Extreme Points in Multi-Dimensional Screening

TL;DR

This paper characterizes the extreme points of incentive-compatible mechanisms in multi-dimensional screening problems, providing a comprehensive framework that includes various economic scenarios and revealing the density of certain mechanisms.

Contribution

It offers a unified characterization of extreme points in multi-dimensional screening, extending to various economic problems and connecting to convex geometry.

Findings

Extreme points are tractable in one-dimensional types.

In multi-dimensional types, extreme points are dense in a subset called exhaustive mechanisms.

Mechanisms that maximize revenue are dense among incentive-compatible mechanisms.

Abstract

We characterize the extreme points of the set of incentive-compatible mechanisms for screening problems with linear utility. Our framework subsumes problems with and without transfers, such as monopoly pricing, principal-optimal bilateral trade and barter exchange, delegation and veto bargaining, or belief elicitation via proper scoring rules. In every problem with one-dimensional types, extreme points admit a tractable description. In every problem with multi-dimensional types, extreme points are dense in a rich subset of incentive-compatible mechanisms, which we call exhaustive mechanisms. Building on these characterizations, we derive parallel conclusions for mechanisms that can be rationalized as (uniquely) optimal under a fixed objective. For example, in the multi-good monopoly problem, mechanisms that uniquely maximize revenue for some type distribution are dense among all incentive-compatible and individually rational mechanisms. The proofs exploit a novel connection between menus of extreme points and indecomposable convex bodies, first studied by Gale (1954).