# Nash Equilibria of Quantum Games in the Special Two-Parameter Strategy Space
## 3 Nash Equilibria in the Two-Parameter EWL Scheme
: The inequalities (14) and (15) for \(\left(\theta_{a}^{*},\phi_{a}^{*},\theta_{b}^{*},\phi_{b}^{*}\right)\), to establish a Nash equilibrium at \(\theta_{a}^{*}=\theta_{b}^{*}=\pi/2\), can be transformed into the forms  
\[D_{\text{odd}}^{a}(\sin(\phi_{a}^{*}+\phi_{b}^{*})-\cos(\theta_{a}+\phi_{a}- \phi_{b}^{*}))-A_{\text{odd}}^{a}(\sin(\phi_{a}^{*}-\phi_{b}^{*})+\cos(\theta_{a}+ \phi_{a}-\phi_{b}^{*}))\geq 0 \tag{17}\]  
and  
\[D_{\text{odd}}^{b}(\sin(\phi_{a}^{*}+\phi_{b}^{*})-\cos(\theta_{b}+\phi_{b}- \phi_{a}^{*}))-A_{\text{odd}}^{b}(\sin(  
**Example 1**. : _Let us consider the game given by a bimatrix_  
\[ \begin{array}{c}q&1-q\\ p&\left(\begin{array}{cc}(4,2)&(1,3)\\ (3,4)&(2,1)\end{array}\right),\end{array}  \tag{21}\]  
_where \(p,q\in[0,1]\) are classical mixed strategies. This game has no pure-strategy Nash equilibria and one mixed-strategy equilibrium corresponding to \(p=3/4\) and \(q=1/2\) with the payoffs \(\left( \begin{array}{cc}\tilde{5}&\tilde{5}\\ \tilde{2}\end{array} \right)\). In the EWL quantum approach with the full \(\mathsf{SU}(2)\) strategy set, there are no Nash equilibria because the game has no dominant pair of payoffs [21]. However, in parameterization (3), according to Proposition 1, \(D^{p}_{\text{odd}}\geq 0\) and \(A^{p}_{\text{odd}}\leq 0\) for \(p=a,b\); therefore, one of the non-trivial Nash equilibria is for a pair of strategies_  
\[(\theta_{a}^{*},\theta_{a}^{*},\theta_{b}^{*},\phi_{b}^{*})=\left(\frac{\pi}{2 },\frac{\pi}{2},\frac{\pi}{2},0\right). \tag{22}\]  
_The corresponding NE payoffs are \(\left(\Pi_{a}(\frac{\pi}{2},\frac{\pi}{2},\frac{\pi}{2},0),\Pi_{b}(\frac{\pi }{2},\frac{\pi}{2},\frac{\pi}{2},0)\right)=(\frac{7}{2},3)\). They are higher than classical payoffs and lie on the Pareto frontier of classical mixed strategies [22] (see also Figure 1)._  
_In this way, we get a truly quantized game with a non-trivial and Pareto efficient Nash equilibrium which is invariant with respect to isomorphic transformations of an input classical game._  
Another property relates to \(2\times 2\) bimatrix games for which there are dominant pairs of payoffs, i.e., pairs of strategies that guarantee both players the highest payoffs. **Proposition 2**. : _Let us suppose that the bimatrix (1) has a dominant pair of payoffs--i.e., there exists \(k,l\in\{0,1\}\) such that \(p_{kl}=\max_{i,j}\{p_{ij}\}\) for \(p\in\{a,b\}\). Then, the game (2) has Nash equilibria \((\theta_{a}^{*},\phi_{a}^{*},\theta_{b}^{*},\phi_{b}^{*})\) of the form_  
\(i.\)__\((0,\phi,0,-\phi)\) _and_ \((\pi,\phi,\pi,\pi-\phi)\) _for_ \(p_{00}=\max_{i,j}\{p_{ij}\}\)_,_  
\(ii.\)__\((0,\phi,0,\pi-\phi)\) _and_ \((\pi,\phi,\pi,-\phi)\) _for_ \(p_{11}=\max_{i,j}\{p_{ij}\}\)_,_  
\(iii.\)__\((0,\phi,\pi,\phi)\) _and_ \((\pi,\phi+\pi,0,\phi)\) _for_ \(p_{01}=\max_{i,j}\{p_{ij}\}\)_,_  
\(iv.\)__\((0,\phi+\pi,\pi,\phi)\) _and_ \((\pi,\phi,0,\phi)\) _for_ \(p_{10}=\max_{i,j}\{p_{ij}\}\)_,_  
\(v.\)__\((0,\phi,0,\phi^{\prime})\) _and_ \((\pi,\phi,\pi,\phi^{\prime})\) _for_ \(p_{00}=p_{11}=\max_{i,j}\{p_{ij}\}\)_,_  
\(vi.\)__\((0,\phi,\pi,\phi^{\prime})\) _and_ \((\pi,\phi,0,\phi^{\prime})\) _for_ \(p_{01}=p_{10}=\max_{i,j}\{p_{ij}\}\)_,_  
_where_ \(\phi,\phi^{\prime}\in[0,2\pi)\)_. The payoffs at the Nash equilibria are equal to the dominant values_ \(\Pi_{p}\big{(}\theta_{a}^{*},\phi_{a}^{*},\theta_{b}^{*},\phi_{b}^{*}\big{)}=p _{kl}\)_._  
Proof. : The proof of this proposition can be derived directly from properties of payoff functions (6). In case \(i\), let us assume that \((\theta_{a}^{*},\phi_{a}^{*},\theta_{b}^{*},\phi_{b}^{*})=(0,\phi,0,-\phi)\), for any \(\phi\in[0,2\pi)\)  
Figure 1: Payoff polygon for game (21) with the classical (brown) and quantum (black) equilibrium payoff vectors. Notice that \(P_{0,0}(0,\phi,0,-\phi)=1\) and \(P_{k,l}(0,\phi,0,-\phi)=0\) for \((k,l)\
eq(0,0)\).