Before revision:  To address these challenges, we first analytically characterize regions that predict equilibria in the first-stage game with possibly more than two players, and ascertain a monotonic pattern of these regions . Based on this finding, we derive bounds on the average treatment effects ( ATE's ) under nonparametric shape restrictions and the existence of excluded exogenous variables.
After revision:  To address these challenges, we first establish a monotonic pattern of the equilibria in the first-stage game with possibly more than two players, and ascertain a monotonic pattern of these regions . Based on this finding, we derive bounds on the average treatment effects ( ATE's ) under nonparametric shape restrictions and the existence of excluded exogenous variables.