Bertrand competitors.

$\mathbf{Remark\;2}$: Bertrand competitors are players that do not cooperate with each other. Their goal is to maximize their own utility. They will not form any type of “coalition" to manipulate the market.

For each player, the Bayesian Nash Equilibirum (BNE) of placement policy can be derived from Equation (\[obj\]). To solve Equation (\[obj\]), we need to know the retail charging prices of all the service providers. In a Bertrand competition, the retail prices for every combination of the charging station placement policies are determined by the first order of conditions (FOC): $$\label{eq1}
\frac{\partial\Pi_1}{\partial p_1}=\sum_{n=1}^N\sum_{j=1}^{L}q_ns_{j,1}\left[\Phi_{j,1}^n+(p_1-c_{j,1})\frac{\partial\Phi_{j,1}^n}{\partial p_1}\right]=0$$ $$\label{eq2}
\frac{\partial\Pi_2}{\partial p_2}=\sum_{n=1}^N\sum_{j=1}^{L}q_ns_{j,2}\left[\Phi_{j,2}^n+(p_2-c_{j,2})\frac{\partial\Phi_{j,2}^n}{\partial p_2}\right]=0$$ $$\label{eq3}
\frac{\partial\Pi_3}{\partial p_3}=\sum_{n=1}^N\sum_{j=1}^{L}q_ns_{j,3}\left[\Phi_{j,3}^n+(p_3-c_{j,3})\frac{\partial\Phi_{j,3}^n}{\partial p_3}\right]=0$$ where $c_{j,1},c_{j,2},$ and $c_{j,3}$ represent the LMP at each charging station candidate.

$\mathbf{Remark\;3}$: For simulation simplicity, we assume the charging stations affiliated to the same service provider have the same retail charging prices ($p_1, p_2, \textrm{ and } p_3$). However, our analysis can be easily generalized to the case where each charging station sets its own retail price. Note that those retail prices obtained from Equations (\[eq1\])-(\[eq3\]) may not be the real-time prices used in practice. They are only the equilibrium prices in this market under the assumption of Bertrand competition. They can be interpreted 