Probabilistic analysis of three-player symmetric quantum games played using the Einstein–Podolsky–Rosen–Bohm setting

Abstract

This paper extends our probabilistic framework for two-player quantum games to the mutliplayer case, while giving a unified perspective for both classical and quantum games. Considering joint probabilities in the standard Einstein-Podolsky-Rosen-Bohm (EPR-Bohm) setting for three observers, we use this setting in order to play general three-player noncooperative symmetric games. We analyze how the peculiar non-factorizable joint probabilities provided by the EPR-Bohm setting can change the outcome of a game, while requiring that the quantum game attains a classical interpretation for factorizable joint probabilities. In this framework, our analysis of the three-player generalized Prisoner's Dilemma (PD) shows that the players can indeed escape from the classical outcome of the game, because of nonfactorizable joint probabilities that the EPR setting can provide. This surprising result for three-player PD contrasts strikingly with our earlier result for two-player PD, played in the same framework, in which even non-factorizable joint probabilities do not result in escaping from the classical consequence of the game.Although multiplayer games have been extensively studied in the game theory literature [1,2] their analysis is often found more complex than for two-player games [3]. Economics [4] and mathematical biology [5][6][7] are the areas where most applications of multiplayer games are discussed.Quantum games first came into prominence following the work of Meyer [8] and Eisert et al [9]. However, the first game-like situations involving many agents were brought to the quantum regime in 1990 by Mermin [10,11]. Mermin analyzed a multiplayer game that can be won with certainty when it is played using spin-half particles in a Greenberger-Horne-Zeilinger (GHZ) state [12], while no classical strategy can achieve this.In 1999, Vaidman [13,14] described the GHZ paradox [12] using a game among three players. Vaidman's game is now well known in the quantum game literature. In a similar vein, others [15,16] have discussed game-like situations involving many players for which quantum mechanics significantly helps their chances of winning.The motivation behind these developments [17] was to use a game framework in order to demonstrate the remarkable, and often counterintuitive, quantum correlations that may arise when many agents interact while sharing quantum resources.Systematic procedures were suggested [18-33] to quantize a given game and earlier work considered noncooperative games in their normal-form [1].The approach towards playing quantum games is distinct in that, instead of inventing games in which quantum correlations help players winning games, it proposes quantization procedures for given and very often well known games. That is, instead of tailoring 'winning conditions' for the invented games, which can be satisfied when the game is played by quantum players, quantum game theory finds how the sharing of quantum resources may replace/displace/change the solution(s) or the outcome(s) of known games. The emphasi...

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