Nash equilibrium mapping vs Hamiltonian dynamics vs Darwinian evolution for some social dilemma games in the thermodynamic limit

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 neighbor. 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 represents the diagonal elements and b,c the off-diagonal elements in a symmetric social dilemma game payoff matrix. When either a + d =/= 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.