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This paper studies relative arbitrage opportunities in a market with infinitely many interacting investors. We establish a conditional McKean-Vlasov system to study the market dynamics coupled with investors. We then provide a theoretical framework to study a mean-field system, where the mean-field terms consist of a joint distribution of wealth and strategies. The optimal relative arbitrage is characterized by the equilibrium of extended mean-field games. We show the conditions on the existence and the uniqueness of the mean field equilibrium, then prove the propagation of chaos results for the finite-player game, and demonstrate that the Nash equilibrium converges to the mean field equilibrium when the population grows to infinity. |
