Table 1.
Payoff table.
Table 2.
Notation list.
Fig 1.
Average abundance of strategy A 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.
Fig 2.
Average abundance of strategy A as a function of the payoff entry a0.
Our simulation results clearly show that strategy A is more abundant than strategy B if a0 is approximately greater than 2.0 for regular networks (k = 2) (Panel (a)). This is in perfect agreement with our calculation based on the theorem, i.e., inequality a0 + 2a1 + a2 > b0 + 2b1 + b2 with a1 = 2, a2 = 1, b0 = 4, b1 = 1 and b2 = 1. Note that the criterion a0 > 2 is valid for aspiration across various distributions. It implies that the criterion to favor one strategy over the other is independent of the individualised aspiration, as stated in the theorem. Furthermore, the criterion also holds beyond regular networks (Panel (a)), namely, on random (Panel (b)) and scale-free networks (Panel (c)). It suggests that the criterion can be extrapolated to general population structures. The details of the simulations are as follows: The minimum degree of all the networks is set to be two such that all the individuals have enough neighbours to play the three-player game with. Personal aspirations are randomly assigned in a population with homogeneous aspiration ei = 2 for all i = 1, 2…N (blue ◯), and a population with heterogeneous aspirations generated based on uniform distribution on the interval [0, 5] (red □), bimodal distribution with (orange ◊), and power-law distribution with probability density function f(x) = 2x−3 (purple △). Here,
stands for the normal distribution with mean 2.5 and standard deviation 0.5. In addition, the minimum value of aspiration sampled from the power-law distribution is 1.0. In the beginning, we randomly set 45% of the population to be of strategy A and the rest to be of strategy B. At each time step, the focal individual randomly chooses two individuals from its neighborhood and play a single three-player game with them to obtain the payoff. Fermi function is employed as the decision making function for all the individuals. Each data point is the mean of the average abundance of strategy A calculated from three independent runs (5 × 109 samples in each run, 1.5 × 1010 samples in total). In each run, we start sampling after a relaxation time of 5 × 107 time steps. The average abundance of strategy A is obtained by averaging the abundance of strategy A over 5 × 109 time steps. The population size N = 1000. The selection intensity β = 0.005.
Table 3.
Estimating the coefficients σks by simulation.
We establish three sets of individualised aspirations based on the uniform distribution in [0, 1] (see S2 File for details). For three-player games with b0 = b1 = b2 = 0, the estimated coefficients σ0, σ1, σ2 are obtained by linear regression model in the form of σ0a0 + σ1a1 + σ2a2 + Intercept. For rings, the regression coefficients σ0,σ1,σ2 and Intercept are close to 1,2,1 and 0, which agrees perfectly with theoretical calculations Eq (10). In addition, it holds for all the three sets of aspirations, validating the theorem. For well-mixed populations, the d coefficients still hold, consistent with [34]. Thus the coefficients are robust to the heterogeneity in aspiration for both ring and well-mixed population. The confidence interval for the corresponding estimated coefficients (EC) are [EC-ME, EC+ME], where ME in the parentheses is calculated with confidence level 95%. Please refer to the Methods to see the details of the simulation. Population size N = 100, selection intensity β = 5 × 10−2. More details of the simulation are found in S1 File.
Fig 3.
Robustness of the evolutionary outcome based on the way of payoff accumulation.
Individuals are now participating in three games for a time. One of the three is organised by itself and the rest two by its two nearest neighbours. The final payoff is the average of the accumulated payoff in the three games. This payoff accumulation is different from that in the theorem, where every individual only plays a single d−player game. Yet we find that the heterogeneity of aspiration still does not change the abundance of strategy A. Thus our theorem might be generalised to other ways of payoff accumulation. Details of the simulation: The aspiration values are randomly selected from the uniform distributions on interval [0, 1] (left panel) or [0, 5] (right panel). Within each distribution, we establish two populations with different individualised aspirations. All the rest parameters are the same as those of (Fig 1).