Large Ranking Games with Diffusion Control

Abstract

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 [Formula: see text] 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. Funding: Support from the Deutsche Forschungsgemeinschaft [Grant AN 1024/5-1] is acknowledged. The research of N. Kazi-Tani is supported by the Agence Nationale de la Recherche [Grant ANR-21-CE46-0002-03]. The research of C. Zhou is supported by the National Natural Science Foundation of China [Grant 11871364], the Singapore Ministry of Education [Grants A-8000453-00-00, A-0004273-00-00 and A-0004277-00-00] and the MERLION 2020 award [Grant A-0004589-00-00].