\begin{align}\label{sec_N_players:brokersproblem}
\mathbb E^{B, \nu^B} \left[  \int_0^T \left\{ \bar Q^B_t \left(b\,\nu^B_t  + \alpha_t \right) + \eta^I \frac 1N \sum_{n=1}^N \left(\nu^n_t \right)^2- \eta^B \left( \nu^B_t\right)^2 - 2\,a^B \bar Q^B_t \left( \nu^B_t - \frac 1N \sum_{n=1}^N \nu^n_t \right) -  \phi^B\left( \bar Q^B_t \right)^2 \right\} \d t\right],
\end{align}
