Large Ranking Games with Diffusion Control

We consider a symmetric stochastic differential game where each player can control the diffusion intensity of an individual dynamic state process, and the players whose states at a deterministic finite time horizon are among the best alpha is an element of (0,1) of all states receive a fixed prize. Within the mean field limit version of the game, we compute an explicit equilibrium, a threshold strategy that consists of choosing the maximal fluctuation intensity when the state is below a given threshold and the minimal intensity otherwise. We show that for large n, the symmetric n-tuple of the threshold strategy provides an approximate Nash equilibrium of the n-player game. We also derive the rate at which the approximate equilibrium reward and the best-response reward converge to each other, as the number of players n tends to infinity. Finally, we compare the approximate equilibrium for large games with the equilibrium of the two-player case.