On Optimal Multi-Dimensional Mechanism Design Citation

We efficiently solve the optimal multi-dimensional mechanism design problem for independentbidders with arbitrary demand constraints when either the number of bidders is a constantor the number of items is a constant. In the first setting, we need that each bidder’s valuesfor the items are sampled from a possibly correlated, item-symmetric distribution, allowingdifferent distributions for each bidder. In the second setting, we allow the values of each bidderfor the items to be arbitrarily correlated, but assume that the distribution of bidder types isbidder-symmetric. These symmetric distributions include i.i.d. distributions, as well as manynatural correlated distributions. E.g., an item-symmetric distribution can be obtained by takingan arbitrary distribution, and “forgetting” the names of items; this could arise when differentmembers of a bidder population have various sorts of correlations among the items, but theitems are “the same” with respect to a random bidder from the population.For all > 0, we obtain a computationally efficient additive -approximation, when the valuedistributions are bounded, or a multiplicative (1− )-approximation when the value distributionsare unbounded, but satisfy the Monotone Hazard Rate condition, covering a widely studiedclass of distributions in Economics. Our running time is polynomial in max{#items,#bidders},and not the size of the support of the joint distribution of all bidders’ values for all items,which is typically exponential in both the number of items and the number of bidders. Ourmechanisms are randomized, explicitly price bundles, and in some cases can also accommodatebudget constraints.Our results are enabled by establishing several new tools and structural properties of Bayesianmechanisms. In particular, we provide a symmetrization technique that turns any truthful mech-anism into one that has the same revenue and respects all symmetries in the underlying value dis-tributions. We also prove that item-symmetric mechanisms satisfy a natural strong-monotonicityproperty which, unlike cyclic-monotonicity, can be harnessed algorithmically. Finally, we pro-vide a technique that turns any given -BIC mechanism (i.e. one where incentive constraintsare violated by ) into a truly-BIC mechanism at the cost of O(√) revenue. We expect ourtools to be used beyond the settings we consider here. Indeed there has already been follow-upresearch [8, 9] making use of our tools. ∗Supported by a Sloan Foundation Fellowship and NSF Award CCF-0953960 (CAREER) and CCF-1101491.†Supported by a NSF Graduate Research Fellowship and a NPSC Graduate Fellowship.arXiv:1112.4006v1[cs.GT]17Dec2011