Sharp Test for Equilibrium Uniqueness in Discrete Games with a Flexible Information Structure

I propose a test for an assumption commonly maintained when estimating static discrete games of incomplete information, i.e. the assumption of equilibrium uniqueness in the data generating process. The test is appealing for several reasons. It allows for discrete common knowledge payoff-relevant unobserved heterogeneity (henceforth unobserved heterogeneity, for short). In that sense, it is more general than tests which attribute all correlation between players’ decisions to multiple equilibria. Furthermore, the test does not require the estimation of payoffs to separate multiplicity of equilibria from unobservable heterogeneity. It is therefore useful in empirical applications leveraging multiple equilibria to identify the model’s primitives when commonly-used exclusion restrictions are notavailable. Finally, it makes no parametric assumption on the payoff functions nor the distribution of players’ private information. The main identifying assumption is the existenceof an observable variable that plays the role of a proxy for the unobserved heterogeneity.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. Simulation evidence is provided to study the test’s properties.