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.
S. J. van Enk and R. Pike in PRA 66, 024306 (2002), argue that the equilibrium solution to a quantum game isn't unique but is already present in the classical game itself. In this work, we debunk this assertion by showing that a random strategy in a particular quantum (Hawk-Dove) game is unique to the quantum game. In other words the equilibrium solution of the quantum Hawk-Dove game can not be obtained in the classical Hawk-Dove game. Moreover, we provide an analytical solution to the quantum
2×2 strategic form Hawk-Dove game using random mixed strategies. The random strategies which we describe are evolutionary stable implying both Pareto optimality and Nash equilibrium with their payoff's classically unobtainable.
本文介绍量子博弈和量子演化博弈的基本概念，并分析鹰鸽博弈的量子策略和量子演化稳定策略。量子策略可以得到鹰鸽博弈的Pareto均衡，我们发现在资源利益比损失多情况下，鹰鹰策略不仅是Nash均衡还是量子演化稳定策略。 We introduce the basic concepts of quantum strategy and quantum evolution stable strategy and use them to analyze the Hawk-Dove game. We find that quantum strategy can realize the Pareto equilibrium by Dove-Dove strategy and the Hawk-Hawk strategy is not only a Nash equilibrium, but also a quantum evolution stable strategy when the re-source gain is larger than the loss in the Hawk-Dove game.
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.