Rank-preserving multidimensional mechanisms

We study the mechanism design problem of a monopolist with multiple heterogeneous indivisible objects. The buyer’s type is multidimensional, with a value for each object. The value for a bundle of objects is the sum of the value of each object in the bundle. We introduce a new property which we call rank preserving. A mechanism is rank preserving if objects with greater buyer values are allocated with (weakly) higher probabilities. We show that in an exchangeable environment (i.e., type space is a n-dimensional cube and the prior is symmetric), there exists an optimal (i.e., revenue maximizing)   mechanism which is rank preserving and symmetric. However, an optimal deterministic mechanism need not be symmetric or rank preserving. We show that an optimal deterministic mechanism that is symmetric exists if and only if an optimal deterministic mechanism that is rank preserving exists. If a  mechanism satisfies aggregate monotonicity, i.e., the sum of all tail allocation probabilities increases as the value of an object increases, then the mechanism is revenue monotone (i.e., higher types pay more). As an application of this result, we show that an almost deterministic mechanism is revenue monotone. We establish an equivalence between symmetric, rank-preserving mechanisms in this heterogeneous objects model and mechanisms for homogeneous objects with decreasing marginal values.

Joint work with Sushil Bikhchandani

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