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Nash Equilibrium Improvement in Quantum Extensions of Symmetric Games

Anna Gorczyca–Goraj, Krzysztof Grzegorz Grzanka, Piotr Frąckiewicz, Remigiusz Smolinski, M. Szopa

Published Aug 6, 2026• 0 citations
Publisher

Abstract

Abstract Quantum strategies can alter the equilibrium structure of classical games, but apparent improvements may depend on arbitrary strategy labels or on the particular quantum operations admitted. Here, we characterize payoff improvement across all permissible three-strategy extensions of renaming-symmetric \(2 \times 2\) games within the Eisert-Wilkens-Lewenstein framework. We preserve the classical game as a subgame and add one fixed unitary strategy, restricting attention to the three extension classes that remain invariant under isomorphic transformations of the classical game. Across six canonical payoff orderings, including coordination, anti-coordination, and dominant-strategy dilemmas, we derive the conditions under which the extended games admit pure Nash equilibria whose payoffs are no lower than those of the corresponding symmetric classical equilibrium. The classical-type \(A_0\) extension produces no strict improvement, while \(B_0\) does so only in restricted regions and generally asymmetrically. By contrast, the genuinely quantum \(C_0\) extension supports non-inferior equilibria across all six classes and yields strict, mutually beneficial improvements in identifiable payoff regions. These results move beyond isolated game examples by showing precisely when representation-invariant quantum extensions can improve equilibrium payoffs and when they cannot. These conditions provide a design map for quantum-enabled multi-agent protocols by identifying the payoff structures under which an added quantum strategy can improve equilibrium outcomes without altering the players' underlying preferences.

DOI
10.21203/rs.3.rs-10471769/v1

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Abstract Quantum strategies can alter the equilibrium structure of classical games, but apparent improvements may depend on arbitrary strategy labels or on the particular quantum operations admitted. Here, we characterize payoff improvement across all permissible three-strategy extensions of renaming-symmetric (2 \times 2) games within the Eisert-Wilkens-Lewenstein framework. We preserve the classical game as a subgame and add one fixed unitary strategy, restricting attention to the three extension classes that remain invariant under isomorphic transformations of the classical game. Across six canonical payoff orderings, including coordination, anti-coordination, and dominant-strategy dilemmas, we derive the conditions under which the extended games admit pure Nash equilibria whose payoffs are no lower than those of the corresponding symmetric classical equilibrium. The classical-type (A_0) extension produces no strict improvement, while (B_0) does so only in restricted regions and generally asymmetrically. By contrast, the genuinely quantum (C_0) extension supports non-inferior equilibria across all six classes and yields strict, mutually beneficial improvements in identifiable payoff regions. These results move beyond isolated game examples by showing precisely when representation-invariant quantum extensions can improve equilibrium payoffs and when they cannot. These conditions provide a design map for quantum-enabled multi-agent protocols by identifying the payoff structures under which an added quantum strategy can improve equilibrium outcomes without altering the players' underlying preferences.
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.
The aim of the paper is to examine pure Nash equilibria in a quantum game that extends the classical bimatrix game of dimension 2. The strategies of quantum players are specific types of two-parameter unitary operations such that the resulting quantum game is invariant under isomorphic transformations of the input classical game. We formulate general statements for the existence and form of Nash equilibria and discuss their Pareto efficiency. We prove that, depending on the payoffs of a classical game, the corresponding quantum game may or may not have Nash equilibria in the set of unitary strategies under study. Some of the equilibria cease to be equilibria if the players’ strategy set is the three-parameter special unitary group.
An evolutionarily stable strategy (ESS) is a strategy that, once adopted by a population, cannot be invaded by any sufficiently rare alternative (mutant) strategy. We analyse evolutionary stability in an Eisert--Wilkens--Lewenstein (EWL) quantization of a symmetric 2×22\times22×2 coordination game, whose classical mixed equilibrium p∗=1/2p^*=1/2p∗=1/2 is not an ESS. In the restricted two-parameter EWL strategy space, the corresponding quantum strategy is a strict symmetric Nash equilibrium and an ESS for every entanglement strength γ>0γ>0γ>0. In the full SU(2)SU(2)SU(2) strategy space it remains a Nash equilibrium, but in the unreduced game, in which every unitary is a distinct strategy, exact payoff duplicates preclude strictness and ESS for every pure strategy. For the agreement-payoff class, the payoff factors through a Bloch map; identifying payoff-equivalent unitaries gives the reduced space U(1)\SU(2)≅S2U(1)\backslash SU(2)\cong S^2U(1)\SU(2)≅S2, where strictness and ESS are restored for γ>0γ>0γ>0. An arbitrarily small perturbation outside this class that preserves p∗=1/2p^*=1/2p∗=1/2 yields a profitable deviation, so the reduced-space result is not generic.
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.
We develop a rigorous mathematical framework for quantum game theory applied to static 2x2 games, extending classical concepts to the quantum setting where players may employ arbitrary unitary operations (pure strategies) or probability measures over the continuous group SU(2) (mixed strategies). The Eisert-Wilkens-Lewenstein protocol is introduced as the standard implementation of quantum 2x2 games. We prove the existence of Nash equilibria for continuous quantum mixed strategies via a fixed-point argument, generalising the classical Nash theorem to the quantum case.
The study focuses on strategic-form games extended in the Eisert–Wilkens–Lewenstein scheme by two unitary operations. Conditions are determined under which the pair of unitary operators, along with classical strategies, form a game invariant under isomorphic transformations of the input classical game. These conditions are then applied to determine these operators, resulting in five main classes of games satisfying the isomorphism criterion, and a theorem is proved providing a practical criterion for this isomorphism. The interdependencies between different classes of extensions are identified, including limit cases in which one class transforms into another.