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Quantum Evolutionary Stable Strategies

The extension of the concept of Evolutionary Stable Strategy (ESS) to quantum domain is investigated. For the pair-wise game of Prisoner's Dilemma played in a population it is shown that a two-parameter quantum strategy can invade a classical ESS. However in this game a quantum ESS cannot be invaded by another two parameter quantum strategy. Game theory has been successfully applied in mod-eling the evolutionary processes in natural world. Certain paradoxical situations[1,2] in animal conflicts have been explained by the game theory. The concept of an Evolutionary Stable Strategy (ESS) was introduced by Maynard Smith and Price [3]. An ESS is a strategy, which if adopted in a conflict by a population , can withstand a small invading group. The ESS is thus stable and persists through time. ESS is a central idea in the Evolutionary Game Theory: a branch of the Game Theory dealing with evolutionary games. An evolutionary game means a model of strategic interaction continuing over time in which higher payoff strategies gradually displace strategies with lower payoffs. There is also some inertia involved to distinguish between evolutionary and revolutionary changes. Inertia means that aggregate behavior does not change too abruptly. What evolution may have got to do with quantum mechanics? The concept of an ESS developed from applying the ideas of game theory to animal conflicts and recently certain ideas of game theory have been extended to quantum domain [4,5]. We think that recent work on quantum theory of games can provide a link between the theory of evolution and quantum mechanics. If the genes engage themselves in selfish games [6] played at molecular level where quantum mechanics decides the rules then it is interesting to speculate about the quantum analogues of ESS's. If the games of survival between animals give rise to ESS's then what about the possibility of quantum games among the molecules of the genes giving rise to quantum strategies that are stable and persist through time. Such Quantum Evolutionary Stable Strategies (QESS's) may have interesting characteristics like their classical counterparts. They may be immune from invasion from other mutant quantum strategies. We consider the question of mutant quantum strategy trying to invade other classical or quantum ESS in a population engaged in a pair-wise game of Prisoner's Dilemma. We will consider the symmetric version of the pair-wise game where all members of the population are indistinguishable and each player is equally likely …