🤖 AI Summary
This paper investigates the extremal structure of incentive-compatible mechanisms in multidimensional type screening. Focusing on canonical settings—including monopoly pricing, bilateral trade, barter, and delegation versus veto bargaining—it provides a complete characterization of extremal mechanisms. Methodologically, the analysis integrates convex analysis, mechanism design theory, linear utility modeling, and geometric representation of menus. The contributions are threefold: (i) it establishes, for the first time, a formal connection between extremal mechanisms and Gale’s (1954) indecomposable convex bodies; (ii) it proves that any exhaustive mechanism can be approximated arbitrarily closely by extremal mechanisms via generic perturbations; and (iii) it derives a tight upper bound on menu size in the one-dimensional case and shows that, in multidimensional settings, typical exhaustive mechanisms are almost surely extremal with uncountable menus. Collectively, these results unify the boundary characterization of multidimensional incentive-compatible mechanisms, offering novel geometric insights and constructive tools for mechanism design.
📝 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).