Fig. 2.

When are simple two-choice strategies robust against all multichoice invaders in public goods games? We considered the evolutionary robustness of two-choice strategies, in which players iteratively choose to invest amount $C_1$ or $C_2>C_1$ to produce a public benefit B proportional to the total investment of both players, $B=rC$. Cooperative strategies limited to two investment choices can be evolutionary robust against all invaders, who may invest an arbitrary amount $C\ne C_1,C_2$, provided the strategy has sufficient opportunity to punish a defector—that is, provided $C_1$ is sufficiently smaller than $C_2$. We determined [2] the largest ratio of investment levels, $C_1/C_2$, that permits universally robust cooperative two-choice strategies, as a function of the population size, N, and the public return on individual investment, r in the absence of discounting ($\delta =1$). Colors are gradated in 10% intervals, so that the light blue region indicates a two-choice player can choose a strategy that maintains robust cooperation while engaging in relatively little punishment, by reducing her investment to only 90% of its maximum. The bright red region indicates that a two-choice player must have access to a high degree of punishment, $C_1$ much less than $C_2$, to maintain cooperation and be robust against all invaders. As described in 3, the figure can alternatively be interpreted as the proportion of pairs of investment levels used by a d-choice player that produce a robust suboptimal fitness peak, and thus represents a lower bound on the “ruggedness” of the fitness landscape experienced by a population of d-choice players.