# On Correlated Equilibria in Marinatto-Weber Type Quantum Games
## 2 Preliminaries for Game Theory  
In this section, we review relevant notion from classical game theory that is needed to follow our work. A reader who is not familiar with that topic is encouraged to see, for example [16].  
The basic model of games studied in game theory is a game in strategic form.  
Definition 1: [16] A game in strategic form (or in normal form) is an ordered triple \((N,(S_{i})_{i\in N},(u_{i})_{i\in N})\), in which  
* \(N=\{1,2,\ldots,r\}\) is a finite set of players;
* \(S_{  
In a Nash equilibrium, the players make their choices independently of one another. A more general solution concept is a correlated equilibrium. It covers situations in which the players can choose their strategies on the basis of the recommended strategy profiles.  
Definition 4: [17] A correlated equilibrium in the bimatrix game (1) is a probability distribution \(P=(p_{ij})\) satisfying  
\[\sum_{j=0}^{n-1}(a_{ij}-a_{kj})p_{ij}\geq 0\quad\text{for all}\quad i,k=0, \ldots,m-1, \tag{5}\] \[\sum_{i=0}^{m-1}(b_{ij}-b_{il})p_{ij}\geq 0\quad\text{for all}\quad j,l=0, \ldots,n-1. \tag{6}\]  
Games with incomplete information concern problems in which players may not be informed about certain elements of the game, for example, about payoff functions of other players.  
Definition 5: [16] A Harasanyi game with incomplete information is a quintuple  
\[\left(N,\left(T_{i}\right)_{i\in N},p,S,\left(s_{t}\right)_{t\in\times i\in N }T_{i}\right), \tag{7}\]  
where:  
* \(N\) is a finite set of players.
* \(T_{i}\) is a finite set of types for player \(i\), for each \(i\in N\). The set of type vectors is denoted by \(T=\times_{i\in N}T_{i}\).
* \(p\in\Delta(T)\) is a probability distribution over the set of type vectors that satisfies \(p(t_{i})=\sum_{t_{-i}\in T_{-i}}p(t_{i},t_{-i})>0\) for every player \(i\in N\) and every type \(t_{i}\in T_{i}\).
* \(S\) is a set of states of nature. Every state of nature \(s\in S\) is a triple \(s=(N,\left(A_{i}\right)_{i\in N},\left(u_{i}\right)_{i\in N})\), where \(A_{i}\) is a nonempty set of actions of player \(i\) and \(u_{i}\colon\times_{i\in N}A_{i}\to\mathbb{R}\) is the payoff function of player \(i\).
* \(s_{t}=(N,\left(A_{i}(t_{i})\right)_{i\in N},\left(u_{i}(t)\right)_{i\in N})\in S\) is the state game for the type vector \(t\), for every \(t\in T\). Thus, player \(i\)'s action set in the state game \(s_{t}\) depends on his type \(t_{i}\) only, and is independent of the types of the other players.  
A Harasanyi game with incomplete information proceeds in the following way [16]:  
* A chance mover chooses a type vector \(t=(t_{1},t_{2},\ldots,t_{n})\in T\) according to the probability distribution \(p\).
* Each player \(i\) knows his type \(t_{i}\), but does not know the types \(t_{-i}=(t_{j})_{j\
eq i}\) of the other players.
* Each player \(i\) chooses an action \(a_{i}\in A_{i}(t_{i})\).
* Each player \(i\) obtains the payoff \(u_{i}(t,a)\), where \(a=(a_{1},a_{2},\ldots,a_{n})\) is the profile of actions chosen by all the players.  
Definition 6: [16] A pure strategy of player \(i\) in a game with incomplete information is a function \(s_{i}\colon T_{i}\to\bigcup_{t_{i}\in T_{i}}A_{i}(t_{i})\) that satisfies  
\[s_{i}(t_{i})\in A_{i}(t_{i}) \tag{8}\]  
for each \(t_{i}\in T_{i}\).  
Let \(s=(s_{i})_{i\in N}\) be a strategy profile. The expected payoff in a game with incomplete information is  
\[U_{i}(s)=\sum_{t\in T}p(t)u_{i}(t,s), \tag{9}\]  
where \(u_{i}(t,s)\) is a payoff resulting from playing the strategy profile \(s\) in the game associated with the type vector \(t\).