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

Consider the scenario wherein the repeated Prisoner\rq{}s dilemma game is played not among two agents but with an infinite number of agents, i.e., in the \textit{thermodynamic} limit. Do we expect cooperative behaviour to manifest among these many agents in such situations? This paper addresses this question by examining the role of noise or selection pressure in the evolution of cooperative behaviour in the repeated Prisoner's Dilemma game, involving strategies such as \textit{Tit-for-Tat}, \textit{Always Defect}, \textit{GRIM}, \textit{Win-Stay, Lose-Shift}, and others. To analyze, we employ the analytical Nash equilibrium mapping (NEM) technique and compare it with an agent based model (ABM), both based on the \textit{1D}-Ising model. We use \textit{game magnetization} as an indicator of cooperative behaviour. A significant finding is that for some repeated games, a discontinuity in the game magnetization value indicates a \textit{first}-order \textit{selection pressure/noise}-driven phase transition, particularly when strategies do not punish opponents for defecting even once. We also observe that in these particular cases, the phase transition critically depends on the number of \textit{rounds} the game is played in the thermodynamic limit. For all five games, we find that ABM and NEM, in conjunction with game magnetization, give clues on the evolution of cooperative behaviour in the thermodynamic limit of the repeated Prisoner's dilemma game.