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arXiv:1503.08851 · Progress of Theoretical and Experimental Physics

General quantum two-player games, their gate operators, and Nash equilibria

Katarzyna Bolonek-Lasoń

Published Feb 23, 2015• 10 citations
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Abstract

Two-player |$N$|-strategy games quantized according to the Eisert–Lewenstein–Wilkens scheme [Phys. Rev. Lett. 83, 3077 (1999)] are considered. Group-theoretical methods are applied to the problem of finding a general form of gate operators (entanglers) under the assumption that the set of classical pure strategies is contained in the set of pure quantum ones. The role of the stability group of the initial state of the game is stressed. As an example, it is shown that maximally entangled games do not admit nontrivial pure Nash strategies. The general arguments are supported by explicit computations performed in the three-strategy case.

DOI
10.1093/ptep/ptv004

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Two-player |NNN|-strategy games quantized according to the Eisert–Lewenstein–Wilkens scheme [Phys. Rev. Lett. 83, 3077 (1999)] are considered. Group-theoretical methods are applied to the problem of finding a general form of gate operators (entanglers) under the assumption that the set of classical pure strategies is contained in the set of pure quantum ones. The role of the stability group of the initial state of the game is stressed. As an example, it is shown that maximally entangled games do not admit nontrivial pure Nash strategies. The general arguments are supported by explicit computations performed in the three-strategy case.
The two-players N strategies games quantized according to the Eisert-Lewenstein-Wilkens scheme [1] are considered. It is shown that in the case of maximal entanglement no nontrivial pure Nash equilibrium exists. The proof relies on simple geometric properties of "chiral" group SU(N)×SU(N)SU (N)\times SU(N)SU(N)×SU(N) and is based on considering the stability subgroup of the initial state of the game. The explicit forms of neither the gate operator nor the payoff matrix are necessary.
We develop a rigorous mathematical framework for quantum game theory applied to static 2x2 games, extending classical concepts to the quantum setting where players may employ arbitrary unitary operations (pure strategies) or probability measures over the continuous group SU(2) (mixed strategies). The Eisert-Wilkens-Lewenstein protocol is introduced as the standard implementation of quantum 2x2 games. We prove the existence of Nash equilibria for continuous quantum mixed strategies via a fixed-point argument, generalising the classical Nash theorem to the quantum case.
This work, based on the author's MA thesis, concentrates on simultaneous move quantum games of two players. A numerical algorithm based on the method of best response functions, designed to search for pure strategy Nash equilibrium in quantum games is constructed. It is then used show that there is a critical degree of entanglement above which a pure strategy Nash Equilibrium does not exist. The behavior of the two payoffs approach the cooperative classical ones. Bayesian quantum games are defined, and it is shown that under certain conditions, there {\it is} a pure strategy Nash equilibrium in such games even when entanglement is maximal. The basic ingredients of a quantum game based on a two-players {\it three} decisions classical games are presented. This requires the definition of trits (instead of bits) and quantum trits (instead of quantum bits). It is shown that in this quantum game, there is no classical commensurability in the sense that the classical strategies are not obtained as a special case of the quantum strategies. An entanglement operator transforming a pure two qutrit state into a maximally entangled one is explicitly constructed.