Abstract
How cooperation evolves and manifests itself in the thermodynamic or infinite player limit of social dilemma games is a matter of intense speculation. Various analytical methods have been proposed to analyze the thermodynamic limit of social dilemmas. In this work, we compare two analytical methods, i.e., Darwinian evolution and Nash equilibrium mapping, with a numerical agent-based approach. For completeness, we also give results for another analytical method, Hamiltonian dynamics. In contrast to Hamiltonian dynamics, which involves the maximization of payoffs of all individuals, in Darwinian evolution, the payoff of a single player is maximized with respect to its interaction with the nearest neighbour. While the Hamiltonian dynamics method utterly fails as compared to Nash equilibrium mapping, the Darwinian evolution method gives a false positive for game magnetization—the net difference between the fraction of cooperators and defectors—when payoffs obey the condition \(a+d=b+c\), wherein a,d represent the diagonal elements and b,c the off-diagonal elements in a symmetric social dilemma game payoff matrix. When either \(a+d \ne b+c\) or when one looks at the average payoff per player, the Darwinian evolution method fails, much like the Hamiltonian dynamics approach. On the other hand, the Nash equilibrium mapping and numerical agent-based method agree well for both game magnetization and average payoff per player for the social dilemmas in question, i.e., the Hawk–Dove game and the Public goods game. This paper thus brings to light the inconsistency of the Darwinian evolution method vis-a-vis both Nash equilibrium mapping and a numerical agent-based approach.
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The data supporting this study’s findings are available within the article and Appendix.
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Acknowledgements
The grants have funded this study: (1) Josephson junctions with strained Dirac materials and their application in quantum information processing, SERB Grant No. CRG/20l9/006258, and (2) Nash equilibrium versus Pareto optimality in N-Player games, SERB MATRICS Grant No. MTR/2018/000070.
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CB gave the initial idea for the project, edited the paper, and wrote the code in Python 3. He also administered the project. He provided the key references and instructed the other author to initiate him into the project. CB replied to the reviewer’s queries and made changes to the manuscript as required. AKUM wrote the first draft of the paper, made the graphs and did the calculations, and wrote code in Python 2.7.
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Appendix A: Agent-based simulation
Appendix A: Agent-based simulation
The python3 code we used for finding the magnetization vs. reward graph for the Hawk–Dove game (see Fig. 1) using agent-based simulation is given below.
The code is the same for simulating the public goods game, except for the energy matrix, where E is negative of the public goods game payoff matrix.
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Benjamin, C., U.M., A.K. Nash equilibrium mapping vs. Hamiltonian dynamics vs. Darwinian evolution for some social dilemma games in the thermodynamic limit. Eur. Phys. J. B 96, 105 (2023). https://doi.org/10.1140/epjb/s10051-023-00573-4
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DOI: https://doi.org/10.1140/epjb/s10051-023-00573-4