Author(s)

Yshai Avishai^1,2

¹Department of Physics, Ben-Gurion University of the Negev, Beer-Sheva, Israel.
²Yukawa Institute for Theoretical Physics, Kyoto, Japan.

This work concentrates on simultaneous move non-cooperating quantum games. Part of it is evidently not new, but it is included for the sake self consistence, as it is devoted to introduction of the mathematical and physical grounds of the pertinent topics, and the way in which a simple classical game is modified to become a quantum game (a procedure referred to as a quantization of a classical game). The connection between game theory and information science is briefly stressed, and the role of quantum entanglement (that plays a central role in the theory of quantum games), is exposed. Armed with these tools, we investigate some basic concepts like the existence (or absence) of a pure strategy and mixed strategy Nash equilibrium and its relation with the degree of entanglement. The main results of this work are as follows: 1) Construction of a numerical algorithm based on the method of best response functions, designed to search for pure strategy Nash equilibrium in quantum games. The formalism is based on the discretization of a continuous variable into a mesh of points, and can be applied to quantum games that are built upon two-players two-strategies classical games, based on the method of best response functions. 2) Application of this algorithm to study the question of how the existence of pure strategy Nash equilibrium is related to the degree of entanglement (specified by a continuous parameter γ ). It is shown that when the classical game G_C has a pure strategy Nash equilibrium that is not Pareto efficient, then the quantum game G_Q with maximal entanglement (γ = π/2) has no pure strategy Nash equilibrium. By studying a non-symmetric prisoner dilemma game, it is found that there is a critical value 0<γ_c<π/2 such that for γ<γ_c there is a pure strategy Nash equilibrium and for γ≥γ_c there is no pure strategy Nash equilibrium. The behavior of the two payoffs as function of γ starts at that of the classical ones at (D, D) and approaches the cooperative classical ones at (C, C) (C = confess, D = don’t confess). 3) We then study Bayesian quantum games and show that under certain conditions, there is a pure strategy Nash equilibrium in such games even when entanglement is maximal. 4) We define the basic ingredients of a quantum game based on a two-player three strategies classical game. This requires the introduction of trits (instead of bits) and quantum trits (instead of quantum bits). It is proved 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.

Two-Players Two Strategies Quantum Game and SU(2) Strategies, Relevance of Entanglement and Bell States, Nash Equilibrium and Its Relation to Entanglement in Pure and Mixed Strategy Quantum Games, Nash Equilibrium and Partial Entanglement, Nash Equilibrium Despite Maximal Entanglement, Two Players Three Strategies Quantum Games: Qutrits and SU(3) Strategies

Avishai, Y. (2023) On Topics in Quantum Games. Journal of Quantum Information Science, 13, 79-130. doi: 10.4236/jqis.2023.133006.