Multiple treatments with strategic substitutes

We develop an empirical framework to identify and estimate the effects of treatments on outcomes of interest when the treatments are the result of strategic interaction (e.g., bargaining, oligopolistic entry, peer effects). We consider a model where agents play a discrete game of complete information and strategic substitutability, whose equilibrium actions (i.e., binary treatments) determine a post-game outcome in a nonseparable model with endogeneity. Due to the simultaneity in the first stage, the model as a whole is incomplete and the selection process fails to exhibit the conventional monotonicity. Without imposing parametric restrictions or large support assumptions, this poses challenges in recovering treatment parameters. To address these challenges, we establish a monotonic pattern of the equilibria in the first-stage game in terms of the number of treatments selected. 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. We show that the instrument variation that compensates strategic substitution helps solve the multiple equilibria problem. We apply our method to data on airlines and air pollution in cities in the U.S. We find that (i) the causal effect of each airline on pollution is positive, and (ii) the effect is increasing in the number of firms but at a decreasing rate.