*3.3. Nash Equilibrium*

According to the above definition of the Nash equilibrium, this section focuses on the mathematical proof of the existence and uniqueness of the Nash equilibrium.

**Lemma 1.** *For each residential community n* ∈ *N, the function R<sup>h</sup> n Lh <sup>n</sup>*, *L<sup>h</sup>* −*n is continuously di*ff*erentiable in L<sup>h</sup> n. For the fixed value of Lh*,*<sup>t</sup>* −*n, the function Rh n Lh <sup>n</sup>*, *L<sup>h</sup>* −*n is concave about L<sup>h</sup> n.*

**Proof.** It is obvious that, *Rh n Lh <sup>n</sup>*, *L<sup>h</sup>* −*n* is continuously differentiable in *L<sup>h</sup> <sup>n</sup>*. As for the concavity of *Rh n Lh <sup>n</sup>*, *L<sup>h</sup>* −*n* , we just need to prove the Hessian matrix of *Rn*(*Ln*, *L*−*n*) is negative definite. The Hessian matrix of *R<sup>h</sup> n Lh <sup>n</sup>*, *L<sup>h</sup>* −*n* is:

$$\nabla\_{L\_n^h}^2 R\_n^h \mathbf{(} L\_n^h, L\_{-n}^h) = \text{diag} \left[ 2 \left( a\_h^t - c\_h^t \right) \right]\_{t=1}^T \tag{13}$$

Due to the negative value of 2 *at <sup>h</sup>* <sup>−</sup> *<sup>c</sup><sup>t</sup> h* , Equation (13) is a diagonal matrix with all diagonal elements being negative. Hence, the Hessian matrix of *R<sup>h</sup> n Lh <sup>n</sup>*, *L<sup>h</sup>* −*n* is negative definite. Consequently, function *Rh n Lh <sup>n</sup>*, *L<sup>h</sup>* −*n* is concave about *L<sup>h</sup> <sup>n</sup>*. -

**Definition 1.** *The variational inequality (VI), denoted by VI*(L, *F*)*, is to find a vector x*<sup>∗</sup> ∈ L *such that:*

$$\mathbb{P}\left(\mathbf{x} - \mathbf{x}^\*\right)^T \mathbf{F}(\mathbf{x}^\*) \le 0 \; \forall \mathbf{x} \in \mathcal{L} \tag{14}$$

According to Lemma 1 and Definition 1, we can obtain the following lemma:

**Lemma 2.** *The optimization problem of the non-cooperative model (10) is equivalent to the VI problem VI*(L, *F*) *where:*

$$F(L^h) = \left[ F\_n(L\_{n'}^h L\_{-n}^h) \right]\_{n=1}^N \tag{15}$$

*where <sup>L</sup><sup>h</sup>* <sup>=</sup> *Lh <sup>n</sup>*, *L<sup>h</sup>* −*n and F<sup>h</sup> n Lh <sup>n</sup>*, *L<sup>h</sup>* −*n are expressed as follows:*

$$\nabla\_n^h \mathcal{L}\_n^h \mathcal{L}\_{-n}^h = \nabla\_{L\_n^h} \mathcal{R}\_n^h \mathcal{L}\_{n'}^h \mathcal{L}\_{-n}^h \tag{16}$$

**Proof.** The proof can be found in [32]. -

Based on Lemma 2, the following proposition can be obtained.

**Proposition 1.** *In the formulated non-cooperative model (10), its Nash equilibrium is unique.*

**Proof.** According to Lemma 2, VI problem VI(L, *F*) has the same solution with the solution of Equation (10). That is, we just need to prove the uniqueness of VI(L, *F*)'s solution, then **Proposition 1** can be proved. According to [33], we know that VI(L, *F*) will have a unique solution when *F Lh* is strictly monotone about feasible set L.

To prove the strict monotone of *F Lh* is to prove:

$$\sum\_{t=1}^{T} \sum\_{n=1}^{N} \left[ \left( \mathbf{x}\_{n}^{h,t} - y\_{n}^{h,t} \right) \left( \nabla\_{\mathbf{x}\_{n}^{h,t}} R\_{n}^{h} \left( \mathbf{x}^{h} \right) - \nabla\_{y\_{n}^{h,t}} R\_{n}^{h} \left( y^{h} \right) \right) \right] > 0 \tag{17}$$

where *<sup>x</sup><sup>h</sup>* <sup>=</sup> *xh n N <sup>n</sup>*=<sup>1</sup> ∈ L*h*,*y<sup>h</sup>* <sup>=</sup> *yh n N <sup>n</sup>*=<sup>1</sup> ∈ L*h*.

Let *l <sup>h</sup>*,*<sup>t</sup>* = *xh*,*<sup>t</sup>* <sup>1</sup> , *<sup>x</sup>h*,*<sup>t</sup>* <sup>2</sup> , ··· , *xh*,*<sup>t</sup> N* and *j <sup>h</sup>*,*<sup>t</sup>* = *yh*,*<sup>t</sup>* <sup>1</sup> , *<sup>y</sup>h*,*<sup>t</sup>* <sup>2</sup> , ··· , *<sup>y</sup>h*,*<sup>t</sup> N* , then Equation (17) can be rewritten as

follows:

$$\sum\_{t=1}^{T} \left[ \left( I^{h,t} - j^{h,t} \right) \left( \nabla\_{I^{h,t}} R\_{nl}^{h,t} \left( I^{h,t} \right) - \nabla\_{j^{h,t}} R\_{nl}^{h,t} \left( j^{h,t} \right) \right) \right] > 0 \tag{18}$$

where:

$$R\_n^{h,t}(I^{h,t}) = p\_h^t \mathbf{x}\_n^{h,t} - \mathbf{u}\_n^{h,t}$$

and:

$$\nabla\_{I^{h,t}} R^{h,t}\_{\boldsymbol{n}} \left( I^{h,t} \right) = \left[ \nabla\_{\boldsymbol{x}^{h,t}\_1} R^{h,t}\_{\boldsymbol{n}} \left( I^{h,t} \right), \nabla\_{\boldsymbol{x}^{h,t}\_2} R^{h,t}\_{\boldsymbol{n}} \left( I^{h,t} \right), \dots, \nabla\_{\boldsymbol{x}^{h,t}\_N} R^{h,t}\_{\boldsymbol{n}} \left( I^{h,t} \right) \right]^T$$

If the following condition is satisfied, then Equation (18) will hold:

$$\left(\mathbf{l}^{h,t} - \mathbf{j}^{h,t}\right)\left(\mathbf{g}\_{h,t}\left(\mathbf{l}^{h,t}\right) - \mathbf{g}\_{h,t}\left(\mathbf{j}^{h,t}\right)\right) > 0 \; \forall t \in T \tag{19}$$

where *gh*,*<sup>t</sup> l h*,*t* = ∇*<sup>l</sup> h*,*tRh*,*<sup>t</sup> n l h*,*t* .

If the Jacobian matrix of *gh*,*<sup>t</sup> l h*,*t* is negative definite, then Equation (19) will be satisfied. Assume that *Gh*,*<sup>t</sup> l h*,*t* = ∇*<sup>l</sup> <sup>h</sup>*,*tgh*,*<sup>t</sup> l h*,*t* , then:

$$\mathbf{G}\_{h,t}\left(\mathbf{l}^{h,t}\right) + \mathbf{G}\_{h,t}\left(\mathbf{l}^{h,t}\right)^T = \mathbf{2}\left(a\_h^t - c\_h^t\right)\left(\mathbf{1}\mathbf{1}^T + \mathbf{I}\right) \tag{20}$$

where **I**is a unit matrix and **1** is a *<sup>N</sup>* × 1 matrix where all elements are 1. Since characteristic values of **11***<sup>T</sup>* + **I** are 1 and *N* + 1, the characteristic values of *Gh*,*<sup>t</sup> l h*,*t* + *Gh*,*<sup>t</sup> l h*,*t <sup>T</sup>* are 2 *at <sup>h</sup>* <sup>−</sup> *<sup>c</sup><sup>t</sup> h* and 2(*N* + 1) *at <sup>h</sup>* <sup>−</sup> *<sup>c</sup><sup>t</sup> h* . Consequently, *Gh*,*<sup>t</sup> l h*,*t* + *Gh*,*<sup>t</sup> l h*,*t <sup>T</sup>* is negative definite. That is, *<sup>F</sup>*(*L*) is a strictly monotone function. Therefore, the Nash equilibrium of the formulated non-cooperative game is unique. -

Based on the above analysis, the Nash equilibrium for the game can be solved. Equation (9) is to search the maximal value of the comprehensive income in all dispatching slots *T*. But, when the comprehensive income achieves the maximal value in each time slot *t* ∈ *T*, then the comprehensive income in all dispatching slots will also achieve the maximal value. That is, Equation (9) can be translated into the following optimization problems:

$$\begin{cases} \text{maximize} \boldsymbol{w}\_{n}^{h,t} = \left[ \left( \boldsymbol{a}\_{h}^{t} - \boldsymbol{c}\_{h}^{t} \right) \boldsymbol{L}\_{n}^{h,t} + \left( \boldsymbol{b}\_{h}^{t} - \boldsymbol{d}\_{h}^{t} \right) + \boldsymbol{a}\_{h}^{t} \boldsymbol{L}\_{-n}^{h,t} \right] \boldsymbol{L}\_{n}^{h,t} \\ \text{ s.t. } \boldsymbol{L}\_{n}^{h,\text{min}} \le \boldsymbol{L}\_{n}^{h,t} \le \boldsymbol{L}\_{n}^{h,\text{max}} \end{cases} \tag{21}$$

Furthermore, when the bidding strategies of other communities *Lh*,*<sup>t</sup>* <sup>−</sup>*<sup>n</sup>* are regarded as fixed values, then the optimal bidding strategy of community *n* can be expressed as follows:

$$\varphi\_n(\mathbf{L}\_{-n}^{h,t}) = \underset{\mathbf{L}\_n^{h,t}}{\text{argmax}} \; w\_n^{h,t}(\mathbf{L}\_n^{h,t}, \mathbf{L}\_{-n}^{h,t}) \; \forall n \in \mathbb{N} \tag{22}$$

where ϕ*<sup>n</sup> Lh*,*<sup>t</sup>* −*n* represents the optimal bidding strategy of community *n* corresponding to strategies *Lh*,*<sup>t</sup>* <sup>−</sup>*n*; *<sup>L</sup>h*,*<sup>t</sup>* <sup>−</sup>*<sup>n</sup>* <sup>=</sup> *Lh*,*<sup>t</sup>* <sup>1</sup> , ··· , *<sup>L</sup>h*,*<sup>t</sup> <sup>n</sup>*−1, *<sup>L</sup>h*,*<sup>t</sup> <sup>n</sup>*+1, ··· , *<sup>L</sup>h*,*<sup>t</sup> N* represents the bidding strategies of other communities in time slot *t*.

#### **4. Evolution Analysis between Groups** *N* **and** *M*

According to the above analysis, the economic compensation of each residential community in group *N* is correlated not only with the bidding price parameters, but also with the total bidding amount in the market. Therefore, a community's economic compensation will be influenced when the population of group *N* changes. In the initial period of the DR project, communities will obtain high economic compensation for participating in DR. Hence, the neighboring communities may be infected to participate in DR for the high economic compensation. However, when the population of group *N* has consistent growth, the economic compensation of each community will be reduced gradually. Consequently, those residential communities who care more about energy consumption satisfaction will not choose to participate in DR anymore. Finally, the population of group *N* and group *M* will reach a dynamic balance. In this section, communities' transition probability model between group *N* and group *M* is formulated, and then the group population is analyzed.
