Selection pressure/Noise driven cooperative behaviour in the thermodynamic limit of repeated games

Consider the scenario where an infinite number of players (i. e. , the thermodynamic limit) find themselves in a Prisoner's dilemma type situation, in a repeated setting. Is it reasonable to anticipate that, in these circumstances, cooperation will emerge? This paper addresses this question by examining the emergence of cooperative behaviour, in the presence of noise (or, under selection pressure), in repeated Prisoner's Dilemma games, involving strategies such as Tit-for-Tat, Always Defect, GRIM, Win-Stay, Lose-Shift, and others. To analyze these games, we employ a numerical Agent-Based Model (ABM) and compare it with the analytical Nash Equilibrium Mapping (NEM) technique, both based on the 1D-Ising chain. We use game magnetization as an indicator of cooperative behaviour. A significant finding is that for some repeated games, a discontinuity in the game magnetization indicates a first-order selection pressure/noise-driven phase transition. The phase transition is particular to strategies where players do not severely punish a single defection. We also observe that in these particular cases, the phase transition critically depends on the number of rounds the game is played in the thermodynamic limit. For all five games, we find that both ABM and NEM, in conjunction with game magnetization, provide crucial inputs on how cooperative behaviour can emerge in an infinite-player repeated Prisoner's dilemma game.