The Equivalence Induced by Unifying Fitness Mappings in Frequency-Dependent Moran Process

As one of the most prevalent microscopic mechanisms, frequency-dependent Moran process is widely adopted to model the traits' evolution of agents in a well-mixed population, where an individual is chosen for reproduction proportional to its fitness. Using a pair of specific fitness mappings, it has been demonstrated that the Moran process leads to identical evolutionary outcomes under weak selection. But does the equivalent relation hold for any two fitness functions? In this paper, by introducing a general mapping that fitness is an arbitrary non-negative function of payoff and selection intensity, the fitness mappings is unified. And accordingly we investigate that whether the unifying fitness mappings can lead to equivalent evolutionary outcomes in frequency-dependent Moran process. By calculating fixation probabilities and fixation times under weak selection, we find that the effect of different fitness mappings on these two quantities just embody in a constant factor under mild conditions. In particular, this constant factor can be absorbed into the selection intensity by proper rescaling or by making the payoff matrix change a scale. Thus, in spite of the scaling factor, any two fitness functions are equivalent under weak selection for frequency-dependent Moran process.