Evolutionarily Stable Strategy (ESS) in classical game theory is a refinement of Nash equilibrium concept. We investigate the consequences when a small group of mutants using quantum strategies try to invade a classical ESS in a population engaged in symmetric bimatrix game of Prisoner’s Dilemma. Secondly we show that in an asymmetric quantum game between two players an ESS pair can be made to appear or disappear by resorting to entangled or unentangled initial states used to play the game even when the strategy pair remains a Nash equilibrium in both forms of the game. I.
An evolutionarily stable strategy (ESS) was originally defined as a static concept but later given a dynamic characterization. A well known theorem in evolutionary game theory says that an ESS is an attractor of replicator dynamics but not every attractor is an ESS. We search for a dynamic characterization of ESSs in quantum games and find that in certain asymmetric bi-matrix games evolutionary stability of attractors can change as the game switches between its two forms, one classical and the other quantum.
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
2×2 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)\SU(2)≅S2, 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.
Quantum evolutionary stable strategies (ESSs) are considered as stable solutions to population games on the molecular level. The distributive diagram of the games is brought out in the light of their rescaled games, so that a convenient way to look for quantum ESSs of 2×2 symmetric games can be obtained. It is found that transpositions may take place when a game is quantized. They are related to the parameters in the classical payoff matrix and those of the initial quantum states. One can easily find quantum ESSs by referring to the tables and the figure shown in the main body. This paper draws an overall outline of NEs (Nash equilibria) and ESSs of this kind of game in the classical version as well as in the quantum version.
We quantize the Hawk–Dove game by using the most general form of a pure initial state to investigate the existence of pure and mixed evolutionarily stable strategies (ESS). An example is considered to draw a comparison between the classical and quantum version of the game. Our choice of the most general initial quantum state enables us to make the game symmetric or asymmetric. We show that for a particular set of game parameters where there exists only mixed ESS in the classical version of the game, quantization allows even a pure strategy to be an ESS for the symmetric game in addition to mixed ESS. On the other hand only pure strategy ESS can exist for the asymmetric quantum version of the Hawk–Dove game.