The evolution of cooperation in finite populations with synergistic payoffs


In a series of papers, Forber and Smead (J Philos 111(3):151–166, 2014, Biol Philos 30(3):405–421, 2015) and Smead and Forber (Evolution 67(3):698–707, 2013) make a valuable contribution to the study of cooperation in finite populations by analyzing an understudied model: the prisoner’s delight. It always pays to cooperate in the one-shot prisoner’s delight, so this model presents a best-case scenario for the evolution of cooperation. Yet, what Forber and Smead find is highly counterintuitive. In finite populations playing the prisoner’s delight, increasing the benefit of cooperation causes selection to favor defection. Here, I extend their model by considering the effects of non-linear payoffs. In particular, I show that interesting subtleties arise when payoffs are synergistic. Indeed, analysis reveals that increasing the benefit of cooperation does not always favor the spread of defection if payoffs are synergistic. I conclude by drawing some general considerations about robustness analysis in evolutionary models.

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Fig. 1


  1. 1.

    Strictly speaking, the range of α(xC) and β(xC) is here the set of natural numbers less than N. So the notions of steepness, inflection point, and other terms that apply to continuous functions are not defined for these functions. Here, I use these terms as a convenient short-hand for what would be the case in the limit of α(xC) and β(xC) as N goes to infinity, but my results do not depend on this. Similar functions have been studied in non-cooperative settings by Archetti and Scheuring (2011).


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I thank Alex Rosenberg, Rory Smead, Hannah Read, and especially two anonymous referees for helpful feedback on previous drafts of this paper.

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Correspondence to Rafael Ventura.

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Ventura, R. The evolution of cooperation in finite populations with synergistic payoffs. Biol Philos 34, 43 (2019).

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  • Cooperation
  • Prisoner’s delight
  • Finite populations
  • Synergy