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arXiv:quant-ph/0010092 · Chinese Physics Letters

Multi-Player and Multi-Choice Quantum Game

Du Jiang-Feng, Hui Li, Xiaodong Xu, Zhou Xian-Yi, Rongdian Han

Published Aug 22, 2002• 24 citations
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Abstract

We investigate a multi-player and multi-choice quantum game. We start from a two-player and two-choice game, and the result is better than its classical version. Then we extend this to N -player and N -choice cases. In the quantum domain, we provide a strategy with which players can always avoid the worst outcome. Also, by changing the value of the parameter of the initial state, the probabilities for players to obtain the best pay-off will be much higher than in its classical version.

DOI
10.1088/0256-307x/19/9/301

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We investigate a multi-player and multi-choice quantum game. We start from a two-player and two-choice game, and the result is better than its classical version. Then we extend this to N -player and N -choice cases. In the quantum domain, we provide a strategy with which players can always avoid the worst outcome. Also, by changing the value of the parameter of the initial state, the probabilities for players to obtain the best pay-off will be much higher than in its classical version.
We give a self contained introduction to a few quantum game protocols, starting with the quantum version of the two-player two-choice game of Prisoners dilemma, followed by a n-player generalization trough the quantum minority games, and finishing with a contribution towards a n-player m-choice generalization with a quantum version of a three-player Kolkata restaurant problem. We have omitted some technical details accompanying these protocols, and instead laid the focus on presenting some general aspects of the field as a whole. This review contains an introduction to the formalism of quantum information theory, as well as to important game theoretical concepts, and is aimed to work as an introduction suiting economists and game theorists with limited knowledge of quantum physics as well as to physicists with limited knowledge of game theory.
Quantum games have demonstrated to propose a new point of view for the solution of the classical problems and dilemmas in game theory. It has been shown in many literatures that quantum games are more efficient than classical games and provide a greater and more efficient solution. As interest in multi-player quantum games grow, methods are proposed to analyze and obtain efficient solutions for these problems. In quantum games with cooperation, analysis so far have taken into consideration all possible scenarios for the multiple players playing different strategies and in the calculation of payoff, setting the payoff of scenarios not possible to be zero. Chapter 5 presents the analysis of multiple players (three players and four players) using this method of analysis. In Chapter 6, a modification of the analysis was done. In Chapter 5, it should be noted that the probabilities of participating scenarios do not sum to unity which is technically not ideal. Hence Chapter 6 addresses this by performing re-normalizing the probabilities of the participating scenarios so that the probabilities sum to unity. The analysis was done for three and four player games and a comparison was made between the un-normalized method and the re-normalized method. It was found that re-normalization provides a more accurate representation of the classical game and hence can be a better solution for the quantum game.
Quantum game theory is a multidisciplinary field which combines quantum mechanics with game theory by introducing non-classical resources such as entanglement, quantum operations and quantum measurement. By transferring two-player two-strategy (2 × 2) dilemma containing classical games into the quantum realm, dilemmas can be resolved in quantum pure strategies if entanglement is distributed between the players who use quantum operations. Moreover, players receive the highest sum of payoffs available in the game, which are otherwise impossible in classical pure strategies. Encouraged by the observation of rich dynamics of physical systems with many interacting parties and the power of entanglement in quantum versions of 2 × 2 games, it became generally accepted that quantum versions can be easily extended to N -player situations by simply allowing N -partite entangled states. In this article, however, we show that this is not generally true because the reproducibility of classical tasks in the quantum domain imposes limitations on the type of entanglement and quantum operators. We propose a benchmark for the evaluation of quantum and classical versions of games, and derive the necessary and sufficient conditions for a physical realization. We give examples of entangled states that can and cannot be used, and the characteristics of quantum operators used as strategies.
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 F2(S1), and vice versa. If the two response functions intersect, the intersection point (S1*, S2*) 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: {αi, βj, θk}, 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, (αi, βj, θk) → xijk. 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.