Decision-making by the two negotiating parties is simulated by a prisoner's dilemma game.The game is formulated in a quantum manner, where players strategies are unitary transformations of qubits built over the basis of opposite decision options.Quantum strategies are correlated through the mechanism of quantum entanglement and the result of the game is obtained by the collapse of the resulting transformed state.The range of strategies allowed for quantum players is richer than in case of a classical game and therefore the result of the game can be better optimized.On the other hand, the quantum game is save against eavesdropping and the players can be assured that this type of quantum arbitration is fair.We show that quantum prisoner's dilemma has more favorable Nash equilibria than its classical analog and they are close to the Pareto optimal solutions.Some economical examples of utilizing quantum game Nash equilibria are proposed.
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.
Theory of games is an important discipline of applied mathematics, which has found numerous applications in psychology, ecology, and social science. While a well-established framework has been established for classical game theory, extending, and generalizing it towards the quantum domain still has some open questions. In this project, we focus on the prisoner’s dilemma-a standard example of a game analyzed in game theory-in a quantum world. Here, we start with the quantum version of prisoner’s dilemma, explain the quantum circuit implementation for describing the behavior of quantum prisoners, and investigate how quantum entanglement can change their strategies. By using the IBM quantum computer, we experimentally study the quantum prisoner dilemma, and we conclude that the best quantum strategies will break the prisoner’s dilemma.
In a normal form, prisoners' dilemma (PD) is represented by a payoff matrix showing players' strategies and payoffs. To obtain the distinguishing trait and strategic form of PD, certain constraints are imposed on the elements of its payoff matrix. We quantize PD by a generalized quantization scheme to analyze its strategic behavior in the quantum domain. The game starts with a general entangled state of the form and the measurement for payoffs is performed in entangled and product bases. We show that for both measurements, there exist respective cutoff values of entanglement of the initial quantum state up to which the strategic form of the game remains intact. Beyond these cutoffs the quantized PD behaves like the chicken game (CG) up to another cutoff value. For the measurement in the entangled basis the dilemma is resolved for sinξ > 1/7 with Q ⊗ Q as a Nash Equilibrium (NE). However, the quantized game behaves like PD when sinξ > 1/3; whereas in the range 1/7 < sinξ < 1/3 it behaves like CG with Q ⊗ Q as an NE. For the measurement in the product basis the quantized PD behaves like classical PD for with D ⊗ D as an NE. In region , the quantized PD behaves like classical CG with C ⊗ D and D ⊗ C as NEs.
A well-known model in game theory, the Volunteer's Dilemma describes a group of
n players who decide whether to volunteer for a collective benefit at a personal cost, or to abstain and risk forfeiting the benefit altogether. A quantum version of this dilemma, developed within the Eisert-Wilkens-Lewenstein framework, allows each player to manipulate one qubit of a shared entangled state, leading to symmetric Nash equilibria with higher expected payoffs than in the classical game. Existing analyses, however, assume maximal entanglement, which is experimentally demanding for hardware-constrained quantum systems. Within the same framework, we introduce a generalized Quantum Volunteer's Dilemma with a tunable entanglement parameter
γ and study the extent to which equilibrium behavior depends on the level of entanglement. We derive explicit conditions relating
γ, the number of players, and the players' strategies under which symmetric Nash equilibria exist, focusing on two canonical strategy profiles: one for
2≤n≤9, and one for even
n. We find that maximal entanglement is not required to sustain symmetric equilibria. Instead, equilibrium behavior persists above a threshold value, which we compute analytically in both cases. We also demonstrate that the threshold value directly depends on system size.