A Geometric Approach to Inference in Set-Identified Entry Games

In this paper, we consider inference procedures for entry games with complete information. Due to the presence of multiple equilibria, we know that such a model may be set identified without imposing further restrictions. We complete the model with the unknown selection mechanism and characterize geometrically the set of predicted choice probabilities, in our case, a convex polytope with many facets. Testing whether a parameter belongs to the identified set is equivalent to testing whether the true choice probability vector belongs to this convex set. Using tools from the convex analysis, we calculate the support function and the extreme points. The calculation yields a finite number of inequalities, when the explanatory variables are discrete, and we characterized them once for all. We also propose a procedure that selects the moment inequalities without having to evaluate all of them. This procedure is computationally feasible for any number of players and is based on the geometry of the set. Furthermore, we exploit the specific structure of the test statistic used to test whether a point belongs to a convex set to propose the calculation of critical values that are computed once and independent of the value of the parameter tested, which drastically improves the calculation time. Simulations in a separate section suggest that our procedure performs well compared with existing methods.