Set Identification in Models with Multiple Equilibria

TL;DR

This paper introduces a computational approach to identify features of models with multiple equilibria, utilizing core inclusion and optimization techniques, with applications to bargaining and oligopoly entry games.

Contribution

It develops efficient algorithms and combinatorial methods for characterizing the identified set in models with multiple equilibria, including normal form games.

Findings

Characterizes the identified set via inequalities and optimization programs.

Provides algorithms for checking parameter inclusion in the identified set.

Illustrates methods with bargaining and oligopoly entry game examples.

Abstract

We propose a computationally feasible way of deriving the identified features of models with multiple equilibria in pure or mixed strategies. It is shown that in the case of Shapley regular normal form games, the identified set is characterized by the inclusion of the true data distribution within the core of a Choquet capacity, which is interpreted as the generalized likelihood of the model. In turn, this inclusion is characterized by a finite set of inequalities and efficient and easily implementable combinatorial methods are described to check them. In all normal form games, the identified set is characterized in terms of the value of a submodular or convex optimization program. Efficient algorithms are then given and compared to check inclusion of a parameter in this identified set. The latter are illustrated with family bargaining games and oligopoly entry games.