# On Correlated Equilibria in Marinatto-Weber Type Quantum Games
## 5 Bimatrix Representation of the MW Scheme  
The MW scheme can be described in terms of classical game theory. The model  
\[\left(\sum_{i=0}^{m-1}\sum_{j=0}^{n-1}\alpha_{ij}|ij\rangle,(S_{1},S_{2}),(u_{ 1},u_{2})\right) \tag{45}\]  
can be viewed as a family of bimatrix games in which the rows and columns of (1) are permuted according to a probability distribution \(P=\big{(}|\alpha_{ij}|^{2}\big{)}\). In the case of the MW approach to a \(2\times 2\) game, the players play one of the following games  
\[ \begin{array}{cccc}V_{0}&V_{1}&V_{0}&V_{1}\\ G_{l_{0},I_{0}}\colon\begin{array}{c}V_{0}\\ V_{1}\end{array}\left(\begin{array}{c}(a_{00},b_{00})\\ (a_{10},b_{10})\\ (a_{11},b_{11})\end{array}\left(\begin{array}{c}(a_{01},b_{01})\\ (a_{11},b_{11})\end{array}\left(\begin{array}{c}(a_{00},b_{00})\\ (a_{11},b_{11})\end{array}\left(\begin{array}{c}(a_{10},b_{10})\\ (a_{11},b_{11})\end{array}\left(\begin{array}{c}(a_{10},b_{10})\\ (a_{10},b_{10})\end{array}\right),\right.\\ V_{0}&V_{1}&V_{0}&V_{1}\end{array} \left( \begin{array}{c}(a_{10},b_{10})\\ (a_{11},b_{11})\end{array} \left( \begin{array}{c}(a_{10},b_{10})\\ (a_{10},b_{10})\end{array} \right). \tag{46}\]  
These games occur with probabilities  
\[ \begin{array}{c|cccc}G&G_{k,I_{0}}&G_{l_{0},I_{1}}&G_{l_{1},I_{0}}&G_{l_{1}, I_{1},I_{1}}&G_{l_{1},I_{1}}\\ \hline p(G)&|\alpha_{00}|^{2}&|\alpha_{01}|^{2}&|\alpha_{10}|^{2}&|\alpha_{11}| ^{2}.\end{array}  \tag{47}\]  
One can check that the expected payoffs in the game defined by (46) and (47) are  
\[u_{1(2)}((V_{k},V_{l}))=a_{k,l}(b_{k,l})|\alpha_{00}|^{2}+a_{k,l+ 21}(b_{k,l+21})|\alpha_{01}|^{2}\\ +a_{k+21,l}(b_{k+21,l})|\alpha_{10}|^{2}+a_{k+21,l+21}(b_{k+21,l+21} )|\alpha_{11}|^{2}. \tag{48}\]  
The same expected payoffs are obtained if we use Formula (20) for the MW approach to \(2\times 2\) game; i.e.,  
\[u_{1(2)}((V_{k},V_{l}))=\text{tr}\,\Big{(}(V_{k}\otimes V_{l})|\Psi\rangle \langle\Psi|(V_{k}\otimes V_{l})^{\dagger}M_{1(2)}\Big{)} \tag{49}\]with  
\[|\Psi\rangle=\sum_{i,j=0}^{1}\alpha_{ij}|ij\rangle\ \ \text{and}\ \ M_{1(2)}=\sum_{i,j=0}^{1}a_{ij}(b_{ij})|ij\rangle \langle ij|. \tag{50}\]  
In what follows, we modify the MW scheme. The idea behind our scheme can also be explained with the use of the game given by (46) and (47). Let us assume in addition to the fact that probability distribution (47) is a common knowledge among the players that player 1 is informed that either the set of games \(\{G_{I_{0},I_{0}},G_{I_{0},I_{1}}\}\) or the set of games \(\{G_{I_{1},I_{0}},G_{I_{1},I_{1}}\}\) has to be taken into consideration. Player 2 knows that either \(\{G_{I_{0},I_{0}},G_{I_{1},I_{0}}\}\) or \(\{G_{I_{0},I_{1}},G_{I_{1},I_{1}}\}\) is actually played. To be more precise, we assume that the players play a game with incomplete information (see Definition (5)) defined as follows:  
* The sets of types are \[T_{1}=\{I_{0},I_{1}\},\quad T_{2}=\{I_{0},II_{1}\}.\] (51)
* The probability distribution \(p\) over \(T=((I_{0},I_{0}),(I_{0},II_{1}),(I_{1},I_{0}),(I_{1},II_{1}))\) \[p=(|\alpha_{00}|^{2},|\alpha_{01}|^{2},|\alpha_{10}|^{2},|\alpha_{11}|^{2})\] (52) such that \[p(I_{0})=p(I_{0},II_{0})+p(I_{0},II_{1})=|\alpha_{00}|^{2}+| \alpha_{01}|^{2},\] (53) \[p(I_{1})=p(I_{1},II_{0})+p(I_{1},II_{1})=|\alpha_{10}|^{2}+| \alpha_{11}|^{2},\] \[p(II_{0})=p(I_{0},II_{0})+p(I_{1},II_{0})=|\alpha_{00}|^{2}+| \alpha_{10}|^{2},\] \[p(II_{1})=p(I_{0},II_{1})+p(I_{1},II_{1})=|\alpha_{01}|^{2}+| \alpha_{11}|^{2}.\]
* States of nature are: \[G_{I_{0},I_{0}},G_{I_{0},I_{1}},G_{I_{1},I_{0}},G_{I_{1},I_{1}}\] (54) Now, from Definition 6, the players' strategies \(\tau_{1}\) and \(\tau_{2}\) are the functions \[u_{1(2)}((V_{k},V_{l}))=\text{tr}\,\Big{(}(V_{k}\otimes V_{l})|\Psi\rangle \langle\Psi|(V_{k}\otimes V_{l})^{\dagger}M_{1(2)}\Big{)} \tag{49}\]0 (55)  
Strategy \(s_{i}\) specifies an action \(V_{0}\) or \(V_{1}\) for player \(i\) depending on his type. Denote by \((V_{k_{0}},V_{k_{1}})\) the strategy of player 1.