We investigate the quantization of non-zero sum games. For the particular case of the Prisoners' Dilemma we show that this game ceases to pose a dilemma if quantum strategies are allowed for. We also construct a particular quantum strategy which always gives reward if played against any classical strategy. 03.65.Bz, 02.50.Le One might wonder what games and physics could have possibly in common. After all, games like chess or poker seem to heavily rely on bluffing, guessing and other activities of unphysical character. Yet, as was shown by von Neumann and Morgenstern [1], conscious choice is not essential for a theory of games. At the most abstract level, game theory is about numbers that entities are efficiently acting to maximize or minimize [2] . For a quantum physicist it is then legitimate to ask what happens if linear superpositions of these actions are allowed for, that is if games are generalized into the quantum domain. There are several reasons why quantizing games may be interesting. First, classical game theory is a well established discipline of applied mathematics [2] which has found numerous applications in economy, psychology, ecology and biology [2, 3] . Since it is based on probability to a large extend, there is a fundamental interest in generalizing this theory to the domain of quantum probabilities. Second, if the "Selfish Genes" [3] are reality, we may speculate that games of survival are being played already on the molecular level where quantum mechanics dictates the rules. Third, there is an intimate connection between the theory of games and the theory of quantum communication. Indeed, whenever a player passes his decision to the other player or the game's arbiter, he in fact communicates information, which -as we live in a quantum world -is legitimate to think of as quantum information. On the other hand it has recently been transpired that eavesdropping in quantum-channel communication [4-6] and optimal cloning [7] can readily be conceived a strategic game between two or more players, the objective being to obtain as much information as possible in a given set-up. Finally, quantum mechanics may well be useful to win some specially designed zerosum unfair games, like PQ penny flip, as was recently demonstrated by Meyer [8], and it may assure fairness in remote gambling [9] . In this letter we consider non-zero sum games where -in contrast to zero-sum games -the two players no longer appear in strict opposition to each other, but may rather benefit from mutual cooperation. A particular instance of this class of games, which has found widespread applications in many areas of science, is the Prisoners' Dilemma. In the Prisoners' Dilemma, each of the two players, Alice and Bob, must independently decide whether she or he chooses to defect (strategy D) or cooperate (strategy C). Depending on their decision taken, each player receives a certain pay-off -see Tab. I. The objective of each player is to maximize his or her individual pay-off. The catch of the dilemma is that D is the dominant strategy, that is, rational reasoning forces each player to defect, and thereby doing substantially worse than if they would both decide to cooperate [10] . In terms of game theory, mutual defection is also a Nash equilibrium [2] : in contemplating on the move DD in retrospect, each of the players comes to the conclusion that he or she could not have done better by unilaterally changing his or her own strategy [11] . In this paper we give a physical model of the Prisoners' Dilemma, and we show that -in the context of this model -the players escape the dilemma if they both resort to quantum strategies. Moreover, we shall demonstrate that (i) there exists a particular pair of quantum strategies which always gives reward and is a Nash equilibrium and (ii) there exist a particular quantum strategy which always gives at least reward if played against any classical strategy. The physical model consists of (i) a source of two bits, one bit for each player, (ii) a set of physical instruments which enable the player to manipulate his or her own bit in a strategic manner, and (iii) a physical measurement device which determines the players' pay-off from the state of the two bits. All three ingredients, the source, the players' physical instruments, and the pay-off physical measurement device are assumed to be perfectly known to both players. TABLE I. Pay-off matrix for the Prisoners' Dilemma. The first entry in the parenthesis denotes the pay-off of Alice and the second number is Bob's pay-off. The numerical values are chosen as in [3] . Referring to Eq. (2) this choice corresponds to r = 3 ("reward "), p = 1 ("punishment"), t = 5 ("temptation"), and s = 0 ("sucker's pay-off ").
An important work in the field of quantum game theory by Eisert et al. REF examined the application of quantum techniques in the prisoners' dilemma game.
Quantum Games and Quantum Strategies
Phys. Rev. Lett. 83, 3077 (1999)
Physics