## Estimating the coefficients *σ*_{k}s 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 *b*_{0} = *b*_{1} = *b*_{2} = 0, the estimated coefficients *σ*_{0}, *σ*_{1}, *σ*_{2} are obtained by linear regression model in the form of *σ*_{0}*a*_{0} + *σ*_{1}*a*_{1} + *σ*_{2}*a*_{2} + Intercept. For rings, the regression coefficients *σ*_{0},*σ*_{1},*σ*_{2} 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.