Pairwise comparison and selection temperature in evolutionary game dynamics

Abstract

Recently, the frequency-dependent Moran process has been introduced in order to describe evolutionary game dynamics in finite populations. Here, an alternative to this process is investigated that is based on pairwise comparison between two individuals. We follow a long tradition in the physics community and introduce a temperature (of selection) to account for stochastic effects. We calculate the fixation probabilities and fixation times for any symmetric 2 x 2 game, for any intensity of selection and any initial number of mutants. The temperature can be used to gauge continuously from neutral drift to the extreme selection intensity known as imitation dynamics. For some payoff matrices the distribution of fixation times can become so broad that the average value is no longer very meaningful.

Figure 1

Fixation probabilities in a population of size N = 20. Simulation results (symbols) obtained from averaging over 10⁶ realizations coincide perfectly with the theoretical result, Eq.(12) (solid lines). Arrows indicate increasing intensity of selection. For neutral selection (diamonds), the fixation probability is given by the fraction of cooperators. In the Prisoner’s Dilemma, fixation of cooperators becomes less likely with increasing intensity of selection, as shown for β = 0.05 (squares) and β = 0.1 (circles). Only for weak selection and a high initial number of cooperators, they have reasonable chances. In the Snowdrift Game, the fixation probability of cooperators increases with increasing intensity of selection, as the internal equilibrium is closer to pure cooperation. Here, the fixation probabilities are shown for β = 0.05 (squares) and β = 0.1 (circles). However, the fixation time of defectors increases accordingly, see Fig. 2 (b = 1, c = 0.5).

Figure 2

Conditional fixation times for fixation of defectors in a population of N = 20. Symbols show simulation results whereas lines depict the fixation times obtained according to Eq. (28). Arrows indicate increasing intensity of selection. For neutral selection (diamonds), the fixation time increases with the initial number of cooperators k, as the distance to the point of fixation increases. In the Snowdrift Game, fixation times increase with increasing selection intensity (squares β = 0.05, circles β = 0.1), as the system spends much time near the internal Nash equilibrium. On the contrary, for the Prisoner’s Dilemma, now stronger selection leads to faster fixation (squares β = 0.05, circles β = 0.1). Here, increasing selection intensity induces opposite behaviour for both games in what concerns the average fixation times and the fixation probability, although this is not the case in general (b = 1, c = 0.5, averages over 10⁶ realizations).

Figure 3

Unconditional fixation times in a population of N = 20. Lines show the theoretical result from Eq. (17) whereas symbols are results from computer simulations. Arrows indicate increasing intensity of selection. For neutral selection (black diamonds), the fixation time increases with increasing distance from the absorbing states. For the Snowdrift Game, fixation times increase with increasing intensity of selection (squares β = 0.05, circles β = 0.1). For the Prisoner’s Dilemma, the fixation time increases with the number of cooperators (squares β =0.05, circles β = 0.1), which results from the high fixation probability in 100% defection. Hence, only close to 100% cooperation, the fixation time decreases (symbols as in Fig. 2, b = 1, c = 0.5, averages over 10⁶ realizations).

Figure 4

Probability distributions of the conditonal fixation times of a single defector in a population of cooperators. While the average fixation times (arrows) agree well with simulations, as shown in Fig. 2, the probability distributions can become extremely broad. For the Prisoner’s Dilemma and for neutral selection, the deviations of the fixation time from the average are comparably small. However, for the Snowdrift game an extremely wide range of fixation times is observed. Hence, the average fixation time is of limited interest, as large deviations are observed with a very high probability (N = 20, β = 0.1).