Quantum Game Theory

Abstract

We present a review of the quantum game theory seen from the author’s perspective. The quantum strategy is an extension of game theoretical strategy in which the classical joint probability representing the choice of players is replaced by quantum joint probability generated from Hilbert vectors or quantum wave functions. The extra environmental parameters appear in quantum strategies, because of the inherent entanglement present in the quantum wave functions. A detailed examination of the contents of quantum strategy reveals the existence of two components, the first of which is the pseudo-classical strategy that is a deformed strategy modified from the original classical strategy by environmental parameters, and the second of which is the purely quantum contribution that originates from the quantum interference effect. The pseudo-classical component is shown to represent the altruistic strategy which is effective in bringing dilemma games to Pareto optimal Nash equilibria. The quantum interference component usually supplies a small correction. In a proper setting of Harsanyi-type game with incomplete information, however, pseudo-classical terms can be made to cancel each other, leaving only quantum effect in the Nash payoffs, to which, a direct link to the quantum breaking of Bell inequality is established. We also discuss possible applications of quantum games to the evolutionary theory of biology.