  Consider the scenario where an infinite number of players (i.e., the
\textit{thermodynamic} limit) find themselves in a Prisoner's dilemma type
situation, in a \textit{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 \textit{noise} (or, under \textit{selection pressure}), in repeated
Prisoner's Dilemma games, involving strategies such as \textit{Tit-for-Tat},
\textit{Always Defect}, \textit{GRIM}, \textit{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 \textit{1D}-Ising chain. 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 indicates a
\textit{first}-order \textit{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 \textit{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.
