  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 1-D 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 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 (QPD) 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.
