# On Correlated Equilibria in Marinatto-Weber Type Quantum Games
## 3 The Generalized Marinatto-Weber Scheme  
In this section, we recall the Marinatto-Weber (MW) scheme for bimatrix games and then we present the generalized model that was introduced in [4].  
The (MW) scheme was originally designed for a \(2\times 2\) game:  
\[\begin{pmatrix}(a_{00},b_{00})&(a_{01},b_{01})\\ (a_{10},b_{10})&(a_{11},b_{11})\end{pmatrix},a_{ij}\in\mathbb{R}. \tag{10}\]  
According to the model, each of the two players acts with the identity matrix \(I\) of size \(2\) and the Pauli matrix \(\sigma_{x}\) on his own qubit of some fixed two-qubit state \(|\Psi\rangle\)  
\[|\Psi_{\rm f}\rangle\langle\Psi_{\rm f}|=pqI\otimes I|\Psi\rangle \langle\Psi|I\otimes I+p(1-q)I\otimes\sigma_{x}|\Psi\rangle\langle\Psi|I \otimes\sigma_{x}\\ +(1-p)q\sigma_{x}\otimes I|\Psi\rangle\langle\Psi|\sigma_{x} \otimes I+(1-p)(1-q)\sigma_{x}\otimes\sigma_{x}|\Psi\rangle\langle\Psi|\sigma _{x}\otimes\sigma_{x}, \tag{11}\]  
where \(p\) and \(q\) are the probabilities of choosing the identity \(I\) by player \(1\) and player \(2\), respectively. Player \(i\)'s payoff \(u_{i}\) depends on \(p\) and \(q\), and through the measurement operators  
\[M_{1}=\sum_{i=0}^{1}\sum_{j=0}^{1}a_{ij},\quad M_{2}=\sum_{i=0}^{1}\sum_{j=0} ^{1}b_{ij}, \tag{12}\]  
it is given by the following formula:  
\[u_{i}(p,q)=\operatorname{tr}(|\Psi_{\rm f}\rangle\langle\Psi_{\rm f}|M_{i}). \tag{13}\]  
The MW scheme imples the classical \(2\times 2\) game by putting \(|\Psi\rangle=|ij\rangle,i,j\in\{0,1\}\). The strategy sets in the generalization of the MW scheme for \(m\times n\) bimatrix games are sets of permutation matrices. It enables one to obtain a classical \(m\times n\) bimatrix game when \(|\Psi\rangle=|ij\rangle,i\in\{0,1,\ldots,m-1\}\) and \(j\in\{0,1,\ldots,n-1\}\).  
Let us consider \(m\times n\) bimatrix game (1). The generalized Marinatto-Weber (gMW) scheme is defined by a triple  
\[\Gamma_{gMW}=(|\Psi\rangle,(S_{1},S_{2}),(u_{1},u_{2})), \tag{14}\]  
where  
* \(|\Psi\rangle\) is a joint state of \(m\)-dimensional and \(n\)-dimensional quantum systems: \[|\Psi\rangle=\sum_{i=0}^{m-1}\sum_{j=0}^{n-1}\alpha_{ij}|ij\rangle,\quad \alpha_{ij}\in\mathbb{C};\] (15)
* \(S_{i}\) is a set of strategies of player \(i\in\{1,2\}\): \[S_{1}=\{V_{k}\colon k=0,\ldots,m-1\},\quad S_{2}=\{V_{k}\colon k=0,\ldots,n-1\};\] (16) and \(V_{k}\) for \(k=0,1,\ldots,r-1\) acts on states of the computational basis \(\{|0\rangle,|1\rangle,\ldots,|r-1\rangle\}\) as follows: \[V_{k}|i\rangle=|i+_{r}k\rangle,\;k=0,1,\ldots,r-1,\] (17) where the symbol \(+_{r}\) denotes addition modulo \(r\);
* \(u_{i}\colon S_{1}\times S_{2}\to\mathbb{R}\) is the payoff of player \(i\in\{1,2\}\) defined as the expected value of the measurement \(M_{i}\), \[M_{1}=\sum_{i=0}^{m-1}\sum_{j=0}^{n-1}a_{ij}|ij\rangle\langle ij|,\quad M_{2}= \sum_{i=0}^{m-1}\sum_{j=0}^{n-1}b_{ij}|ij\rangle\langle ij|\] (18)on the final state  
\[|\Psi_{\rm f}\rangle=\left(V_{k}\otimes V_{l}\right)|\Psi\rangle; \tag{19}\]  
i.e.,  
\[u_{i}\left(V_{k},V_{l}\right)={\rm tr}\left(|\Psi_{\rm f}\rangle\langle\Psi_{ \rm f}|M_{i}\right) \tag{20}\]  
for \(k=0,1,\ldots,m-1\) and \(l=0,1,\ldots,n-1\).  
This scheme can certainly reproduce a classical \(m\times n\) bimatrix game. If \(|\Psi\rangle=|00\rangle\), the form of the final state \(|\Psi_{\rm f}\rangle\) is determined by each pair of \(\left(V_{k_{i}},V_{l_{j}}\right)\) and is given in the following matrix:  
\[\begin{array}{cccc}&V_{0}&V_{1}&\cdots&V_{n-1}\\ V_{0}&\left(\begin{array}{cccc}|00\rangle&|01\rangle&\cdots&|0,n-1\rangle\\ |10\rangle&|11\rangle&\cdots&|1,n-1\rangle\\ \vdots&\vdots&\ddots&\vdots\\ |m-1,0\rangle&|m-1,1\rangle&\vdots&|m-1,n-1\rangle\end{array}\right)\end{array} \tag{21}\]  
Formula (20) applied to each entry of (21) results in the classical game (1). The same bimatrix up to the order of players' strategies could also be obtained if one uses any \(|\Psi\rangle=|ij\rangle\). Note also that the original MW scheme for a \(2\times 2\) game introduced in [3] can be obtained by taking \(m=n=2\).