On optimal multidimensional mechanism design

Abstract

We efficiently solve the optimal multi-dimensional mechanism design problem for independent bidders with arbitrary demand constraints when either the number of bidders is a constant or the number of items is a constant. In the first setting, we need that each bidder's values for the items are sampled from a possibly correlated, item-symmetric distribution, allowing different distributions for each bidder. In the second setting, we allow the values of each bidder for the items to be arbitrarily correlated, but assume that the distribution of bidder types is bidder-symmetric. These symmetric distributions include i.i.d. distributions, as well as many natural correlated distributions. E.g., an item-symmetric distribution can be obtained by taking an arbitrary distribution, and "forgetting" the names of items; this could arise when different members of a bidder population have various sorts of correlations among the items, but the items are "the same" with respect to a random bidder from the population.For all > 0, we obtain a computationally efficient additive -approximation, when the value distributions are bounded, or a multiplicative (1− )-approximation when the value distributions are unbounded, but satisfy the Monotone Hazard Rate condition, covering a widely studied class 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. Our mechanisms are randomized, explicitly price bundles, and in some cases can also accommodate budget constraints.Our results are enabled by establishing several new tools and structural properties of Bayesian mechanisms. In particular, we provide a symmetrization technique that turns any truthful mechanism into one that has the same revenue and respects all symmetries in the underlying value distributions. We also prove that item-symmetric mechanisms satisfy a natural strong-monotonicity property which, unlike cyclic-monotonicity, can be harnessed algorithmically. Finally, we provide a technique that turns any given -BIC mechanism (i.e. one where incentive constraints are violated by ) into a truly-BIC mechanism at the cost of O( √ ) revenue. We expect our tools to be used beyond the settings we consider here. Indeed there has already been follow-up research [8,9] making use of our tools.

Optimal Design of Robust Combinatorial Mechanisms for Substitutable Goods

“…The equivalence of the constraint sets given by (6)- (7) and F can be seen by observing that the lefthand side of (6) is equal to u j , thus by subtracting u i from both sides of this constraint, and expanding the right-hand side, we obtain the inequality defining (9). We note that this new constraint is a special case of more general condition that incentive-compatible allocations must correspond to utility functions that are convex continuous (see, e.g., [5]).…”

Near-Optimal Scheduling Mechanisms for Deadline-Sensitive Jobs in Large Computing Clusters

“…We note that while extensions from single-value to multivalue domains do exist (see, e.g., Cai et al [2011] and references therein), we provide here a general framework for truthfulness, which covers our model as special case. To conclude this section, we apply this framework to the bounded flexible scheduling problem, and argue that the greedy algorithms from Section 3 are truthful.…”

Near-optimal scheduling mechanisms for deadline-sensitive jobs in large computing clusters

“…The framework requires extending well-known results for single-parameter settings, where the private information held by each user consists of a single scalar. While extensions to multi-parameters auctions do exist (see, e.g., [8] and references therein), we provide here a general framework for truthfulness, which covers our model as special case.…”