Distributionally robust mechanism design

Abstract

We study a mechanism design problem where an indivisible good is auctioned to multiple bidders, for eachof whom it has a private value that is unknown to the seller and the other bidders. The agents perceive theensemble of all bidder values as a random vector governed by an ambiguous probability distribution, whichbelongs to a commonly known ambiguity set. The seller aims to design a revenue maximizing mechanism thatis not only immunized against the ambiguity of the bidder values but also against the uncertainty about thebidders’ attitude towards ambiguity. We argue that the seller achieves this goal by maximizing the worst-caseexpected revenue across all value distributions in the ambiguity set and by positing that the bidders haveKnightian preferences. For ambiguity sets containing all distributions supported on a hypercube, we showthat the Vickrey auction is the unique mechanism that is optimal, efficient and Pareto robustly optimal. Ifthe bidders’ values are additionally known to be independent, then the revenue of the (unknown) optimalmechanism does not exceed that of a second price auction with only one additional bidder. For ambiguitysets under which the bidders’ values are dependent and characterized through moment bounds, on the otherhand, we provide a new class of randomized mechanisms, the highest-bidder-lotteries, whose revenues cannotbe matched by any second price auction with a constant number of additional bidders. Moreover, we showthat the optimal highest-bidder-lottery is a 2-approximation of the (unknown) optimal mechanism, whereasthe best second price auction fails to provide any constant-factor approximation guarantee.