A non-cooperative Pareto-efficient solution to a one-shot Prisoners Dilemma

The Prisoner’s Dilemma is a simple model that captures the essential contradiction between individual rationality and global rationality. Although the one-shot Prisoner’s Dilemma is usually viewed simple, in this paper we will categorize it into five different types. For the type-4 Prisoner’s Dilemma game, we will propose a self-enforcing algorithmic model to help non-cooperative agents obtain Pareto-efficient payoffs. The algorithmic model is based on an algorithm using complex numbers and can work in macro applications.

💡 Research Summary

The paper revisits the classic one‑shot Prisoner’s Dilemma (PD), a game in which two rational players each choose either “Cooperate” or “Defect”. In the standard formulation the unique Nash equilibrium is (Defect, Defect), yielding the payoff (P, P), which is Pareto‑inferior to the cooperative outcome (R, R). Recent quantum‑game approaches, notably the Eisert‑Wilkens‑Lewenstein (EWL) model, have shown that by allowing players to act on an entangled quantum state they can reach a quantum Nash equilibrium that delivers the Pareto‑optimal payoff (R, R). However, the EWL model has been criticized for (1) introducing new rules that do not map onto the original PD, (2) treating the quantum state as a binding contract, (3) lacking a quantum Nash equilibrium in the full three‑parameter strategy space, and (4) requiring the arbitrator to perform quantum measurements—an unrealistic demand for macro‑level applications such as economics or politics.

To overcome these objections, the authors propose an “Amended EWL” (A‑EWL) framework that can be simulated entirely with classical computation. Each player possesses (i) a quantum coin (qubit) that can be represented by a two‑dimensional complex vector, (ii) a classical card whose two sides encode “Cooperate” and “Defect”, and (iii) a communication channel to a neutral arbitrator. The protocol proceeds in five steps: (1) initialise both qubits in the state |C C⟩, (2) apply an entangling unitary J(γ) (γ∈