# Many-player games of optimal consumption and investment under relative performance criteria

## TL;DR

This paper analyzes a multi-agent portfolio optimization problem where agents' utilities depend on their absolute and relative wealth and consumption, providing explicit solutions for both finite and mean field games.

## Contribution

It derives a unique closed-form equilibrium solution for a multi-player and mean field game with relative performance criteria, extending classical portfolio optimization models.

## Key findings

- Equilibrium solutions are explicit and unique under constant investment and time-dependent consumption.
- Competitive agents' behaviors are highly nonlinear and depend on risk tolerance and competitiveness.
- High risk-tolerance agents may behave like low risk-tolerance agents in competitive settings.

## Abstract

We study a portfolio optimization problem for competitive agents with CRRA utilities and a common finite time horizon. The utility of an agent depends not only on her absolute wealth and consumption but also on her relative wealth and consumption when compared to the averages among the other agents. We derive a closed form solution for the $n$-player game and the corresponding mean field game. This solution is unique in the class of equilibria with constant investment and continuous time-dependent consumption, both independent of the wealth of the agent. Compared to the classical Merton problem with one agent, the competitive model exhibits a wide range of highly nonlinear and non-monotone dependence on the agents' risk tolerance and competitiveness parameters. Counter-intuitively, competitive agents with high risk tolerance may behave like non-competitive agents with low risk tolerance.