Agent-based Modelling of Quantum Prisoner's Dilemma

TL;DR

This work analyzes the emergence of cooperation in an infinite population playing the one-shot Quantum Prisoner's Dilemma (QuPD) by comparing analytical Nash equilibrium mapping (NEM) to a numerical Agent-based Modelling (ABM) approach. It employs five thermodynamic-analogue indicators—$\mu$, $\chi_{\gamma}$, $\mathfrak{c}_j$, $\langle\Lambda\rangle$, and $\wp_C$—to characterize behavior as a function of entanglement $\gamma$ and payoffs $(B,C)$ in the thermodynamic limit. A key finding is the existence of two first-order phase transitions at $\gamma_A$ and $\gamma_B$, with $\gamma_A=\tan^{-1}\sqrt{C/B}$ and $\gamma_B=\pi-\gamma_A$, separating Defect and Quantum phases; within $[\gamma_A, \gamma_B]$ quantum strategies dominate and align with Nash equilibrium and Pareto optimality. The results demonstrate a strong, consistent agreement between ABM and NEM across all indicators, including under finite noise $\beta$ and payoff variations, highlighting entanglement as a controllable resource for cooperation in quantum social dilemmas. This work thus provides a robust framework for analyzing cooperative phenomena in quantum games in the thermodynamic limit and informs potential applications in quantum networks and distributed decision-making.

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.

Figures (6)

• Figure 1: NEM/ABM: Visualization of QuPD in thermodynamic (or, infinite-player) limit. The QuPD game is mapped to a $1D$-Ising chain, where at each site, two entangled quantum players reside, and they play the two-strategy QuPD. The quantum players also interact with their nearest neighbours via a classical coupling $\mathcal{T}$, in the presence of an external uniform field $\mathcal{F}$ and noise$\beta$.
• Figure 3: ABM and NEM (in insets): $\mu^{ABM/NEM}$ vs changing cooperation bonus$\mathbb{B}$ for fixed cost$\mathbb{C}=2.0$ and entanglement$\gamma = \frac{\pi}{6}$ in QuPD.
• Figure 4: ABM and NEM (in insets): Entanglement susceptibility$\chi_\gamma$ vs entanglement$\gamma$ (a) for $\beta=1, 2, 3$ and (b) for $\beta\rightarrow \infty$. Other parameters common to both are: reward$\mathbb{R}= (\mathbb{B-C}) =3.0$, sucker's payoff$\mathbb{S}= \mathbb{-C}= -2.0$, temptation$\mathbb{T}= \mathbb{B}=5.0$, punishment$\mathbb{P}=0.0$, $\gamma_A = 0.5639$ and $\gamma_B=2.5777$ in QuPD.
• Figure 5: ABM and NEM (in insets): Correlation$\mathfrak{c}_j$ vs $\gamma$ for (a) $\beta=1, 2, 3$ and (b) $\beta\rightarrow \infty$. Other parameters common to both are distance$j=11$, reward$\mathbb{R}= (\mathbb{B-C}) =3.0$, sucker's payoff$\mathbb{S}= \mathbb{-C}= -2.0$, temptation$\mathbb{T}= \mathbb{B}=5.0$, punishment$\mathbb{P}=0.0$, $\gamma_A = 0.5639$ and $\gamma_B=2.5777$ in QuPD.
• Figure 6: ABM and NEM (in insets): Player's payoff average$\langle\Lambda\rangle$ vs $\gamma$ for (a) $\beta= 1, 2, 3$and (b)$\beta\rightarrow \infty$. Other parameters common to both are reward$\mathbb{R}= (\mathbb{B-C}) =3.0$, sucker's payoff$\mathbb{S}= \mathbb{-C}= -2.0$, temptation$\mathbb{T}= \mathbb{B}=5.0$, punishment$\mathbb{P}=0.0$, $\gamma_A = 0.5639$ and $\gamma_B=2.5777$ in QuPD.