The role of mixed strategies in spatial evolutionary games

We study three-strategy evolutionary games on a square lattice when the third strategy is a mixed strategy of the first and second ones. It is shown that the resultant three-strategy game is a potential game as well as its two-strategy version. Evaluating the potential we derive a phase diagram on a two-dimensional plane of rescaled payoff parameters that is valid in the zero noise limit of the logit dynamical rule. The mixed strategy is missing in this phase diagram. The effects of two different dynamical rules are analyzed by Monte Carlo simulations and the results of imitation dynamics indicate the dominance of the mixed strategy within the region of the hawk-dove game where it is an evolutionarily stable strategy. The effects and consequences of the different dynamical rules on the final stationary states and phase transitions are discussed.