Quantum games: Numerical approach

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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 x 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 x J x 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.

Original languageEnglish
Title of host publicationProceedings - 2012 8th International Conference on Natural Computation, ICNC 2012
Pages898-903
Number of pages6
DOIs
StatePublished - 18 Sep 2012
Event2012 8th International Conference on Natural Computation, ICNC 2012 - Chongqing, China
Duration: 29 May 201231 May 2012

Publication series

NameProceedings - International Conference on Natural Computation
ISSN (Print)2157-9555

Conference

Conference2012 8th International Conference on Natural Computation, ICNC 2012
Country/TerritoryChina
CityChongqing
Period29/05/1231/05/12

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