\ge J_i\Big( \ldots, (\pi^{i-1,*},c^{i-1,*}), (\pi, c), (\pi^{i+1,*},c^{i+1,*}), \ldots\Big).$$ An equilibrium $(\pi^{i,*}, c^{i,*})_{i=1}^n$ is called a *strong equilibrium* if, for each $i$, the process $c^{i,*}$ is deterministic and continuous, and the process $\pi^{i,*}$ is deterministic and constant.[^1]

The main result is the following, which gives the explicit form of an equilibrium:

\[thmN\] Let $n \ge 2$. Assume that for all $i = 1, ..., n$, we have $x_0^i > 0$, $\delta_i > 0$, $\theta_i \in [0,T]$, $\epsilon_i > 0$, $\mu_i > 0$, $\sigma_i \ge 0$, $\nu_i \ge 0$, and $\sigma_i + \nu_i > 0$. Then there is a unique strong equilibrium $(\pi^{i,*}, c^{i,*})_{i=1}^n$, and it takes the following form: $$\begin{aligned}
 \pi^{i,*} &= \frac{\delta_i \mu_i }{\sigma_i^2 + \nu^2_i(1 + (\delta_i -1)\theta_i/n)} -  \frac{\theta_i (\delta_i -1)\sigma_i}{\sigma_i^2 + \nu_i^2(1 + (\delta_i -1)\theta_i/n)} \frac{\phi}{1 + \psi} \label{optcontroleq} \\
c^{i,*}_t &= \begin{cases}
            \left(\frac{1}{\newomega_i} + \left(\frac{1}{\newC_i} - \frac{1}{\newomega_i} \right)e^{- \newomega_i (T - t)} \right)^{-1} & \mathrm{if } \  \beta_i  \neq 0 \\
            (T - t + \newC_i^{-1})^{-1} & \mathrm{if } \ \beta_i = 0. 
        \end{cases} \label{optconsumption}\end{aligned}$$ The constants $(\phi,\psi)$ and $(\newomega_i,\newC_i)_{i=1}^n$ are given by $$\begin{aligned}
\phi &= \frac{1}{n} \sum_{k = 1}^n \delta_k \frac{\mu_k \sigma_k}{\sigma_k^2 + \nu_k^2(1 + (\delta_k-1)\theta_k/n)}, \nonumber \\
\psi &= \frac{1}{n} \sum_{k = 1}^n  \theta_k 