# On Correlated Equilibria in Marinatto-Weber Type Quantum Games
## 5 Bimatrix Representation of the MW Scheme
The first element of the pair is the action for player 1 of type \(I_{0}\), the second one is the action of type \(I_{1}\). Similarly, denote by \((V_{l_{0}},V_{l_{1}})\) the strategy of player 2. To find the expected payoff in the game (51)-(55) we use Formula (9). For example, if the strategy profile is \(((V_{0},V_{1}),(V_{1},V_{1}))\), the action profile \((V_{0},V_{1})\) is played in \(G_{I_{0},I_{0}}\) and \(G_{I_{0},I_{1}}\), and \((V_{1},V_{1})\) in games \(G_{I_{0},I_{0}}\) and \(G_{I_{1},II_{1}}\). Since games \(G_{I_{0},I_{0}}\), \(G_{I_{0},I_{1}}\), \(G_{I_{1},I_{1},I_{0}}\) and \(G_{I_{1},II_{1}}\) occur with the probability \(|\alpha_{00}|^{2},|\alpha_{01}|^{2},|\alpha_{10}|^{2}\), and \(|\alpha_{11}|^{2}\), respectively, the payoffs resulting from playing \(((V_{0},V_{1}),(V_{1},V_{1}))\) are  
\[u_{1(2)}((V_{k},V_{l}))=\text{tr}\,\Big{(}(V_{k}\otimes V_{l})|\Psi\rangle \langle\Psi|(V_{k}\otimes V_{l})^{\dagger}M_{1(2)}\Big{)} \tag{49}\]1 \[u_{1(2)}((V_{k},V_{l}))=\text{tr}\,\Big{(}(V_{k}\otimes V_{l})|\Psi\rangle \langle\Psi|(V_{k}\otimes V_{l})^{\dagger}M_{1(2)}\Big{)} \tag{49}\]2  
In general, the expected payoffs in the game (51)-(55) can be given by the following formula:  
\[u_{1(2)}((V_{k},V_{l}))=\text{tr}\,\Big{(}(V_{k}\otimes V_{l})|\Psi\rangle \langle\Psi|(V_{k}\otimes V_{l})^{\dagger}M_{1(2)}\Big{)} \tag{49}\]3 (57) \[u_{1(2)}((V_{k},V_{l}))=\text{tr}\,\Big{(}(V_{k}\otimes V_{l})|\Psi\rangle \langle\Psi|(V_{k}\otimes V_{l})^{\dagger}M_{1(2)}\Big{)} \tag{49}\]4  
where  
* \(|\Psi\rangle\) is defined as in (15);
* \(B_{i}\) is the set of strategies of player \(i\in\{1,2\}\): \[u_{1(2)}((V_{k},V_{l}))=\text{tr}\,\Big{(}(V_{k}\otimes V_{l})|\Psi\rangle \langle\Psi|(V_{k}\otimes V_{l})^{\dagger}M_{1(2)}\Big{)} \tag{49}\]5 (59) with typical elements \((V_{k_{0}},V_{k_{1}},\ldots,V_{k_{m-1}})\) and \((V_{l_{0}},V_{l_{1}},\ldots,V_{l_{n-1}})\), respectively;
* \(U_{i}\colon B_{1}\times B_{2}\to\mathds{R}\) is the payoff of player \(i\in\{1,2\}\) defined as the expected value of the measurements (18) on the final state \[u_{1(2)}((V_{k},V_{l}))=\text{tr}\,\Big{(}(V_{k}\otimes V_{l})|\Psi\rangle \langle\Psi|(V_{k}\otimes V_{l})^{\dagger}M_{1(2)}\Big{)} \tag{49}\]6 (60) i.e., \[u_{1(2)}((V_{k},V_{l}))=\text{tr}\,\Big{(}(V_{k}\otimes V_{l})|\Psi\rangle \langle\Psi|(V_{k}\otimes V_{l})^{\dagger}M_{1(2)}\Big{)} \tag{49}\]7 (61) We now verify that (61) coincides with (57) for \(m,n=2\). First note that \(|\Psi\rangle\langle\Psi|\) can be written as  
\[u_{1(2)}((V_{k},V_{l}))=\text{tr}\,\Big{(}(V_{k}\otimes V_{l})|\Psi\rangle \langle\Psi|(V_{k}\otimes V_{l})^{\dagger}M_{1(2)}\Big{)} \tag{49}\]8  
Let us determine each term of  
\[u_{1(2)}((V_{k},V_{l}))=\text{tr}\,\Big{(}(V_{k}\otimes V_{l})|\Psi\rangle \langle\Psi|(V_{k}\otimes V_{l})^{\dagger}M_{1(2)}\Big{)} \tag{49}\]9  
For  
* \(i=0,j=0\) \[|\Psi\rangle=\sum_{i,j=0}^{1}\alpha_{ij}|ij\rangle\ \ \text{and}\ \ M_{1(2)}=\sum_{i,j=0}^{1}a_{ij}(b_{ij})|ij\rangle \langle ij|. \tag{50}\]0 (64)
* \(i=0,j=1\) \[|\Psi\rangle=\sum_{i,j=0}^{1}\alpha_{ij}|ij\rangle\ \ \text{and}\ \ M_{1(2)}=\sum_{i,j=0}^{1}a_{ij}(b_{ij})|ij\rangle \langle ij|. \tag{50}\]1 (65)
* \(i=1,j=0\) \[|\Psi\rangle=\sum_{i,j=0}^{1}\alpha_{ij}|ij\rangle\ \ \text{and}\ \ M_{1(2)}=\sum_{i,j=0}^{1}a_{ij}(b_{ij})|ij\rangle \langle ij|. \tag{50}\]2 (66)
* \(i=1,j=1\) \[|\Psi\rangle=\sum_{i,j=0}^{1}\alpha_{ij}|ij\rangle\ \ \text{and}\ \ M_{1(2)}=\sum_{i,j=0}^{1}a_{ij}(b_{ij})|ij\rangle \langle ij|. \tag{50}\]3 (67) Measuring \(\rho_{\rm f}\) with respect to \(M_{1}\) and \(M_{2}\) given by (50) yields the same expected payoff as (57). It is commonly required in quantum game theory that a given scheme must coincide with the classical game under specific settings. The classical game in the MW game is obtained by taking \(|00\ldots 0\rangle\) in (15). Another way of deriving the classical game can be carried out by a restriction of the players' strategies.