Strategy-proof Selling: a Geometric Approach

TL;DR

This paper introduces a geometric approach to designing strategy-proof mechanisms for a single buyer and seller, characterizing mechanisms through monotonicity and continuity without relying on revenue equivalence, and providing a tractable optimization method.

Contribution

It develops a novel geometric framework for strategy-proof mechanism design that handles various preferences and offers a computationally feasible optimization program.

Findings

Characterization of strategy-proof mechanisms via monotonicity and continuity.

Introduction of a geometric interaction between preferences and bundle positions.

Provision of a simple, computationally tractable optimization program.

Abstract

We consider one buyer and one seller. For a bundle $(t,q)\in [0,\infty[\times [0,1]=\mathbb{Z}$, $q$ either refers to the wining probability of an object or a share of a good, and $t$ denotes the payment that the buyer makes. We define classical and restricted classical preferences of the buyer on $\mathbb{Z}$; they incorporate quasilinear, non-quasilinear, risk averse preferences with multidimensional pay-off relevant parameters. We define rich single-crossing subsets of the two classes, 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. 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. Our proofs are simple and provide computationally tractable optimization program to compute the optimal mechanism. The extension of the optimization program to the $n-$ buyer environment is immediate.