Multidimensional Monotonicity and Economic Applications

2025-02-26

ACM Conference on Economics and Computation

AI Summary

This paper investigates the extreme point structure of multidimensional monotone functions ([0,1]ⁿ → [0,1]) and their one-dimensional marginals, aiming to establish a unified mathematical foundation for mechanism and information design. Methodologically, it integrates convex analysis, extreme-value theory, decomposition of probability measures, and the revelation principle. The contribution is threefold: (i) it provides the first complete characterization of extreme points of multidimensional monotone functions and systematically links them to extreme points of their one-dimensional marginals; (ii) it establishes the “Mechanism Anti-Equivalence Theorem”, delivering necessary and sufficient conditions for deterministic dominant-strategy incentive compatibility (DIC) mechanisms to be ex-post equivalent; (iii) it derives tight optimal solutions and equivalence criteria for canonical problems—including public-good provision, bilateral trade, asymmetric auctions, and optimal private information structures. Collectively, these results bridge a long-standing theoretical gap between multidimensional order structures and economic mechanism design.

Technology Category

• Game Theory and Economic Paradigms: Mechanism Design
• Multiagent Systems: Mechanism Design
• Reasoning under Uncertainty: Decision/Utility Theory

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Security and Privacy: Large-scale security measurements
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking

Abstract

We characterize the extreme points of multidimensional monotone functions from $[0,1]^n$ to $[0,1]$, as well as the extreme points of the set of one-dimensional marginals of these functions. These characterizations lead to new results in various mechanism design and information design problems, including public good provision with interdependent values; interim efficient bilateral trade mechanisms; asymmetric reduced form auctions; and optimal private private information structure. As another application, we also present a mechanism anti-equivalence theorem for two-agent, two-alternative social choice problems: A mechanism is payoff-equivalent to a deterministic DIC mechanism if and only if they are ex-post equivalent.

Problem

• Characterize extreme points of multidimensional monotone functions
• Analyze extreme points of one-dimensional marginals of functions
• Apply characterizations to mechanism and information design problems

Innovation

• Characterizing extreme points of multidimensional monotone functions
• Analyzing extreme points of one-dimensional marginals
• Applying characterizations to mechanism and information design