{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,12]],"date-time":"2026-05-12T09:34:39Z","timestamp":1778578479356,"version":"3.51.4"},"reference-count":35,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2025,7,15]],"date-time":"2025-07-15T00:00:00Z","timestamp":1752537600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>The concept of Nash equilibria in pure strategies for quantum extensions of the general form of the Prisoner\u2019s 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\u2019 impact on traditional game-theoretical problems.<\/jats:p>","DOI":"10.3390\/e27070755","type":"journal-article","created":{"date-parts":[[2025,7,15]],"date-time":"2025-07-15T15:01:05Z","timestamp":1752591665000},"page":"755","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Nash Equilibria in Four-Strategy Quantum Extensions of the Prisoner\u2019s Dilemma Game"],"prefix":"10.3390","volume":"27","author":[{"given":"Piotr","family":"Fr\u0105ckiewicz","sequence":"first","affiliation":[{"name":"Institute of Exact and Technical Sciences, Pomeranian University in S\u0142upsk, ul. Bohater\u00f3w Westerplatte 64, 76-200 S\u0142upsk, Poland"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8222-0987","authenticated-orcid":false,"given":"Anna","family":"Gorczyca-Goraj","sequence":"additional","affiliation":[{"name":"Department of Operations Research, University of Economics in Katowice, ul. Bogucicka 3, 40-287 Katowice, Poland"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7228-4768","authenticated-orcid":false,"given":"Krzysztof","family":"Grzanka","sequence":"additional","affiliation":[{"name":"Department of Operations Research, University of Economics in Katowice, ul. Bogucicka 3, 40-287 Katowice, Poland"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2837-8481","authenticated-orcid":false,"given":"Katarzyna","family":"Nowakowska","sequence":"additional","affiliation":[{"name":"Institute of Exact and Technical Sciences, Pomeranian University in S\u0142upsk, ul. Bohater\u00f3w Westerplatte 64, 76-200 S\u0142upsk, Poland"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7897-7004","authenticated-orcid":false,"given":"Marek","family":"Szopa","sequence":"additional","affiliation":[{"name":"Department of Operations Research, University of Economics in Katowice, ul. 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