Optimal Investment Under Relative Performance Concerns

We consider the problem of optimal investment when agents take into account their relative performance by comparison to their peers. Given N interacting agents, we consider the following optimization problem for agent i , : urn:x-wiley:09601627:media:mafi12034:mafi12034-math-0002 where is the utility function of agent i , his portfolio, his wealth, the average wealth of his peers, and is the parameter of relative interest for agent i . Together with some mild technical conditions, we assume that the portfolio of each agent i is restricted in some subset . We show existence and uniqueness of a Nash equilibrium in the following situations: unconstrained agents, constrained agents with exponential utilities and Black–Scholes financial market. We also investigate the limit when the number of agents N goes to infinity. Finally, when the constraints sets are vector spaces, we study the impact of the s on the risk of the market.