Multi-Player Quantum Games

Game theory is a mature field of applied mathematics. It formalizes the conflict between competing agents, and has found applications ranging from economics through to biology [1,2]. Quantum information is a young field of physics. Regarding information as a physical quantity, rather than mathematical entity, has lead to concepts such as quantum computation [3]. Recently the first efforts have been made to combine these fields; the fusion may lead to new insights into the nature of information [4–6]. For two-player games, it has been found that when the allowed ‘moves’ are extended to include everything quantum mechanically possible, then the predominant strategies in the game can disappear, and only reappear if the players degrade the quantum coherence. Here we present the first study of quantum games with more than two players. We demonstrate that such games can exhibit ‘coherent’ equilibrium strategies which have no analogue in classical games, or even in two-player quantum games. These equilibria are generally of a cooperative nature: quantum players can exploit their environment highly efficiently through the use of collaborative strategies. It has been known for some time that various quantum processes can be usefully thought of as games. Quantum cryptography, for example, is readily cast as a game between the individuals who wish to communicate, and those who wish to eavesdrop [7]. Quantum cloning has been thought of as a physicist playing a game against nature [8], and indeed even the measurement process itself may be thought of in these terms [9]. Furthermore, Meyer [10,11] has pointed out that quantum algorithms may be usefully thought of as games between classical and quantum agents. Against this background, it is natural to seek a unified theory of games and quantum mechanics. Such a theory might lend insight into biological and chemical processes occurring in the quantum regime; it would certainly provide a fuller understanding of the physics of information [3]. The fundamental unit of classical information is the bit. The corresponding unit of quantum information is the ‘qubit’ – a general quantum superposition of ‘0’ and ‘1’, α0|0〉+ α1|1〉. In multi-qubit systems, superposition gives rise to entanglement: qubits are entangled if their states cannot be defined independently from one another. Whereas a pair of classical bits must be in one of the four states {00, 01, 10, 11}, a pair of qubits can be in a state, such as 1 √ 2 (|0〉⊗|0〉+|1〉⊗|1〉), which cannot be factorized into two separate qubit states. The interdependence remains even when the two qubits are far apart this is the origin of ‘non-local’ effects in quantum mechanics. Although the effect cannot directly transfer information, it has been identified as a crucial resource in quantum communication, quantum computation and error-correction, and some forms of quantum cryptography [3]. Here we will see that when the resources controlled by competing agents are entangled, they can cooperate to perfectly exploit their environment (i.e. the ‘game’). Formally a game involves of a number of agents or players, who are allowed a certain set of moves or actions. The payoff function $() specifies how the players will be rewarded after they have performed their actions. The i player’s strategy, si, is her procedure for deciding which action to play, depending her information. A strategy space, S = {si}, is the set of strategies available to her. A strategy profile s = (s1, s2, .., sN) is an assignment of one strategy to each player. An equilibrium is a strategy profile with a degree of stability: e.g. in a Nash equilibrium no player can improve her expected payoff by unilaterally changing her strategy. The study of equilibria is fundamental in understanding a game. The games we consider here are static: they are played only once so that there is no history for the players to consider. Moreover, each player has complete knowledge of the game’s structure. Thus the set of allowed actions corresponds directly to the space of deterministic strategies. Our procedure for quantizing games is a generalization of the elegant scheme introduced by Eisert et al. [12,13]. We reason as follows. Game theory, being a branch of applied mathematics, defines games without reference to the physical universe. However, quantum mechanics is a physical theory, and must be applied to a physical system. We therefore begin by creating a physical model for the games of interest. A very natural way to do this is by considering the flow of information, see Fig 1(a).