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We study the design of optimal mechanisms when the designer is uncertain both about the form of information held by the agents and also about which equilibrium will be played. Theguaranteeof a mechanism is its worst performance across all information structures and equilibria. Thepotentialof an information structure is its best performance across all mechanisms and equilibria. We formulate a pair of linear programs, one of which is a lower bound on the maximum guarantee across all mechanisms, and the other of which is an upper bound on the minimum potential across all information structures. In applications to public expenditure, bilateral trade, and optimal auctions, we use the bounding programs to characterize guarantee‐maximizing mechanisms and potential‐minimizing information structures and show that the max guarantee is equal to the min potential. 

A profit‐maximizing seller has a single unit of a good to sell. The bidders have a pure common value that is drawn from a distribution that is commonly known. The seller does not know the bidders' beliefs about the value and thinks that beliefs are designed adversarially by Nature to minimize profit. We construct a strong maxmin solution to this joint mechanism design and information design problem, consisting of a mechanism, an information structure, and an equilibrium, such that neither the seller nor Nature can move profit in their respective preferred directions, even if the deviator can select the new equilibrium. The mechanism and information structure solve a family of maxmin mechanism design and minmax information design problems, regardless of how an equilibrium is selected. The maxmin mechanism takes the form of a proportional auction : each bidder submits a one‐dimensional bid, the aggregate allocation and aggregate payment depend on the aggregate bid, and individual allocations and payments are proportional to bids. We report a number of additional properties of the maxmin mechanisms, including what happens as the number of bidders grows large and robustness with respect to the prior over the value.