Bounding Equilibria in Counterfactual Analysis

In many economic settings, counterfactual analysis can be difficult for two reasons: (i) we do not know how to compute the equilibrium of the game, or (ii) even if we know how to compute one equilibrium, the game might feature multiple equilibria, which are difficult to exhaustively characterize. I propose a new methodology to allow for counterfactual analysis even when these problems might arise. The method relies on determining valid (conservative) bounds to counterfactual outcomes that contain any outcome that could be sustained in equilibrium, i.e., any outcome that can be supported by a set of equilibrium constraints. To ensure that all potential solutions are considered, I propose to reframeequilibrium constraints as a relaxed mixed-integer linear program. I show that the framework can alsobe used to narrow down equilibria, by imposing additional equilibrium constraints. I provide examples related to discrete choice models, dynamic games and multi-unit auctions, which exemplify how the method can be used in a practical context.