  What happens when an infinite number of players play a quantum game? In this
paper, we will answer this question by looking at the emergence of cooperation
in the presence of noise in a one-shot quantum Prisoner's dilemma (QuPD). We
will use the numerical Agent-based model (ABM) and compare it with the
analytical Nash equilibrium mapping (NEM) technique. To measure cooperation, we
consider five indicators, i.e., game magnetization, entanglement
susceptibility, correlation, player's payoff average, and payoff capacity,
respectively. In quantum social dilemmas, entanglement plays a non-trivial role
in determining the players' behavior in the thermodynamic limit, and we
consider the existence of bipartite entanglement between neighboring players.
For the five indicators in question, we observe \textit{first}-order phase
transitions at two entanglement values, and these phase transition points
depend on the payoffs associated with the QuPD game. We numerically analyze and
study the properties of both the \textit{Quantum} and the \textit{Defect}
phases of the QuPD via the five indicators. The results of this paper
demonstrate that both ABM and NEM, in conjunction with the chosen five
indicators, provide insightful information on cooperative behavior in the
thermodynamic limit of the one-shot quantum Prisoner's dilemma.
