*3.2. Non-Cooperative Game Formulation*

The bidding price in the market is determined by the total bidding amount of all communities in group *N*. Therefore, the economic compensation of community *n* is determined not only by its own bidding strategy, but also the bidding strategies of other communities in group *N*. That is, community *n* has to take other communities' strategies into consideration when it makes its bidding strategy. Hence, the bidding strategy problem for residential communities belongs to the typical non-cooperative game. Based on the objective Equation (9), the non-cooperative game can be formulated as follows [31]:

*Energies* **2019**, *12*, 3727


$$R\_n^h \Big(L\_n^h, L\_{-n}^h\Big) = \sum\_{t=1}^T \left[ \left(a\_h^t - c\_h^t\right)L\_n^{h,t} + \left(b\_h^t - d\_h^t\right) + a\_h^t L\_{-n}^{h,t} \right] L\_n^{h,t} \tag{10}$$

where *L<sup>h</sup>* <sup>−</sup>*<sup>n</sup>* <sup>=</sup> *Lh* <sup>1</sup>, ··· , *<sup>L</sup><sup>h</sup> <sup>n</sup>*−1, *<sup>L</sup><sup>h</sup> <sup>n</sup>*+1, ··· , *<sup>L</sup><sup>h</sup> N* represents the bidding strategy set of other communities except *n* in group *N*; *Lh*,*<sup>t</sup>* <sup>−</sup>*<sup>n</sup>* represents the total bidding amount of *N* − 1 communities with:

$$L\_{-n}^{h,t} = \sum\_{i=1, i \neq n}^{N} L\_i^{h,t} \tag{11}$$

All residential communities will constantly update their own strategy for the higher economic compensation based on payoff Equation (10). Once all communities in group *N* obtain their own maximal profit, no one will change the strategy. Such an equilibrium state is called the Nash equilibrium, which can be expressed as:

$$\mathbb{R}\_n^h(\mathbf{L}\_n^{h\*}, \mathbf{L}\_{-n}^{h\*}) \ge \mathbb{R}\_n^h(\mathbf{L}\_n^h, \mathbf{L}\_{-n}^{h\*}) \tag{12}$$

where  *Lh n* ∗ , *Lh*<sup>∗</sup> −*n* represents the Nash equilibrium of a formulated non-cooperative game.
