TY - GEN

T1 - Quantum games

T2 - 2012 8th International Conference on Natural Computation, ICNC 2012

AU - Avishai, Yshai

PY - 2012/9/18

Y1 - 2012/9/18

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=84866150106&partnerID=8YFLogxK

U2 - 10.1109/ICNC.2012.6234560

DO - 10.1109/ICNC.2012.6234560

M3 - Conference contribution

AN - SCOPUS:84866150106

SN - 9781457721311

T3 - Proceedings - International Conference on Natural Computation

SP - 898

EP - 903

BT - Proceedings - 2012 8th International Conference on Natural Computation, ICNC 2012

Y2 - 29 May 2012 through 31 May 2012

ER -