  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.
