We discuss a natural game of competition and solve the corresponding mean field game with common noise when agents’ rewards are rank-dependent. We use this solution to provide an approximate Nash equilibrium for the finite player game and obtain the rate of convergence.
"A rank-based mean field game in the strong formulation." Electron. Commun. Probab. 21 1 - 12, 2016. https://doi.org/10.1214/16-ECP24