Optimal Strategy-proof Mechanisms on Single-crossing Domains

TL;DR

This paper characterizes strategy-proof mechanisms in a single-crossing domain with complex preferences, providing a geometric approach and a computational method for optimal mechanisms without relying on traditional revenue equivalence tools.

Contribution

It introduces a novel geometric characterization of strategy-proof mechanisms in rich single-crossing domains and offers a tractable optimization program for finite-range mechanisms.

Findings

Characterization of strategy-proof mechanisms via monotonicity and continuity.

Development of a computationally tractable optimization program.

Extension of analysis to multi-buyer environments and qualitative variables.

Abstract

We consider an economic environment with one buyer and one seller. For a bundle $(t,q)\in [0,\infty[\times [0,1]=\mathbb{Z}$, $q$ refers to the winning probability of an object, and $t$ denotes the payment that the buyer makes. We consider continuous and monotone preferences on $\mathbb{Z}$ as the primitives of the buyer. These preferences can incorporate both quasilinear and non-quasilinear preferences, and multidimensional pay-off relevant parameters. We define rich single-crossing subsets of this class and characterize strategy-proof mechanisms by using monotonicity of the mechanisms and continuity of the indirect preference correspondences. We also provide a computationally tractable optimization program to compute the optimal mechanism for mechanisms with finite range. We do not use revenue equivalence and virtual valuations as tools in our proofs. Our proof techniques bring out the geometric interaction between the single-crossing property and the positions of bundles $(t,q)$s in the space $\mathbb{Z}$. We also provide an extension of our analysis to an $n-$buyer environment, and to the situation where $q$ is a qualitative variable.