Extended Data Fig. 1: Super-additive cooperation when initial conditions and group mixing are relatively unfavourable for the evolution of cooperation.

The graphs show the mean surplus per individual per ingroup interaction from the final generations of evolutionary simulations. Error bars show 95% confidence intervals, which are calculated by bootstrapping over 50 independently simulated populations and omitted when extremely narrow. RI signifies the repeated interactions scenario with n = 100 interactions per ingroup pair. GC(1) indicates the group competition scenario as competition outcomes vary in sensitivity to group differences (λ). GC(100) combines the two component mechanisms into the joint scenario, with the bars from left to right corresponding to increasing λ. When the joint scenario is super-additive, the mean surplus is decomposed (supporting information section 3²⁶) into the repeated interactions effect (orange), the group competition effect (purple), and the super-additive effect (white). Initial conditions consist of a population of unconditionally selfish individuals (All selfish), and group mixing is as low as possible (Ξ = 0), both of which are relatively unfavourable for ingroup cooperation. That said, results are nearly identical when initial conditions consist of a population of perfect reciprocators (supporting figure 163²⁶), which reveals that initial conditions actually have no meaningful effect. The life cycle either decouples (a, c) or couples (b, d) game play and individual selection. The number of migrants per group per generation (m_j) is either relatively low (a, b) or high (c, d). Cancellation effects at the group level⁸ hinder cooperation in general. Moreover, the initial population consists entirely of selfish individuals, and cooperative strategies must actually invade to become established. Super-additivity is nonetheless common (a, b, d) and in some cases the result of extremely strong positive interactions (d).