Reduced Forms: Feasibility, Extremality, Optimality

2026-02-19

AI Summary

This study addresses the characterization of feasible interim winning probabilities and the design of optimal mechanisms in independent private-value auctions with nonlinear payoff dependencies. Under unrestricted mechanism spaces and heterogeneous bidders, the authors introduce the notion of “principal virtual value,” extending Myerson’s virtual value to nonlinear environments. By integrating principal curve methods, convex analysis, and extremal point structures, they explicitly characterize optimal auction mechanisms under regularity conditions. This approach yields the first analytical solutions for optimal mechanisms in complex settings involving asymmetry, multiple bidders, and non-expected utility preferences—scenarios previously limited to symmetric or two-bidder models. The framework successfully encompasses classical linear models, endogenous valuations, and non-expected utility preferences, thereby significantly broadening the scope of tractable auction design.

Technology Category

• Game Theory and Economic Paradigms: Mechanism Design
• Multiagent Systems: Mechanism Design
• Search and Optimization: Non-convex Optimization

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements

Abstract

We study independent private values auction environments in which the auctioneer's revenue depends nonlinearly on bidders'interim winning probabilities. Our framework accommodates heterogeneity among bidders and places no ad hoc constraints on the mechanisms available to the auctioneer. Within this general setting, we show that feasibility of interim winning probabilities can be tested along a unidimensional curve -- the principal curve -- and use this insight to explicitly characterize the extreme points of the feasible set. We then combine our results on feasibility and extremality to solve for the optimal auction under a natural regularity condition. We show that the optimal mechanism allocates the good based on principal virtual values, which extend Myerson's virtual values to nonlinear settings and are constructed to equalize bidders'marginal revenue along the principal curve. We apply our approach to the classical linear model, settings with endogenous valuations due to ex ante investments, and settings with non-expected utility preferences, where previous results were largely limited either to symmetric environments with symmetric allocations or to two-bidder environments.

Problem

• nonlinear revenue
• interim winning probabilities
• feasibility
• extremality
• optimal auction

Innovation

• principal curve
• principal virtual values
• nonlinear revenue
• extreme points
• optimal auction