# On Correlated Equilibria in Marinatto-Weber Type Quantum Games
## 5 Bimatrix Representation of the MW Scheme
The final state \(\rho_{\rm f}\) resulting from playing a general strategy profile is \[\rho_{\rm f}=\frac{1}{6}\sum_{i=0}^{1}\sum_{j=0}^{2}(V_{k_{i}} \otimes V_{l_{j}})|ij\rangle\langle ij|(V_{k_{i}}\otimes V_{l_{j}})^{\dagger}\\ =\frac{1}{6}\Big{(}|k_{0},l_{0}\rangle\langle k_{0},l_{0}|+|k_{0}, l_{1}+_{3}1\rangle\langle k_{0},l_{1}+_{3}1|+|k_{0},l_{2}+_{3}2\rangle\langle k_{0}, l_{1}+_{3}2|\\ +|k_{1}+_{2}1,l_{0}\rangle\langle k_{1}+_{2}1,l_{0}|+|k_{1}+_{2}1, l_{1}+_{3}1\rangle\langle k_{1}+_{2}1,l_{1}+_{3}1|\\ +|k_{1}+_{2}1,l_{2}+_{3}2\rangle\langle k_{1}+_{2}1,l_{2}+_{3}2| \Big{)}. \tag{72}\]  
_Hence_  
\[(U_{1},U_{2})((V_{k_{0}},V_{k_{1}}),(V_{l_{0}},V_{l_{1}},V_{l_{2}}))\\ =\frac{1}{6}\Big{(}(a_{k_{0},l_{0}},b_{k_{0},l_{0}})+(a_{k_{0},l_ {1}+_{3}1},b_{k_{0},l_{1}+_{3}1})+(a_{k_{0},l_{2}+_{3}2},b_{k_{0},l_{2}+_{3}2 })+(a_{k_{1}+_{2}1,l_{0}},b_{k_{1}+_{2}1,l_{0}})\\ +(a_{k_{1}+_{2}1,l_{1}+_{3}1},b_{k_{1}+_{2}1,l_{1}+_{3}1})+(a_{k_ {1}+_{2}1,l_{2}+_{3}2},b_{k_{1}+_{2}1,l_{2}+_{3}2})\Big{)}. \tag{73}\]  
_By substituting the payoffs of (22) we deduce that the payoff of player 1 that corresponds to the strategy profile \(((V_{0},V_{0}),(V_{0},V_{0},V_{0}))\) is_  
\[U_{1}(((V_{0},V_{0}),(V_{0},V_{0},V_{0})))=\frac{13}{6}. \tag{74}\]  
_If player 1 deviates to \((V_{1},V_{0})\), the resulting payoff is_  
\[U_{1}(((V_{1},V_{0}),(V_{0},V_{0},V_{0})))=\frac{7}{3}. \tag{75}\]  
_Thus, the strategy profile \(((V_{0},V_{0}),(V_{0},V_{0},V_{0}))\) is not a Nash equilibrium in the game \(\Gamma_{cMW}\)._  
The next proposition states that the equivalence between the correlated equilibrium and the Nash equilibrium can be generalized to every bimatrix game. **Proposition 3**. : _Let \((p_{ij})\) be a probability distribution over the outcomes of bimatrix game (1), and let \(\Gamma_{cMW}=\left(\sum_{ij}\sqrt{p_{ij}}\mathrm{e}^{\mathrm{i}\theta_{ij}}|ij \rangle,(B_{1},B_{2}),(U_{1},U_{2})\right)\) be associated with (1). Then \((p_{ij})\) is a correlated equilibrium in (1) if and only if the strategy profile \((\tau^{*},\tau^{*})=\left(\prod_{i=0}^{m-1}V_{0}^{(i)},\prod_{j=0}^{n-1}V_{0}^{ (j)}\right)\) is a Nash equilibrium in \(\Gamma_{cMW}\)._  
Proof. : Let us consider the expression  
\[U_{1}(\tau^{*},\tau^{*})\geq U_{1}((V_{k_{0}},V_{0},V_{0},\ldots,V_{0}),\tau^{ *}). \tag{76}\]  
According to (61),  
\[U_{1}\left(\left(V_{k_{0}},V_{0},V_{0},\ldots,V_{0}\right),\tau^{ *}\right)\\ =\mathrm{tr}\left(\sum_{j=0}^{n-1}(V_{k_{0}}\otimes\mathds{1})P_{ 0,j}|\Psi\rangle\langle\Psi|\left((V_{k_{0}}\otimes\mathds{1})P_{0,j}\right)^{ \dagger}M_{1}\right)+\mathrm{tr}\left(\sum_{i=1}^{m-1}\sum_{j=0}^{n-1}P_{ij}| \Psi\rangle\langle\Psi|P_{ij}^{\dagger}M_{1}\right). \tag{77}\]Hence,  
\[U_{1}\left(\tau^{*},\tau^{*}\right)-U_{1}\left(\left(V_{k_{0}},V_{0},V_{0}, \ldots,V_{0}\right),\tau^{*}\right)\\ =\text{tr}\left(\sum_{j=0}^{n-1}P_{0,j}|\Psi\rangle\langle\Psi|P_{0 j}^{\ddagger}M_{1}\right)-\text{tr}\left(\sum_{j=0}^{n-1}\left(V_{k_{0}}\otimes \mathds{1}\right)P_{0,j}|\Psi\rangle\langle\Psi|P_{0,j}^{\ddagger}\left(V_{k_ {0}}\otimes\mathds{1}\right)^{+}M_{1}\right). \tag{78}\]  
Note that  
\[\sum_{j=0}^{n-1}(V_{k_{0}}\otimes\mathds{1})P_{0,j}|\Psi\rangle\langle\Psi|P_{0,j}^{\ddagger}\left(V_{k_{0}}\otimes\mathds{1}\right)^{+}=\sum_{j=0}^{n-1}p_{0,j}|k_{0,j}\rangle\langle k_{0,j}|. \tag{79}\]  
Therefore, Inequality (76) is equivalent to \(\sum_{j=0}^{n-1}p_{0,j}(a_{0,j}-a_{k_{0},j})\) for \(k_{0}\in\{0,1,\ldots,n-1\}\). In general,  
\[(U_{1},U_{2})((V_{i},V_{(m-1)+=i},V_{(m-2)+=i}\ldots,V_{1+=i}),(V_{j},V_{(n-1) +=i},V_{(n-2)+=i}\ldots,V_{1+=j}))=(a_{ij},b_{ij}). \tag{71}\]0  
for \(i,k_{i}\in\{0,1,\ldots,m-1\}\). Analysis similar to the above one shows that  
\[(U_{1},U_{2})((V_{i},V_{(m-1)+=i},V_{(m-2)+=i}\ldots,V_{1+=i}),(V_{j},V_{(n-1) +=i},V_{(n-2)+=i}\ldots,V_{1+=j}))=(a_{ij},b_{ij}). \tag{71}\]1  
for \(j,l_{j}\in\{0,1,\ldots,n-1\}\). In this way we have shown that the correlated equilibrium in a bimatrix \(m\times n\) game is a necessary condition for the existence of the Nash equilibrium \((\tau^{*},\tau^{*})\) in \(\Gamma_{CMW}\).