  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 analyse the thermodynamic
limit of social dilemmas. In a previous work [Chaos Solitons and fractals 135,
109762(2020)] involving one among us, two of those methods, Hamiltonian
Dynamics(HD) and Nash equilibrium(NE) mapping were compared. The inconsistency
and incorrectness of HD approach vis-a-vis NE mapping was brought to light. In
this work we compare a third analytical method, i.e, Darwinian evolution(DE)
with NE mapping and a numerical agent based approach. For completeness, we give
results for HD approach as well. In contrast to HD which involves maximisation
of payoffs of all individuals, in DE, payoff of a single player is maximised
with respect to its nearest neighbour. While, HD utterly fails as compared to
NE mapping, DE method gives a false positive for game magnetisation -- 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 symmetric social dilemma games. When either a+d
=/= b+c or, when one looks at average payoff per player, DE method fails much
like the HD approach. NE mapping and numerical agent based method on the other
hand agree really well for both game magnetisation as well as average payoff
per player for the social dilemmas in question, i.e., Hawk-Dove game and Public
goods game. This paper thus bring to light the inconsistency of the DE method
vis-a-vis both NE mapping as well as a numerical agent based approach.
