Evolutionary stability and payoff equivalence in an Eisert-Wilkens-Lewenstein quantum game

Abstract

An evolutionarily stable strategy (ESS) is a strategy that, once adopted by a population, cannot be invaded by any sufficiently rare alternative (mutant) strategy. We analyse evolutionary stability in an Eisert--Wilkens--Lewenstein (EWL) quantization of a symmetric $2\times2$ coordination game, whose classical mixed equilibrium $p^*=1/2$ is not an ESS. In the restricted two-parameter EWL strategy space, the corresponding quantum strategy is a strict symmetric Nash equilibrium and an ESS for every entanglement strength $γ>0$. In the full $SU(2)$ strategy space it remains a Nash equilibrium, but in the unreduced game, in which every unitary is a distinct strategy, exact payoff duplicates preclude strictness and ESS for every pure strategy. For the agreement-payoff class, the payoff factors through a Bloch map; identifying payoff-equivalent unitaries gives the reduced space $U(1)\backslash SU(2)\cong S^2$, where strictness and ESS are restored for $γ>0$. An arbitrarily small perturbation outside this class that preserves $p^*=1/2$ yields a profitable deviation, so the reduced-space result is not generic.