# Nash Equilibria of Quantum Games in the Special Two-Parameter Strategy Space
## 2 The Eisert-Wilkens-Lewenstein (EWL) Scheme
The invariance criterion is crucial for the quantum game to be a true extension of the classical game. If this condition is not met, two isomorphic forms of the classical game can lead to non-equivalent quantum extensions. Then. it is not clear which of them should be considered as an extension of the classical game. This problem occurs when, for example, we use the parameterization of unitary operators studied in [4]:  
\[U(\theta,\phi)= \begin{pmatrix}e^{i\phi}\cos\frac{\theta}{2}&\sin\frac{\theta}{2} \\ -\sin\frac{\theta}{2}&e^{-i\phi}\cos\frac{\theta}{2}\end{pmatrix} . \tag{9}\]  
Then two bimatrix games that differ in the order of the strategy of one player imply different quantum games [20]. The parameterization by the group \(\mathsf{SU}(2)\) is also invariant under isomorphic transformations of the input game [20], but it allows for the existence of Nash equilibria only in trivial cases, where there is a dominant pair of payoffs. The advantage of parameterization (3) is that it allows for the existence of nontrivial Nash equilibria [21]. To show the invariance of the quantum game in parameterization (3) with respect to the reordering of classical game strategies, one can present payoffs (4) and (5) of the game (1) in the symmetrized form  
\[\Pi_{p}(\theta_{a},\phi_{a},\theta_{b},\phi_{b})= D^{p}_{\text{even}}(1+\cos\theta_{a}\cos\theta_{b})/2+ \tag{10}\] \[D^{p}_{\text{odd}}(\cos(\phi_{a}+\phi_{b})(\cos\theta_{a}+\cos \theta_{b})+\sin(\phi_{a}+\phi_{b})\sin\theta_{a}\sin\theta_{b})/2+\] \[A^{p}_{\text{even}}(1-\cos\theta_{a}\cos\theta_{b})/2+\] \[A^{p}_{\text{odd}}(\cos(\phi_{a}-\phi_{b})(\cos\theta_{a}-\cos \theta_{b})-\sin(\phi_{a}-\phi_{b})\sin\theta_{a}\sin\theta_{b})/2.\]  
The coefficients \(D\) (diagonal) and \(A\) (anti-diagonal) are defined by  
\[ \begin{array}{l}D^{p}_{\text{even}}=(p_{00}+p_{11})/2,\\ D^{p}_{\text{odd}}=(p_{00}-p_{11})/2,\\ A^{p}_{\text{even}}=(p_{01}+p_{10})/2,\\ A^{p}_{\text{odd}}=(p_{01}-p_{10})/2,\end{array}  \tag{11}\]  
and correspond to even or odd combinations of the \(p\)th player payoffs \(p_{ij}\) of the initial bimatrix (1), \(p\in\{a,b\}\). Thanks to symmetrized payoffs, it is easy to prove the invariance of the quantum game with respect to the reordering of the classical game strategies. If, e.g., in the classical game (1), the payoffs of the second player are swapped, the new matrix becomes  
\[(A^{\prime},B^{\prime})= \begin{pmatrix}(a_{01},b_{01})&(a_{00},b_{00})\\ (a_{11},b_{11})&(a_{10},b_{10})\end{pmatrix} . \tag{12}\]  
The (even)odd diagonal payoffs are replaced by antidiagonal and vice versa--\(D^{\prime p}_{\text{even}}=(p_{01}+p_{10})/2=A^{p}_{\text{even}}\); and analogously, \(A^{\prime p}_{\text{even}}=D^{p}_{\text{even}}\), \(D^{\prime p}_{\text{odd}}=A^{p}_{\text{odd}}\) and \(A^{\prime p}_{\text{odd}}=D^{p}_{\text{odd}}\). To see the invariance of the quantum games payoffs  
\[\Pi^{\prime}_{p}(\theta^{\prime}_{a},\phi^{\prime}_{a},\theta^{\prime}_{b}, \phi^{\prime}_{b})=\Pi_{p}(\theta_{a},\phi_{a},\theta_{b},\phi_{b}), \tag{13}\]  
it is enough to make the following substitutions \(\theta^{\prime}_{a}=-\theta_{a}\), \(\phi^{\prime}_{a}=\phi_{a}\), \(\theta^{\prime}_{b}=\pi-\theta_{b}\) and \(\phi^{\prime}_{b}=-\phi_{b}\). Another advantage of quantum payoffs in the form (10), useful in the context of Nash equilibria, is that they depend on \(\phi_{a}\) and \(\phi_{b}\) only through odd coefficients \((D)A^{p}_{\text{odd}}\).