Identification of Information Structures in Bayesian Games

To what extent can an external observer observing an equilibrium action distribution in an incomplete information game infer the underlying information structure? We investigate this issue in a general linear-quadratic-Gaussian framework. A simple class of canonical information structures is offered and proves rich enough to rationalize any possible equilibrium action distribution that can arise under an arbitrary information structure. We show that the class is parsimonious in the sense that the relevant parameters can be uniquely pinned down by an observed equilibrium outcome, up to some qualifications. Our result implies, for example, that the accuracy of each agent's signal about the state is identified, as measured by how much observing the signal reduces the state variance. Moreover, we show that a canonical information structure characterizes the lower bound on the amount by which each agent's signal can reduce the state variance, across all observationally equivalent information structures. The lower bound is tight, for example, when the actual information structure is uni-dimensional, or when there are no strategic interactions among agents, but in general, there is a gap since agents' strategic motives confound their private information about fundamental and strategic uncertainty.