Abstract
What outcomes can be implemented by the choice of an information structure in binary-action super-modular games? An outcome is partially implementable if it satisfies obedience (Bergemann and Morris (2016)). We characterize when an outcome is smallest equilibrium implementable (induced by the smallest equilibrium) and fully implementable (induced by all equilibria). Smallest equilibrium implementation requires a stronger sequential obedience condition: there is a stochastic ordering of players under which players are prepared to switch to the high action even if they think only those before them will switch. Full implementation requires sequential obedience in both directions. Our characterization of smallest equilibrium implementation can be used to solve the information design problem with adversarial equilibrium selection.
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