Multiple Equilibria in Empirical Games of Incomplete Information

Multiple equilibria in the data can confound identification in games of incomplete information. The standard approach is to assume “one equilibrium in the data†(OED) in spite of the fact that model generating the data exhibits multiplicity. I first show that games of incomplete information are non parametrically identified without the OED assumption. Identification is obtained using the unconditional probability distribution over player choices which, unlike equilibrium choice probabilities, are identified under mild conditions even when OED fails. The unconditional probabilities operate as a proxy measure of the equilibrium probabilities, and have a key property: independence from the difference between itself and the equilibrium probability. In this way, games of incomplete information can be expressed as a standard linear errors in variables model where the “measurement error†is uncorrelated with the “proxy†probability distribution. However, this approach is invalid in the presence of payoff relevant unobservables. I provide an alternative identification approach, robust to the presence of such unobservables. In particular I show that if equilibrium multiplicity disappears at extreme points in the support of the observables, and a conditional exclusion restriction on the distribution of the payoff relevant unobservables is satisfied, the model can be identified. In particular, binary choice games with only two players are identified if there are at least five points in the support of the excluded variable. Finally, regardless of whether or not there are payoff relevant unobservables, once payoffs are identified everywhere, so are equilibria, and, letting A be the number of choice alternatives, N the number of players and K the number of equilibria, as long as AN ≥ K, Ishow that equilibrium selection probabilities are also identified.