Mean Field Games for Optimal Investment Under Relative Performance Criteria

In this paper, we study the portfolio optimization problem formulated by Lacker and Soret. They formulate a finite time horizon model that allows agents to be competitive, measuring their utility not only by their absolute wealth but also relative performance compared to the average of other agents. While the finite population or $n$-player game is tractable in some cases, the authors present the Mean Field Game framework to solve this problem. Here, we seek to use this framework to clearly detail the optimal investment and consumption strategies in the CRRA utility case as was briefly outlined in Lacker and Soret, but also derive a solution in the CARA utility case.