Extreme Points in Multi-Dimensional Screening

Patrick Lahr; Axel Niemeyer

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arXiv · 2412.00649

Extreme Points in Multi-Dimensional Screening

Patrick Lahr·Axel Niemeyer

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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).

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Patrick Lahr, Axel Niemeyer. 2025-10-22. Extreme Points in Multi-Dimensional Screening. https://arxiv.org/abs/2412.00649

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