## 4 Bayesian Game among Community Operators
### Bayesian Game Formulation  
In the above formulated game model, each community operator has the full information of other operators about DR resource's grade and knows their payoff function completely. However, it is difficult to conduct the above complete game in reality because much information of other opponents is unknown to community operator. Consequently, the above game theory with complete information cannot be applied in such case. For that reason, Bayesian game theory is employed to describe the game behavior among community operators with incomplete information. Bayesian game, which was first proposed by [PERSON], is an effective approach to formulate the game model with incomplete information via introducing Bayesian formula [31]. Different from the complete information game, except for three basic elements (i.e., players, strategies, and payoffs), Bayesian game has also considered other two elements, that is, the types of player and the probability distribution of the types. In this section, a detailed description about Bayesian game formulation will be given combining the proposed scenario. Community operators with different auxiliary equipment will have different characteristic of DR resource, which can be divided into different types. Assume that the types of community operators are divided into \(\mathcal{J}=[1,2\cdots J]\) kinds. Accordingly, suppose that the type space of community operator \(n\) is \(j_{n}\) and the actual type of operator \(n\) is \(j_{n}\). Furthermore, the type space combination for all community operators is \(J=J_{1}\times J_{2}\times\cdots\times J_{N}\) and the actual type combination is \(j=\left[j_{1},\cdots,j_{n},\cdots,j_{N}\right]\). Due to the lack of information about other operators' types, each community operator will estimate the types of these operators on basis of the probability distribution of the types. Accordingly, the following formula can be employed  
\[\Pr\left(j_{-n}|j_{n}\right)=\frac{\Pr\left(j_{-n},j_{n}\right)}{\Pr\left(j_{n }\right)}=\frac{\prod\limits_{n=1}^{N}\Pr\left(j_{n}\right)}{\Pr\left(j_{n} \right)} \tag{13}\]  
where \(j_{-n}=\left[j_{1},\cdots,j_{n-1},j_{n+1}\cdots,j_{N}\right]\) denotes the actual type combination of other \(N-1\) community operators; \(J_{-n}=J_{1}\times\cdots\times J_{n-1}\times J_{n+1}\cdots\times J_{N}\) denotes the type space combination of other \(N-1\) community operators; \(\Pr\left(j_{-n}|j_{n}\right)\) denotes the conditional probability for the type combination \(j_{-n}\) of other \(N-1\) community operators when operator n belongs to type \(j_{n}\); \(\Pr\left(j_{-n},j_{n}\right)=\Pr\left(j\right)\) denotes the joint probability for the type combination \(j=\left[j_{1},\cdots,j_{n},\cdots,j_{N}\right]\) of all community operators. The value of probability \(\Pr\left(j_{n}\right)\) can be obtained according to the proportion of community operators with type \(j\) in all operators. For example, assume that there are total \(M\) community operators in the grid, and in these operators, \(m\) operators belong to type \(j\). Then, if we choose any one from \(M\) operators as operator \(n\), the probability of operator \(n\) being type \(j\) can be considered as \(\Pr\left(j_{n}\right)=m/M\). Here, it needs to note that, although there are \(M\) community operators in the grid, not all operators will participate in day-ahead market of the same day. In this paper, it is assumed that \(N\) community operators will participate in the bidding market of the same day. According to the formula (13), the incomplete information game can be divided into multiple complete information games by ascertaining the actual types of community operators. The number of complete information games is determined by the number of elements in the type space combination \(J_{-n}\).