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arXiv:2404.02216 · APL Quantum

Agent-based modeling of quantum prisoner’s dilemma

Colin Benjamin, Rajdeep Tah

Published Jan 6, 2025• 3 citations
arXivHTMLPDF

Abstract

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 behavior of the quantum players (or qubits) in the thermodynamic limit, and for QuPD, we consider the existence of bipartite entanglement between neighboring quantum players. For the five indicators in question, we observe 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 quantum and the 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 an infinite-player one-shot quantum prisoner’s dilemma.

DOI
10.1063/5.0239154

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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 behavior of the quantum players (or qubits) in the thermodynamic limit, and for QuPD, we consider the existence of bipartite entanglement between neighboring quantum players. For the five indicators in question, we observe 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 quantum and the 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 an infinite-player one-shot quantum prisoner’s dilemma.
A quantized version of the prisoner’s dilemma game based on the Marinatto–Weber quantization scheme is introduced. A four-qubit-basis initial entangled quantum state is assumed. Two domains for Nash equilibria are observed in terms of the degree of entanglement of the game. Each Nash equilibrium point is found to be Pareto optimal in its corresponding domain. Therefore, there is a unique solution to the game dilemma in each domain of the entanglement parameter. The maximally entangled limit is studied, and the players’ payoffs are found to equal the average of the payoff matrix independently of their choices. Consequently, any combination of players’ choices is a solution to the dilemma of the game. The nonentangled limits are also studied and found to coincide with the classical mixed-strategy game. Our results are equivalent to those of the noisy schemes of the quantum prisoner’s dilemma game. In our scheme, entanglement can be interpreted as a noise.
In this paper, we extend the quantum game theory of Prisoner’s Dilemma to the N-player case. The final state of quantum game theory of N-player Prisoner’s Dilemma is derived, which can be used to investigate the payoff of each player. As demonstration, two cases (2-player and 3-player) are studied to illustrate the superiority of quantum strategy in the game theory. Specifically, the non-unique entanglement parameter is found to maximize the total payoff, which oscillates periodically. Finally, the optimal strategic set is proved to depend on the selection of initial states.
The authors proposed a quantum Prisoner's Dilemma (PD) game as a natural extension of the classic PD game to resolve the dilemma. Here, we establish a new Nash equilibrium principle of the game, propose the notion of convergence and discover the convergence and phase-transition phenomena of the evolutionary games on networks. We investigate the many-body extension of the game or evolutionary games in networks. For homogeneous networks, we show that entanglement guarantees a quick convergence of super cooperation, that there is a phase transition from the convergence of defection to the convergence of super cooperation, and that the threshold for the phase transitions is principally determined by the Nash equilibrium principle of the game, with an accompanying perturbation by the variations of structures of networks. For heterogeneous networks, we show that the equilibrium frequencies of super-cooperators are divergent, that entanglement guarantees emergence of super-cooperation and that there is a phase transition of the emergence with the threshold determined by the Nash equilibrium principle, accompanied by a perturbation by the variations of structures of networks. Our results explore systematically, for the first time, the dynamics, morphogenesis and convergence of evolutionary games in interacting and competing systems.
This work concentrates on simultaneous move non-cooperating quantum games. Part of it is evidently not new, but it is included for the sake self consistence, as it is devoted to introduction of the mathematical and physical grounds of the pertinent topics, and the way in which a simple classical game is modified to become a quantum game (a procedure referred to as a quantization of a classical game). The connection between game theory and information science is briefly stressed, and the role of quantum entanglement (that plays a central role in the theory of quantum games), is exposed. Armed with these tools, we investigate some basic concepts like the existence (or absence) of a pure strategy and mixed strategy Nash equilibrium and its relation with the degree of entanglement. The main results of this work are as follows: 1) Construction of a numerical algorithm based on the method of best response functions, designed to search for pure strategy Nash equilibrium in quantum games. The formalism is based on the discretization of a continuous variable into a mesh of points, and can be applied to quantum games that are built upon two-players two-strategies classical games, based on the method of best response functions. 2) Application of this algorithm to study the question of how the existence of pure strategy Nash equilibrium is related to the degree of entanglement (specified by a continuous parameter γ ). It is shown that when the classical game GC has a pure strategy Nash equilibrium that is not Pareto efficient, then the quantum game GQ with maximal entanglement (γ = π/2) has no pure strategy Nash equilibrium. By studying a non-symmetric prisoner dilemma game, it is found that there is a critical value 0γc such that for γγc there is a pure strategy Nash equilibrium and for γ≥γc there is no pure strategy Nash equilibrium. The behavior of the two payoffs as function of γ starts at that of the classical ones at (D, D) and approaches the cooperative classical ones at (C, C) (C = confess, D = don’t confess). 3) We then study Bayesian quantum games and show that under certain conditions, there is a pure strategy Nash equilibrium in such games even when entanglement is maximal. 4) We define the basic ingredients of a quantum game based on a two-player three strategies classical game. This requires the introduction of trits (instead of bits) and quantum trits (instead of quantum bits). It is proved that in this quantum game, there is no classical commensurability in the sense that the classical strategies are not obtained as a special case of the quantum strategies.
Cooperation is a fundamental aspect of biological systems and human society. In this work, we propose a generalized prisoner's dilemma quantum game and study the game on Watts-Strogatz small-world network (WS model) with different values of entanglement. Here entanglement is used as a metaphor to study relationship between players. In a single step, each player has three strategies to choose: cooperation, defection and adopting a special quantum strategy first introduced by Eisert (denoted by Q). The person who chooses the quantum strategy Q can be regarded as a super cooperator. In prisoner's dilemma game, cooperation is very hard to emerge on relatively homogeneous networks, such as random graphs and WS model. Our experimental results show that in generalized prisoners dilemma quantum game, entanglement can guarantee emergence of super cooperators and greatly increase the payoff of the whole network. Even on homogeneous networks with very large inducement value(b is close to 2), the frequency of cooperators can still exceeds 70%. This work provides a new insight into understanding cooperation when considering the role of relationship between cooperators.