Quantum games: Numerical approach

Yshai Avishai

doi:10.1109/icnc.2012.6234560

Abstract

In classical (standard) game theory, a useful algorithm for searching Nash equilibrium in games of two players, is to determine the best response functions. For each strategy S1 of player 1 player 2 finds a best response function F 2 (S 1 ), and vice versa. If the two response functions intersect, the intersection point (S 1 *, S 2 *) is a candidate for Nash equilibrium. This method is especially useful when the strategy space of each player is determined by a single variable (discrete or continuous). In the last decade, the concept of quantum games has been developed (hence we distinguish between classical and quantum games). In a quantum game with two players the strategy space of each player is composed of 2 × 2 complex unitary matrices with unit determinant. That is the group SU(2). The corresponding strategy space is characterized by three continuous variables represented by angles: 0 ≤ α ≤ 2π, 0 ≤ β ≤ 2π, 0 ≤ θ ≤ π. That turns the use of response functions impractical. In the present contribution we suggest a method for alleviating this problem by discretizing the variables as: {α i , β j , θ k }, i= 1, 2, ..., I; j = 1, 2, ..., J; k = 1, 2, ... K. This enables the representation of every such triple by a single discrete variable, (α i , β j , θ k ) → x ijk . Thereby, the strategy space is characterized by a single discrete variable taking I × J × K values and the method of response functions is feasible. We use it to show the following two results: 1) A two players quantum game with partially entangled initial state has a pure strategy Nash equilibrium. 2) A two player quantum Bayesian game with fully entangled initial state has a pure strategy Nash equilibrium.

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