The NE of the game is the energy consumption scheduling
vector xn that minimizes the electricity payment of each user
n ∈ N. In the following, we formulate the problem of finding
the NE as a binary linear programming problem. To do so, we
first define a new auxiliary binary decision vector yn which is
the same size as xn and is equal to 1 when xn = 0 and is equal
to 0 otherwise. Considering DAP pricing strategy with price
vector f = [f1, ..., f24], which includes the price for different
hours of day, the NE of the game is reduced to the solution
of the following binary linear programming problem.