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The Root of Revenue Continuity
The Root of Revenue Continuity
Sergiu Hart and Noam Nisan
Abstract
In the setup of selling one or more goods, various papers have shown,
in various forms and for various purposes, that a small change
in the distribution of a buyer's valuations may cause
only a small change in the possible revenue
that can be extracted. We prove a simple,
clean, convenient, and general statement to this effect:
let X and Y be random valuations on k additive goods,
let Rev(X) and Rev(Y) denote their optimal revenues,
and let
W(X,Y) be the Wasserstein (or "earth mover's") distance between them; then
| √Rev(X) -
√Rev(Y) | ≤
√W(X,Y) .
This further implies that a simple explicit modification of any
optimal mechanism for X,
namely, "uniform discounting,"
is guaranteed to be almost optimal for any Y that is
close to X in the Wasserstein distance.
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