Average abundance of strategy <i>A</i> for a three-player game with individualised aspirations.

Personal aspirations are randomly assigned based on a uniform distributions on the interval [0, 1] (left panels) or [0, 5] (right panels). Within each class, two sets of aspirations are assigned to represent two populations with different aspirations from the same distribution. In this way, we established two kinds of heterogeneity in aspiration: one resorts to the underlying distribution, and the other is based on the actual values sampled from the same distribution. Intuitively, both of the two heterogeneities would alter the evolutionary outcome. Simulations in line with our theorem, however, show that both kinds of the heterogeneity in aspiration lead to the identical abundance in A for both the well-mixed population (upper panels) and that on rings (lower panels). We illustrate other details of the simulation in the following. In well-mixed populations, the focal individual randomly chooses two individuals from the rest of the population, whereas it plays only with the nearest two neighbours on the ring. In the beginning, we randomly set 5% of the population to be of strategy A and the rest to be of strategy B. For each data point, it is the mean of 20 independent runs. In each run, we iterate the evolutionary process for 1 × 108 generations. The average abundance of strategy A is obtained by averaging abundance of strategy A over the last 5 × 107 generations. The population size N = 100. And the payoff entries are a0 = 3, a1 = 2, a2 = 1, b0 = 4, b1 = 1 and b2 = 1, respectively.