Game susceptibility, Correlation and Payoff capacity as a measure of Cooperative behavior in the thermodynamic limit of some Social dilemmas

Analytically, finding the origins of cooperative behavior in infinite-player games is an exciting topic of current interest. In this paper, we compare three analytical methods, i.e., Nash equilibrium mapping (NEM), Darwinian selection (DS) and Aggregate selection (AS), with a numerical Agent based method (ABM) via the game susceptibility, correlation, and payoff capacity as indicators of cooperative behaviour. While the analytical NEM model shows excellent agreement with the numerical ABM, the other analytical models, like AS and DS, show notable divergence with ABM in the thermodynamic limit for the indicators in question. Previously, cooperative behavior was studied by considering game magnetization and individual players' average payoff as indicators. This paper shows that game susceptibility, correlation, and payoff capacity can aid in understanding cooperative behavior in social dilemmas in the thermodynamic limit. The results obtained via NEM and ABM are in good agreement for all three indicators in question, for both Hawk-Dove and the Public goods games. After comparing the results obtained for all five indicators, we see that individual players' average payoff and payoff capacity serve as the best indicators to study cooperative behavior among players in the thermodynamic limit.