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arXiv:2405.07380 · Quantum Information Processing

Permissible four-strategy quantum extensions of classical games

Piotr Frąckiewicz, Anna Gorczyca–Goraj, M. Szopa

Published Jun 30, 2025• 4 citations
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Abstract

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.

DOI
10.1007/s11128-025-04808-4

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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.
The aim of the paper is to draw attention to a special class of two parameter unitary strategies in the Eisert-Wilkens-Lewenstein scheme for quantum games. We identify the players’ strategies with basis change matrices. Then we prove that the resulting quantum game is invariant with respect to isomorphic transformations of the input game. Moreover, it is shown that the game so obtained may not be trivial with respect to pure Nash equilibria, compared with the model with strategies being the special unitary group SU (2).
The aim of this paper is to bring together the notions of quantum game and game isomorphism. The work is intended as an attempt to introduce a new criterion for quantum game schemes. The generally accepted requirement forces a quantum scheme to generate the classical game in a particular case. Now, given a quantum game scheme and two isomorphic classical games, we additionally require the resulting quantum games to be isomorphic as well. We are concerned with the Eisert-Wilkens-Lewenstein quantum game scheme and the strong isomorphism between games in strategic form.
The aim of this paper is to bring together the notions of quantum game and game isomorphism. The work is intended as an attempt to introduce a new criterion for quantum game schemes. The generally accepted requirement forces a quantum scheme to generate the classical game in a particular case. Now, given a quantum game scheme and two isomorphic classical games, we additionally require the resulting quantum games to be isomorphic as well. We are concerned with the Eisert-Wilkens-Lewenstein quantum game scheme and the strong isomorphism between games in strategic form.
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 2×22 \times 22×2 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 A0A_0A0​ extension produces no strict improvement, while B0B_0B0​ does so only in restricted regions and generally asymmetrically. By contrast, the genuinely quantum C0C_0C0​ 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.
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 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.