# Nash Equilibria of Quantum Games in the Special Two-Parameter Strategy Space
## 3 Nash Equilibria in the Two-Parameter EWL Scheme  
In this section, we consider two categories of games that have Nash equilibria in pure quantum strategies. A Nash equilibrium in a two-person game is a pair of strategies that meets some kind of stability criterion: each strategy in the equilibrium is a best response to the other strategy. For the quantum schema (2), the formal definition is as follows:  
**Definition 1**. : _The pair of strategies \(\left((\theta_{a}^{*},\phi_{a}^{*}),(\theta_{b}^{*},\phi_{b}^{*})\right)\in D_{a} \times D_{b}\) is a Nash equilibrium (NE) of the game (2) if_  
\[\Pi_{a}(\theta_{a}^{*},\phi_{a}^{*},\theta_{b}^{*},\phi_{b}^{*}) \geq\Pi_{a}(\theta_{a},\phi_{a},\theta_{b}^{*},\phi_{b}^{*})\text{ for }(\theta_{a},\phi_{a})\in D_{a}\text{ and} \tag{14}\] \[\Pi_{b}(\theta_{a}^{*},\phi_{a}^{*},\theta_{b}^{*},\phi_{b}^{*}) \geq\Pi_{b}(\theta_{a}^{*},\phi_{a}^{*},\theta_{b},\phi_{b})\text{ for }(\theta_{b},\phi_{b})\in D_{b}, \tag{15}\]  
_where we identify \(\left((\theta_{a}^{*},\phi_{a}^{*}),(\theta_{b}^{*},\phi_{b}^{*})\right)\) with \(\left(\theta_{a}^{*},\phi_{a}^{*},\theta_{b}^{*},\phi_{b}^{*}\right)\)._  
In other words, no player has a profitable unilateral deviation from his strategy. The existence and form of Nash equilibria are closely related to the values of the game payoffs. The proposition below determines sufficient conditions for the existence of Nash equilibria \(\left(\theta_{a}^{*},\phi_{a}^{*},\theta_{b}^{*},\phi_{b}^{*}\right)\) for \(\theta_{a}^{*}=\theta_{b}^{*}=\frac{\pi}{2}\). **Proposition 1**. : _If \(D_{\text{odd}}^{a}D_{\text{odd}}^{b}\geq 0\) and \(A_{\text{odd}}^{a}A_{\text{odd}}^{b}\geq 0\), then the game (2) has a Nash equilibrium \(\left(\theta_{a}^{*},\phi_{a}^{*},\theta_{b}^{*},\phi_{b}^{*}\right)\) if the form \(\left(\pi/2,\phi_{a}^{*},\pi/2,\phi_{b}^{*}\right)\), where the range of \(\phi_{a}^{*},\phi_{b}^{*}\) depends on constraints imposed on \(D_{\text{odd}}^{p}\) and \(A_{\text{odd}}^{p}\):_  
_i._ \((\phi_{a}^{*},\phi_{b}^{*})\in\{(0,\frac{\pi}{2}),(\pi,\frac{\pi}{2})\}\) _for_ \(D_{\text{odd}}^{p}\geq 0\) _and_ \(A_{\text{odd}}^{p}\geq 0\)_,_  
_ii._ \((\phi_{a}^{*},\phi_{b}^{*})\in\{(\frac{\pi}{2},0),(\frac{3\pi}{2},\pi)\}\) _for_ \(D_{\text{odd}}^{p}\geq 0\) _and_ \(A_{\text{odd}}^{p}\leq 0\)_,_  
_iii._ \((\phi_{a}^{*},\phi_{b}^{*})\in\{(\frac{3\pi}{2}),(0,\frac{\pi}{2},\pi)\}\) _for_ \(D_{\text{odd}}^{p}\leq 0\) _and_ \(A_{\text{odd}}^{p}\geq 0\)_,_  
_iv._ \((\phi_{a}^{*},\phi_{b}^{*})\in\{(0,\frac{3\pi}{2}),(\pi,\frac{\pi}{2})\}\) _for_ \(D_{\text{odd}}^{p}\leq 0\) _and_ \(A_{\text{odd}}^{p}\leq 0\)_,_  
_v._ \(\phi_{a}^{*}+\phi_{b}^{*}=\frac{\pi}{2}\) _for_ \(D_{\text{odd}}^{p}\geq 0\) _and_ \(A_{\text{odd}}^{p}=0\)_,_  
_vi._ \(\phi_{a}^{*}+\phi_{b}^{*}=\frac{3\pi}{2}\) _for_ \(D_{\text{odd}}^{p}\leq 0\) _and_ \(A_{\text{odd}}^{p}=0\)_,_  
_vii._ \(\phi_{a}^{*}-\phi_{b}^{*}=\frac{3\pi}{2}\) _for_ \(D_{\text{odd}}^{p}=0\) _and_ \(A_{\text{odd}}^{p}\geq 0\)_,_  
_viii._ \(\phi_{a}^{*}-\phi_{b}^{*}=\frac{\pi}{2}\) _for_ \(D_{\text{odd}}^{p}=0\) _and_ \(A_{\text{odd}}^{p}\leq 0\)_,_  
_ix._ \(\phi_{a}^{*},\phi_{b}^{*}\in[0,2\pi)\) _for_ \(D_{\text{odd}}^{p}=0\) _and_ \(A_{\text{odd}}^{p}=0\)_._  
_The payoffs at the Nash equilibrium are equal to_  
\[ \begin{split}&\Pi_{a}\left(\theta_{a}^{*},\phi_{a}^{*},\theta_{b}^{*}, \phi_{b}^{*}\right)=\left(D_{\text{even}}^{a}+|D_{\text{odd}}^{a}|+A_{\text{ even}}^{a}+|A_{\text{odd}}^{a}|\right)/2,\\ &\Pi_{b}\left(\theta_{a}^{*},\phi_{a}^{*},\theta_{b}^{*},\phi_{b} ^{*}\right)=\left(D_{\text{odd}}^{b}|+|D_{\text{odd}}^{b}|+A_{\text{even}}^{b} +|A_{\text{odd}}^{b}|\right)/2.\end{split}  \tag{16}\]  
Proof.