Thermodynamic susceptibility as a measure of cooperative behavior in social dilemmas

The emergence of cooperation in the thermodynamic limit of social dilemmas is an emerging field of research. While numerical approaches (using replicator dynamics) are dime a dozen, analytical approaches are rare. A particularly useful analytical approach is to utilize a mapping between the spin-1/2 Ising model in 1D and the social dilemma game and calculate the magnetization, which is the net difference between the fraction of cooperators and defectors in a social dilemma. In this paper, we look at the susceptibility, which probes the net change in the fraction of players adopting a certain strategy, for both classical and quantum social dilemmas. The reason being, in statistical mechanics problems, the thermodynamic susceptibility as compared to magnetization is a more sensitive probe for microscopic behavior, e.g., observing small changes in a population adopting a certain strategy. In this paper, we find the thermodynamic susceptibility for reward, sucker's payoff, and temptation in classical Prisoner's Dilemma to be positive, implying that the turnover from defect to cooperate is greater than vice versa, although the Nash equilibrium for the two-player game is to defect. In the classical hawk-dove game, the thermodynamic susceptibility for resource suggests that the number of players switching to hawk from dove strategy is dominant. Entanglement in Quantum Prisoner's Dilemma has a non-trivial role in determining the behavior of thermodynamic susceptibility. At maximal entanglement, we find that sucker's payoff and temptation increase the number of players switching to defect. In the zero-temperature limit, we find that there are two second-order phase transitions in the game, marked by a divergence in the susceptibility. This behavior is similar to that seen in type-II superconductors wherein also two second-order phase transitions are seen.