$x,\delta > 0$. Agent $i$ seeks to maximize the expected utility $$\begin{aligned}
J_i((\pi^i,c^i)_{i=1}^n) = \E\left[\int_0^T   U\left(c^i_tX^i_t  (\overline{cX}_t)^{- \theta_i}; \delta_i \right)dt + \epsilon_i U\left(X^i_T \overline{X}_T^{- \theta_i}; \delta_i \right)\right], \label{def:utility}\end{aligned}$$ defined for any vector of admissible strategies $(\pi^i,c^i)_{i=1}^n$ where $(\pi^i,c^i) \in \A$ for each $i=1,\ldots,n$. Here $\overline{X}_T = \left( \prod_{k = 1}^{n} X_T^k \right)^{1/n}$ and $\overline{cX}_t = \left( \prod_{k = 1}^{n} (c_tX_t)^k \right)^{1/n}$ are the population (geometric) average wealth and consumption rate, respectively. The parameters $\delta_i > 0$ and $\theta_i \in [0,1]$ represent respectively the $i^{\mathrm{th}}$ agent’s risk tolerance and competition weight. We apply the same utility function to both wealth and consumption for tractability reasons, but we scale the utility of wealth with the parameter $\epsilon_i > 0$ to capture the relative importance that the agent assigns to terminal wealth compared to consumption.

Our choice to work with geometric averages instead of arithmetic averages is also motivated by tractability: Just as the (arithmetic) average of independent Brownian motions is again a Brownian motion, the geometric average of independent geometric Brownian motions is again a geometric Brownian motion. It is worth emphasizing another form of the utility function, revealed by writing the terms inside the utility function as $$\begin{aligned}
c^i_tX^i_t  (\overline{cX}_t)^{- \theta_i} 