On the Structure of Informationally Robust Optimal Auctions

We study the design of profit-maximizing mechanisms in environments with interdependent values. A single unit of a good is for sale. There is a known joint distribution of the bidders' values for the good. Two programs are considered: (i) Max (over mechanisms) min (over information structures and equilibria) profit; (ii) Min (over information structures) max (over mechanisms and equilibria) profit. We show that it is without loss to restrict attention to solutions of (i) and (ii) in which actions and signals belong to the same linearly ordered space, equilibrium actions are equal to signals, and the only binding equilibrium constraints are those associated with local deviations. Under such restrictions, the non-linear programs (i) and (ii) become linear programs that are dual to one another in an approximate sense. In particular, the restricted programs have the same optimal value, which we term the profit guarantee. These results simplify the task of computing and characterizing informationally robust optimal auctions and worst-case information structures with general value distributions. The framework can be generalized to include additional feasibility constraints, multiple goods, and ambiguous value distributions.