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The purpose of this paper is to study 2-person zero-sum stochasticdifferential games, in which one player is a major one and the other player isa group of $N$ minor agents which are collectively playing, statisticallyidentical and have the same cost-functional. The game is studied in a weakformulation; this means in particular, we can study it as a game of the type"feedback control against feedback control". The payoff/cost functional isdefined through a controlled backward stochastic differential equation, forwhich driving coefficient is assumed to satisfy strict concavity-convexity withrespect to the control parameters. This ensures the existence of saddle pointfeedback controls for the game with $N$ minor agents. We study the limitbehavior of these saddle point controls and of the associated Hamiltonian, andwe characterize the limit of the saddle point controls as the unique saddlepoint control of the limit mean-field stochastic differential game.

Nonlinear stochastic differential games involving a major player and a large number of collectively acting minor agents