## 4 Non-Cooperative Game of Customers
### Formulation of Non-Cooperative Game  
Game theory is the study of conflicts and cooperation among rational decision makers [48], and has been used to describe customers' behaviors in DR in recent studies [36]. In the proposed DR scheme, each customer rationally and selfishly chooses a DR strategy to maximize their own welfare, while the total DR rebate is limited. As a result, we can formulate a non-cooperative game and find its solution, that is, the Nash equilibrium. It should be noted that the game process is not controlled by LSE and we formulate this game to study how the customers will respond to our proposed DR scheme. Considering that the theory about non-cooperative game has been introduced in many studies, this paper just uses this theory but does not describe the basic theory again. The non-cooperative game for customers can be formulated as follows:  
* _Players_: customer \(i\in\mathcal{I}\);
* _Strategy_: the ideal rebate value ratio \(k_{i}(t)\) for customer \(i\);
* _Utility_: the welfare function of customer given in (10) and limited to (11). Then, we need to study whether there exist Nash equilibriums in the proposed scheme. A study (36) gave a lemma about the existence of Nash equilibriums:  
If the following conditions are satisfied, there exist Nash equilibriums in the game:* The player set is finite. * The strategy sets are closed, bounded, and convex. * The utility functions are continuous and quasi-concave in the strategy space. First, a customer who participates in DR is finite in the proposed DR scheme. Additionally, the strategy for each player is a variable that belongs to a linear constant set according to (11), so the strategy set is closed, bounded and convex. For the welfare function, there is:  
\[ \begin{array}{l}\frac{\partial w_{i}^{DR}(t)}{\partial k_{i}(t)}=\frac{ \partial u_{i}(t)}{\partial k_{i}(t)}-\frac{\partial\mathcal{D}_{(t)}(t)}{ \partial k_{i}(t)}\pi_{r}+\frac{\partial k_{i}^{*}(t)}{\partial k_{i}(t)}\pi_ {R}\\ =\frac{\partial D_{i}(t)}{\partial k_{i}(t)}[b_{i}(t)-a_{i}(t)D_{i}(t)-\pi_{r} ]+\frac{\partial k_{i}^{*}(t)}{\partial k_{i}(t)}\pi_{R}\\ =\left\{-a_{i}(t)P_{\Delta CDL}^{2}(t)k_{i}^{*}(t)+[b_{i}(t)-a_{i}(t)D_{i}(t- 1)-\pi_{r}]P_{\Delta CDL}(t)+\pi_{R}\right\}\frac{\partial k_{i}^{*}(t)}{ \partial k_{i}(t)}\end{array}  \tag{12}\]  
Because customers will estimate \(K(t)\) before making a decision, \(\frac{\partial k_{i}^{*}(t)}{\partial k_{i}(t)}\) can be thought as a constant when customers choose ideal \(k_{i}(t)\). To express it easily, combined with (11), (12) can also be written as an abstract way:  
\[\frac{\partial w_{i}^{DR}(t)}{\partial k_{i}(t)}=M(t)k_{i}(t)+N\left(t\right) \qquad\quad k_{i}(t)\in[\underline{k}_{i}(t),\overline{k}_{i}(t)] \tag{13}\]  
where \(M(t)\), \(N(t)\), \(k_{i}(t)\) and \(\overline{k}_{i}(t)\) are all constant. Because  
\[M(t)=\left\{ \begin{array}{cc}-a_{i}(t)P_{\Delta CDL}^{2}(t)&K(t)\leq 1\\ -\frac{a_{i}(t)P_{\Delta CDL}^{2}(t)}{K(t)}&K(t)>1\end{array} \right. \tag{14}\]  
it is easy to know \(M(t)\) < 0, and  
\[\frac{\partial^{2}w_{i}^{DR}(t)}{\partial k_{i}^{2}(t)}=M(t)<0 \tag{15}\]  
We can find that the utility in this game is a concave function. Let \(\frac{\partial w_{i}^{DR}(t)}{\partial k_{i}(t)}=0\), we can get the pole point:  
\[\hat{k}(t)=-N(t)/M(t) \tag{16}\]  
The first-order derivative of \(w_{i}^{DR}(t)\) maybe have 3 cases, including:  
* If \(\hat{k}(t)>\overline{k}_{i}(t)\), \[\frac{\partial w_{i}^{DR}(t)}{\partial k_{i}(t)}>0\] (17) which means that the utility in this game is an increasing function. * If \(\hat{k}(t)<\underline{k}_{i}(t)\), \[\frac{\partial w_{i}^{DR}(t)}{\partial k_{i}(t)}<0\] (18) which means that the utility in this game is a decreasing function. * If \(\underline{k}_{i}(t)\leq\hat{k}(t)\leq\overline{k}_{i}(t)\) \[\frac{\partial w_{i}^{DR}(t)}{\partial k_{i}(t)}\left\{ \begin{array}{cc}>0& \underline{k}_{i}(t)\leq k_{i}(t)<\hat{k}\\ =0&k_{i}(t)=\hat{k}\\ <0&\hat{k}<k_{i}(t)<\overline{k}_{i}(t)\end{array}