Role of the effective payoff function in evolutionary game dynamics

In most studies regarding evolutionary game dynamics, the effective payoff, a quantity that translates the payoff derived from game interactions into reproductive success, is usually assumed to be a specific function of the payoff. Meanwhile, the effect of different function forms of effective payoff on evolutionary dynamics is always left in the basket. With introducing a generalized mapping that the effective payoff of individuals is a non-negative function of two variables on selection intensity and payoff, we study how different effective payoff functions affect evolutionary dynamics in a symmetrical mutation-selection process. For standard two-strategy two-player games, we find that under weak selection the condition for one strategy to dominate the other depends not only on the classical {\sigma}-rule, but also on an extra constant that is determined by the form of the effective payoff function. By changing the sign of the constant, we can alter the direction of strategy selection. Taking the Moran process and pairwise comparison process as specific models in well-mixed populations, we find that different fitness or imitation mappings are equivalent under weak selection. Moreover, the sign of the extra constant determines the direction of one-third law and risk-dominance for sufficiently large populations. This work thus helps to elucidate how the effective payoff function as another fundamental ingredient of evolution affect evolutionary dynamics.