# Nash Equilibria of Quantum Games in the Special Two-Parameter Strategy Space
## 1 Introduction  
Game theory, which was initiated in 1928 by [PERSON] in the [1] and developed in [2], is one of the younger branches of mathematics. The aim of this theory is the mathematical modeling of participants' behavior in conflict situations, generally assuming that they seek to maximize their profits. The theory of quantum games, developed on the border of game theory and quantum mechanics, is an extension of this theory. It is an interdisciplinary research area in which it is assumed that games and decision problems are played with the use of quantum objects. The non-classical properties of these objects are essential for the way the game is played and the result of the game. The work widely recognized as the beginning of quantum game theory is [3]. [PERSON] showed on the example of a simple extensive game in which a player equipped with unitary strategies has a winning strategy. Among the other fundamental works on quantum game theory, the work [4] is of particular importance. The scheme introduced by [PERSON], [PERSON] and [PERSON] was the first formal protocol for playing a quantum game. The presented model uses the formalism of quantum computing for the description of any \(2\times 2\) bimatrix game. According to the main assumptions of the scheme, the players' strategies are unitary operations dependent on two parameters, and the players act with these operators on the maximal entangled two-qubit state. An alternative model of the quantum playing of the bimatrix game is the scheme introduced by [PERSON] and [PERSON] in [5]. In this case, the players' strategies were restricted to two unitary matrices (the identity and the Pauli matrix \(X\)), with the initial state being any fixed two-qubit state. Quantum game theory also includes quantum models for games with infinite sets of strategies. The work [6] presents a minimalist model for the quantum playing of the Cournot duopoly--a two person normal-form game, in which the set ofstrategies of each player is identified with the interval \([0,\infty)\). The theory of quantum games continues to develop dynamically, as evidenced by recent works [7; 8; 9; 10]. The discussion on various parameterizations of unitary strategies has already started with the first scheme of quantum playing of the \(2\times 2\) game [4]. The authors defined a quantum approach to the prisoner's dilemma game. In this quantum game, the set of strategies of both players was a two-parameter subset of unitary operators from \(\mathsf{SU}(2)\). The quantum game defined in this way was characterized by a unique and pareto-optimal Nash equilibrium. It turned out that the unique result of the game presented in the paper [4] is closely related to the choice of the set of unitary strategies available to the players. In the comment [11], it was shown that the quantum prisoner's dilemma does not have the pareto-optimal Nash equilibrium if the sets of players' strategies are equal to full \(\mathsf{SU}(2)\). Moreover, in such a case, there is not even a Nash equilibrium in pure strategies. The stability predicted by the equilibrium is only obtained when the players choose their strategies according to a certain probability distribution [11; 12; 13]. In the paper [14], other types of two-parameter strategies were considered, and their significant influences on the final result of the game, in particular, on the form of Nash equilibrium, were discussed. However, the dominant trend in the literature on the EWL scheme was the free use of various types of two-parameter strategies [15; 16; 17; 18; 19]. The works [20; 21] attempted to answer the question of what types of sets of unitary operations are appropriate in the EWL game. For this purpose, it was examined whether pairs of classical isomorphic games remain isomorphic when played in the quantum way. One of the main results of the paper [20] was to show that the application of the EWL scheme and certain types of unitary two-parameter operations to a pair of isomorphic games may imply different sets of Nash equilibria for a pair of isomorphic games.