Approximate Revenue from Finite Range Mechanisms

TL;DR

This paper addresses revenue approximation in a seller–buyer setting with an indivisible good, where the buyer's probability of acquisition is $q$ and payment is $t$, under CM continuous and monotone single-crossing preferences with private information. It proves that the expected revenue of any strategy-proof and individually rational mechanism defined on rich single-crossing domains can be arbitrarily well approximated by a sequence of mechanisms with finite range, enabling lossless restriction to finite-range designs. The core contribution is a constructive theorem, using simple increasing function approximations $s_n$ of $t_F$ to build finite-range mechanisms $G^{s_n}$ that converge in revenue to $F$'s revenue, with a corollary that a finite-range optimum is globally optimal. The result highlights how single-crossing geometry unifies and simplifies mechanism design across both quasilinear and non-quasilinear domains, with practical implications for designing tractable, implementable mechanisms.

Abstract

We consider an economic environment where a seller wants to sell an indivisible unit of good to a buyer. We show that revenue from any strategy-proof and individually rational mechanism defined on closed intervals of rich single crossing domains considered in \citep{Goswami1}, can be approximated by the revenue from a sequence of strategy-proof and individually rational mechanisms with finite range. Thus while studying optimal mechanisms without loss of generality we can study mechanisms with finite range.