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

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 analyze evolutionary stability in an Eisert–Wilkens–Lewenstein (EWL) quantization of a symmetric 2× 2 2 × 2 coordination game, whose classical mixed equilibrium p^*=1/2 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 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) SU(2) S² U ( 1 ) \ S U ( 2 ) ≅ S 2 , where strictness and ESS are restored for γ >0 γ > 0 . An arbitrarily small perturbation outside this class that preserves p^*=1/2 p ∗ = 1 / 2 yields a profitable deviation, so the reduced-space result is not generic.