Agent-based Modelling of Quantum Prisoner's Dilemma (2404.02216v2)

Abstract: What happens when an infinite number of players play a quantum game? In this tutorial, 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 behaviour of the quantum players (or, \textit{qubits}) in the thermodynamic limit, and for QuPD, we consider the existence of bipartite entanglement between neighbouring quantum 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 tutorial demonstrate that both ABM and NEM, in conjunction with the chosen five indicators, provide insightful information on cooperative behaviour in an infinite-player one-shot quantum Prisoner's dilemma.