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Entropy

Quantum Negotiation Games: Toward Ethical Equilibria

Remigiusz Smolinski, Piotr Frąckiewicz, Krzysztof Grzegorz Grzanka, M. Szopa

Published Dec 31, 2025• 1 citation
Publisher

Abstract

This paper applies quantum game theory to three ethical dilemmas that frequently arise in negotiation: cooperation versus competition, self-interest versus equity, and honesty versus deception. Using quantum extensions of selected games such as the Prisoner's Dilemma, the Ultimatum Game, the Battle of the Sexes, and the Buyer-Seller Game, we examine whether quantization can generate equilibria that improve classical outcomes while also aligning more closely with ethical principles such as fairness, cooperation, and honesty. The analysis shows that quantum strategies, through entanglement and superposition, can sustain cooperative, fair, or honest behaviour as stable equilibria, outcomes that are typically unstable or unattainable in classical settings. The specific outcomes depend on the chosen quantization method, but across cases, the analysis consistently shows that quantum formulations expand the range of solutions in which efficiency and ethical principles co-exist.

DOI
10.3390/e28010051

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This paper applies quantum game theory to three ethical dilemmas that frequently arise in negotiation: cooperation versus competition, self-interest versus equity, and honesty versus deception. Using quantum extensions of selected games such as the Prisoner's Dilemma, the Ultimatum Game, the Battle of the Sexes, and the Buyer-Seller Game, we examine whether quantization can generate equilibria that improve classical outcomes while also aligning more closely with ethical principles such as fairness, cooperation, and honesty. The analysis shows that quantum strategies, through entanglement and superposition, can sustain cooperative, fair, or honest behaviour as stable equilibria, outcomes that are typically unstable or unattainable in classical settings. The specific outcomes depend on the chosen quantization method, but across cases, the analysis consistently shows that quantum formulations expand the range of solutions in which efficiency and ethical principles co-exist.
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.
This paper investigates Nash equilibria in pure strategies for quantum approach to the Prisoner's Dilemma. The quantization process involves extending the classical game by introducing two additional unitary strategies. We consider five classes of such quantum games, which remain invariant under isomorphic transformations of the classical game. For each class, we identify and analyse all possible Nash equilibria. Our results reveal the complexity and diversity of strategic behaviour in the quantum setting, providing new insights into the dynamics of classical decision-making dilemmas. In the case of the standard Prisoner's Dilemma, the resulting Nash equilibria of quantum extensions are found to be closer to Pareto optimal solutions than those of the classical equilibrium.
The concept of Nash equilibria in pure strategies for quantum extensions of the general form of the Prisoner's Dilemma game is investigated. The process of quantization involves incorporating two additional unitary strategies, which effectively expand the classical game. We consider five classes of such quantum games, which remain invariant under isomorphic transformations of the classical game. The resulting Nash equilibria are found to be more closely aligned with Pareto-optimal solutions than those of the conventional Nash equilibrium outcome of the classical game. Our results demonstrate the complexity and diversity of strategic behavior in the quantum setting, providing new insights into the dynamics of classical decision-making dilemmas. In particular, we provide a detailed characterization of strategy profiles and their corresponding Nash equilibria, thereby extending the understanding of quantum strategies' impact on traditional game-theoretical problems.
To explore the influence of quantum information on the common social problem of honesty and trickery, we propose a Bayesian model for the quantum prisoners’ dilemma game. In this model, the players’ strategy formation is regarded as a negotiation of their move contract based on their types of decision policies, honesty or trickery. Although the implementation of quantum information cannot eliminate tricky players, players in our model can always end up with higher payoffs than in the classical game. For a good proportion of a credibility parameter value, a rational player will take an honest action, which is in remarkable contrast to the observation that players tend to defect in the classical prisoners’ dilemma game. This research suggests that honesty will be promoted to enhance cooperation with the assistance of quantum information resources.
Nash equilibria and correlated equilibria of classical and quantum games are investigated in the context of their Pareto efficiency. The examples of the prisoner's dilemma, battle of the sexes and the game of chicken are studied. Correlated equilibria usually improve Nash equilibria of games but require a trusted correlation device susceptible to manipulation. The quantum extension of these games in the Eisert-Wilkens-Lewenstein formalism and the Frąckiewicz-Pykacz parameterization is analyzed. It is shown that the Nash equilibria of these games in quantum mixed Pauli strategies are closer to Pareto optimal results than their classical counter-parts. The relationship of mixed Pauli strategies equilibria and correlated equilibria is also studied.
We present a novel formulation of quantum game theory based on the Schmidt decomposition, which has the merit that the entanglement of quantum strategies is manifestly quantified. We apply this formulation to 2-player, 2-strategy symmetric games and obtain a complete set of quantum Nash equilibria. Apart from those available with the maximal entanglement, these quantum Nash equilibria are extensions of the Nash equilibria in classical game theory. The phase structure of the equilibria is determined for all values of entanglement, and thereby the possibility of resolving the dilemmas by entanglement in the game of Chicken, the Battle of the Sexes, the Prisoners' Dilemma, and the Stag Hunt, is examined. We find that entanglement transforms these dilemmas with each other but cannot resolve them, except in the Stag Hunt game where the dilemma can be alleviated to a certain degree.