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The proposed scheme is then applied to the Prisoner\u2019s Dilemma game. Rational strategy profiles are examined in the presence of limited awareness of the players. In particular, the paper considers two cases of a classical player against a quantum player game: the first case when the classical player does not know that his opponent is a quantum one and the second case, when the classical player is aware of it. To this end, the notion of unawareness was used, and the extended Nash equilibria were determined.<\/jats:p>","DOI":"10.1007\/s11128-020-02743-0","type":"journal-article","created":{"date-parts":[[2020,7,29]],"date-time":"2020-07-29T06:03:44Z","timestamp":1596002624000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["A quantum approach to twice-repeated $$2\\times 2$$ game"],"prefix":"10.1007","volume":"19","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-8032-7251","authenticated-orcid":false,"given":"Katarzyna","family":"Rycerz","sequence":"first","affiliation":[]},{"given":"Piotr","family":"Fr\u0105ckiewicz","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2020,7,29]]},"reference":[{"key":"2743_CR1","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9781139525343","volume-title":"Quantum Information Theory","author":"M Wilde","year":"2013","unstructured":"Wilde, M.: Quantum Information Theory. 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