# Nash Equilibria of Quantum Games in the Special Two-Parameter Strategy Space
## 3 Nash Equilibria in the Two-Parameter EWL Scheme
Therefore, for both \(p=a,b\), we have  
\[\Pi_{p}(0,\phi,0,-\phi)=\sum_{k,l=0}^{1}P_{k,l}(0,\phi,0,-\phi)p_{kl}=p_{00}\geq \Pi_{p}(\theta_{a},\phi_{a},\theta_{b},\phi_{b}), \tag{23}\]  
for arbitrary \((\theta_{a},\phi_{a},\theta_{b},\phi_{b})\in D_{a}\times D_{b}\). The last inequality follows from the assumption that \(p_{00}=\max_{i,j}\{p_{ij}\}\) and from normalization (8) of the payoffs coefficients. Thus, Equation (23) proves conditions (14) and (15) and yields the values of the NE payoffs at \(p=a,b\). Similar inequalities hold for the second part of case i., where also \(P_{0,0}(\pi,\phi,\pi,\pi-\phi)=1\), which ends the proof of case \(i\). The cases _ii., iii._ and _iv._, can be proven in a similar way. In the case _v._, let us notice that \(P_{0,0}(0,\phi,0,\phi^{\prime})+P_{1,1}(0,\phi,0,\phi^{\prime})=1\) for any \(\phi,\phi^{\prime}\in[0,2\pi)\). Thus, for both \(p=a,b\) we have  
\[\Pi_{p}(0,\phi,0,\phi^{\prime})=\sum_{k,l=0}^{1}P_{k,l}(0,\phi,0,\phi^{\prime })p_{kl}=p_{00}=p_{11}\geq\Pi_{p}(\theta_{a},\phi_{a},\theta_{b},\phi_{b}), \tag{24}\]  
for arbitrary \((\theta_{a},\phi_{a},\theta_{b},\phi_{b})\in D_{a}\times D_{b}\). An analogous inequality holds in the second part of the case _v._, because \(P_{0,0}(\pi,\phi,\pi,\phi^{\prime})+P_{1,1}(\pi,\phi,\pi,\phi^{\prime})=1\). The last case can be proven analogously. Note that the 2-parameter set \(D_{p}\) of players' strategies (3) is a subset of the set \(\mathsf{SU}(2)\) of all (3-parameter) unitary transformations. Therefore, it is clear from the above proof that the dominant pair of payoffs will also result in NE in the full \(\mathsf{SU}(2)\) parameterization. **Example 2**. : _Let us consider the game given by bimatrix_  
\[ \begin{array}{c}q&1-q\\ p&(\begin{array}{cc}(3,3)&(0,3)\\ (3,0)&(1,1)\end{array}).\end{array}  \tag{25}\]  
_This game has two Nash equilibria in pure strategies \((p,q)=(0,0)\) and \((1,1)\) with the payoffs \((1,1)\) and \((3,3)\), respectively. It has a dominant pair of payoffs \(p_{00}=(3,3)\); therefore, according to Proposition 2, it has also Nash equilibria for two pairs of strategies:_  
\[(\theta_{a}^{*},\phi_{a}^{*},\theta_{b}^{*},\phi_{b}^{*})=(0,\phi,0,-\phi)\text { and }(\pi,\phi,\pi,\pi-\phi),\quad\phi\in[0,2\pi). \tag{26}\]  
_The corresponding NE payoffs are equal to \((3,3)\). The payoff pair \((1,1)\) is not accessible within this type of quantum NE._  
Propositions 1 and 2 provide sufficient conditions for the existence of Nash equilibria in the center \(\theta_{a}^{*}=\theta_{b}^{*}=\pi/2\) or at the boundary \(\theta_{a}^{*},\theta_{b}^{*}\in\{0,\pi\}\) of \(\theta\)-parts of players' strategy spaces \(D_{p}\), respectively. It is also possible that the game has both types of NE. As an example, let us consider the matrix (21) from Example 1, in which the payoffs \(a_{00}\) and \(a_{10}\) are swapped. The resulting payoffs satisfy the assumptions of both propositions, and therefore, have both types of NE--the first defined in Proposition 1._ii._ and the second defined in Proposition 2._iv._ We speculate that within parameterization (3), Propositions 1 and 2 describe the only strategy profiles that allow for the existence of pure Nash equilibria.