Optimal Multi-Dimensional Mechanisms are not Local
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Consider the problem of implementing a revenue-optimal, Bayesian Incentive Compatible auction when buyers' values are drawn from distributions ×_i D_i on a particular instance v⃗. Optimal single-dimensional mechanisms are local: in order to allocate the item correctly on a particular valuation profile v⃗, only Õ(1) bits are needed from each player (specifically, their Myerson virtual value [Mye81]), rather than the entire distribution. We show that optimal multi-dimensional mechanisms are not local: in order to allocate the item correctly on a particular valuation profile v⃗, one still needs to know (essentially) the entire distribution. Specifically, if the distributions have support-size n, then Ω(n) bits are necessary from each bidder. We show that this phenomenon already occurs with just two bidders, even when one bidder is single-dimensional, and even when the other bidder is barely multi-dimensional. More specifically, the multi-dimensional bidder is "inter-dimensional" from the FedEx setting with just two days [FGKK16].
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