Inference of Multiple Equilibria in Discrete Games with Correlated Private Information

Résumé

We propose several tests for multiple equilibria in static and dynamic discrete games with incomplete information where agents' private information are correlated conditional on observable states. First, in static Bayesian games, we propose a test that exploits a concept of "adjacent games'', or clusters of games in which individuals are known to follow the same or affiliated equilibrium strategies. The decisions made by individuals participating in different but adjacent games are uncorrelated under the null of single equilibrium. On the other hand, under the alternative such pairs of decisions are correlated as we pool observations across all pairs of adjacent games in a sample. Based on these insights, we propose a simple test for multiple equilibria by partitioning the sample into adjacent games and testing for the correlation between individual decisions across adjacent games. Second, we propose two tests for multiple Markovian Perfect equilibria (MPE) when individual information set are correlated through time-varying game-level unobserved heterogeneity. In this case, the distribution of choice and state history admits a finite-mixture representation, in which the number of components is determined by three factors: the length of history, the cardinality of the support of unobserved heterogeneity, and the multiplicity of MPE. With a proper partition of the outcome history, and by conditioning on a carefully designed set of events, we utilize the relation between the identifiable cardinality of mixture components and the three factors above to separately recover the cardinality of unobserved heterogeneity and the multiplicity of MPE. We propose two tests that can be implemented through rank estimation, depending on the numbers of players and choices. We present monte carlo evidence that these tests perform well in finite samples. (Joint with Aureo de Paula)