Fixation Probabilities of Strategies for Trimatrix Games and Their Applications to Triadic Conflict

This study extends existing formulae of the fixation probabilities of strategies for symmetric games and bimatrix games in finite populations and derives a counterpart for trimatrix games. This allows us to describe the stochastic evolutionary game dynamics when three players are assigned different roles and therefore are not interchangeable. Following previous studies, we also derived two types of stochastic stability conditions based on the obtained fixation probabilities; "strong stochastic stability," which requires that for any initial frequencies of strategies, the fixation probability of a combination of specific strategies is higher than that under neutrality and those of any other combinations are lower than neutrality; and "stochastic stability," which only requires that the fixation probability of a specific strategy combination be higher than that under neutrality for any initial frequencies of strategies. Thus, for the former, we obtain a clear correspondence with bimatrix games, but not necessarily for the latter. The results of applying our findings to triadic conflicts (the Impartial person and mediator game and the Fish in troubled waters game), the volunteer's dilemma, and coordination games are also reported.