article published yes Jakub Svoboda author 130759D2-D7DD-11E9-87D2-DE0DE66974250000-0002-1419-3267 Krishnendu Chatterjee author 2E5DCA20-F248-11E8-B48F-1D18A9856A870000-0002-4561-241X KrCh department Formal Methods for Stochastic Models: Algorithms and Applications project Evolutionary games provide a flexible mathematical framework for many problems in biology and social evolution. Prisoners’ dilemma, and in particular, the important special case of donation games, represents social dilemmas where cooperation is mutually beneficial, yet defection is preferred by selfish agents. In evolutionary games on networks, the agents interact over a population structure. The existence of population structures that promote cooperative behavior is a fascinating and active research topic. Previous research establishes structures promoting cooperation in the limit of weak selection where the benefit-to-cost ratio β exceeds 1.5. The existence of such structures for medium and strong selection for 1 < ß < 2 and for weak selection for 1 < ß < 1.5 has been a long-standing open question. First, we answer the open questions in the affirmative: For every selection strength and every ß > 1, we construct networks promoting cooperation. Second, we present a robustness result with respect to β and selection strength: Our structures promote cooperation for a range of these parameter values rather than specific parameter values. Finally, we supplement our theoretical results with simulation results on small population structures that show the effectiveness of our construction over well-studied population structures. https://research-explorer.ista.ac.at/download/20857/20860/2025_PNAS_Svoboda.pdf application/pdfno National Academy of Sciences2025 eng 1091-6490 4139713610.1073/pnas.2524109122 12251e2524109122 Svoboda J, Chatterjee K. 2025. Promoters of cooperation in evolutionary games. Proceedings of the National Academy of Sciences. 122(51), e2524109122. Svoboda J, Chatterjee K. Promoters of cooperation in evolutionary games. <i>Proceedings of the National Academy of Sciences</i>. 2025;122(51):e2524109122. doi:<a href="https://doi.org/10.1073/pnas.2524109122">10.1073/pnas.2524109122</a> J. Svoboda and K. Chatterjee, “Promoters of cooperation in evolutionary games,” <i>Proceedings of the National Academy of Sciences</i>, vol. 122, no. 51. National Academy of Sciences, p. e2524109122, 2025. J. Svoboda, K. Chatterjee, Proceedings of the National Academy of Sciences 122 (2025) e2524109122. Svoboda, Jakub, and Krishnendu Chatterjee. “Promoters of Cooperation in Evolutionary Games.” <i>Proceedings of the National Academy of Sciences</i>, vol. 122, no. 51, National Academy of Sciences, 2025, p. e2524109122, doi:<a href="https://doi.org/10.1073/pnas.2524109122">10.1073/pnas.2524109122</a>. Svoboda, J., &#38; Chatterjee, K. (2025). Promoters of cooperation in evolutionary games. <i>Proceedings of the National Academy of Sciences</i>. National Academy of Sciences. <a href="https://doi.org/10.1073/pnas.2524109122">https://doi.org/10.1073/pnas.2524109122</a> Svoboda, Jakub, and Krishnendu Chatterjee. “Promoters of Cooperation in Evolutionary Games.” <i>Proceedings of the National Academy of Sciences</i>. National Academy of Sciences, 2025. <a href="https://doi.org/10.1073/pnas.2524109122">https://doi.org/10.1073/pnas.2524109122</a>. 208572025-12-28T23:01:26Z2026-05-20T08:13:17Z