Optimal Strategy-proof Mechanisms on Single-crossing Domains

TL;DR

This paper develops a novel framework for strategy-proof mechanism design on rich single-crossing domains (CMS) where buyer preferences over bundles $(t,q)$ are monotone, continuous, and potentially multidimensional or non-quasilinear. It characterizes SP mechanisms through monotonicity and the continuity of the indirect preference correspondence, and provides a constructive finite-range optimization program to compute optimal mechanisms without relying on revenue equivalence. The analysis yields deterministic, threshold-based optimal mechanisms in the quasilinear case and extends to quality interpretations and more complex domains, while also offering a principled extension to $n$ buyers via lower-efficiency. The results deliver a geometric, topology-backed view of mechanism design that is computationally tractable and broadly applicable to non-quasilinear, multidimensional settings with single-crossing structure.

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.