist the players in sampling their moves, which is a major drawback in many practical applications.

A computational solution to this problem was proposed by Dodis, Halevi and Rabin [DHR00]. They extended the original game by adding a preamble stage, where the players communicate with each other and then they perform the original game. For this extended game, they show that the players can achieve payoffs at least as high as in any correlated equilibrium, provided that the players are computationally bounded and can communicate before the game.

The introduction of cryptography with computational security in game theory is of great interest both from a theoretical and more importantly from a practical point of view. However, the main game-theoretic question remained open: can we achieve any correlated equilibrium for 2-player games without a trusted mediator and also unconditionally?

In this paper, we provide a positive answer to this question. We show that if the players can communicate via a quantum channel before the game, then for 2-player games, payoffs at least as high as in any correlated equilibrium can be achieved, without a trusted mediator and unconditionally. This provides another example of a major advantage of quantum information processing: quantum communication enables players to achieve a real correlated equilibrium unconditionally, a task which is impossible in the classical world.

More precisely, we prove that for any correlated equilibrium p of a strategic game G, there exists an extended game (with a quantum communication initial stage) Q with an efficiently computable approximate Nash equilibrium σ, such that the expected payoff for both players in σ is at least as high as in p.

The main cryptographic tool used in the construction is the quantum weak coin flipping protocol of Mochon [Moc07].

Most of the work was done when the authors visited Centre of Quantum Technologies (CQT), Singapore in early January, 2011, under the support of CQT. I.K.’s research was also supported by French projects ANR-09-JCJC-0067-01, ANR-08-EMER-012 and the project QCS (grant 255961) of the E.U. S.Z.’s research was supported by China Basic Research Grant 2011CBA00300 (sub-project 2011CBA00301), Research Grants Council of Hong Kong (Project no. CUHK418710, CUHK419011), and benefited from research trips under the support of China Basic Research Grant 2007CB807900 (sub-project 2007CB807901).