Sharp Test for Equilibrium Uniqueness in Discrete Games With Private Information and Common Knowledge Unobserved Heterogeneity

Abstract

This paper proposes a test of the single equilibrium in the data assumption commonly maintained when estimating static discrete games of incomplete information. By allowing for discrete common knowledge payoff-relevant unobserved heterogeneity, the test generalizes existing methods attributing all correlation between players’ decisions to multiple equilibria. It does not require the estimation of payoffs and is therefore useful in empirical applications leveraging multiple equilibria to identify the model’s primitives. The procedure boils down to testing the emptiness of the set of data generating processes that can rationalize the sample through a single equilibrium and a finite mixture over unobserved heterogeneity. Under verifiable conditions, this testable implication is generically sufficient for degenerate equilibrium selection. The main identifying assumption is the existence of an observable variable that plays the role of a proxy for the unobservable heterogeneity. Examples of such proxies are provided based on empirical applications from the existing literature.