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Evolutionary path from a prisoner’s dilemma to a harmony game via hawk–dove games

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Abstract

In most evolutionary matrix game models, the payoffs are assumed to be constant. Here, we develop a toy model of hunting in pairs in which interaction outcomes are determined not only by the behavioural strategy choice of the players but also a second, evolvable physical trait of them. We analyse the evolutionary dynamics of this model in a limiting case of the classical monomorphic framework introduced by Maynard Smith and Price characterised by rare mutations in physical trait, moderately frequent mutations in behaviour, and very fast selection in both. We show that this process can transform the evolutionarily stable strategy and, as a result, even the nature of the game played in the population: We present an illustrative numerical example in which the concurrent evolution of the behavioural and trait components of the phenotype turns a social dilemma into a harmony game via an intermediate phase of coexistence.

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All data generated or analysed during this study are included in this published article and its supplementary information files.

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Acknowledgements

T.V. discloses the following: This work was supported by the Hungarian National Research, Development and Innovation Office NKFIH, Hungary KKP 129877. The study was funded by the National Research, Development and Innovation Office in Hungary (RRF-2.3.1-21-2022-00006). This work was supported by project TKP2021-NVA-09. Project no TKP2021-NVA-09 has been implemented with the support provided by the Ministry of Culture and Innovation of Hungary from the National Research, Development and Innovation Fund, financed under the TKP2021-NVA funding scheme.

Funding

Open access funding provided by HUN-REN Centre for Energy Research.

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Authors and Affiliations

  1. HUN-REN Centre for Energy Research, Institute of Technical Physics and Materials Science, Konkoly-Thege Miklós út 29–33, H-1121, Budapest, Hungary

    Balázs Király

  2. Bolyai Institute, University of Szeged, Aradi vértanúk tere 1, H-6720, Szeged, Hungary

    Tamás Varga

  3. National Laboratory for Health Security, University of Szeged, Aradi vértanúk tere 1, H-6720, Szeged, Hungary

    Tamás Varga

  4. HUN-REN Centre for Ecological Research, Institute of Evolution, Konkoly-Thege Miklós út 29–33, H-1121, Budapest, Hungary

    József Garay

Authors

  1. Balázs Király
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  2. Tamás Varga
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  3. József Garay
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Contributions

J.G. and B.K. conceptualized and designed the study. B.K. and T.V. carried out the necessary investigations and analyses. B.K. prepared the figures. All authors contributed to the writing of, read, and approved the final manuscript.

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Balázs Király.

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Király, B., Varga, T. & Garay, J. Evolutionary path from a prisoner’s dilemma to a harmony game via hawk–dove games.
Sci Rep (2026). https://doi.org/10.1038/s41598-026-47565-9

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  • DOI: https://doi.org/10.1038/s41598-026-47565-9

Keywords

  • Matrix game
  • Trait evolution
  • Payoff evolution
  • Social dilemma
  • Evolutionarily stable strategy transition

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