Abstract. This paper characterizes extreme points of the set of incentive-compatible

mechanisms for screening problems with linear utility. Extreme points are exhaustive

mechanisms, meaning their menus cannot be scaled and translated to make additional

feasibility constraints binding. In problems with one-dimensional types, extreme points

admit a tractable description with a tight upper bound on their menu size. In problems with

multi-dimensional types, every exhaustive mechanism can be transformed into an extreme

point by applying an arbitrarily small perturbation. For mechanisms with a finite menu, this

perturbation displaces the menu items into general position. Generic exhaustive mechanisms

are extreme points with an uncountable menu. Similar results hold in applications to

delegation, veto bargaining, and monopoly problems, where we consider mechanisms that

are unique maximizers for specific classes of objective functionals. The proofs involve a novel

connection between menus of extreme points and indecomposable convex bodies, first studied

by Gale (1954).

JEL Codes: D82, D44, D86, C78, C65

Keywords: Multi-Dimensional Types, Extreme Points, Exposed Points, Indecomposable

Convex Bodies, Multi-Good Monopoly Problem, Linear Delegation, Linear Veto Bargaining