## 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).