Quantum mechanics gives stability to a Nash equilibrium

Abstract

We consider a slightly modified version of the Rock-Scissors-Paper (RSP) game from the point of view of evolutionary stability. In its classical version the game has a mixed Nash equilibrium (NE) not stable against mutants. We find a quantized version of the RSP game for which the classical mixed NE becomes stable.Comment: Revised on referee's criticism, submitted to Physical Review

An Introduction to Quantum Game Theory

“…We have already mentioned how the selection of different initial superpositions can lead to different quantum games [21,25,27]. Is it possible to construct a quantum game where, by choosing a suitable superposition for the initial state, the resulting games can be individually losing but the superposition of the results produces a positive payoff?…”

“…Another way of achieving similar results is simply to dispense with the entanglement operators and simply hypothesize various initial states, an approach first used by Marinatto and Weber [21] and since used by other authors [25,27]. The essential difference to Eisert's scheme is the absence of a disentangling operator.…”

“…The game of rock-scissors-paper, where the players have three choices, has been examined by Iqbal and Toor [27]. However, to make the game amenable to analysis, the authors do not allow the players the full range of unitary operations, but rather restrict the strategies to mixtures ofÎ and two operators that involve the interchange of a pair of states.…”

“…The classical game (with quantum operators representing mixed classical strategies) is obtained by selecting |ψ = |00 , while an initial state that is maximally entangled gives rise to the maximum quantum effects. In references [21,27] the authors restrict the available strategies to probabilistic mixtures of the identity and bit flip operators forcing the players to play a mixed classical strategy. The absence of theĴ † gate still leads to different results than playing the game entirely classically.…”

“…Since then more work has been done on quantum prisoners' dilemma [13,14,15,16,17,18,19,20] and a number of other games have been converted to the quantum realm including the battle of the sexes [20,21,22,23], the Monty Hall problem [24,25,26], rock-scissors-paper [27], and others [12,28,29,30,31,32,33,34,35].…”