Reduced Forms: Feasibility, Extremality, Optimality

TL;DR

This paper develops a general framework for analyzing optimal auctions with nonlinear revenue dependencies, introducing principal virtual values and characterizing feasibility and extremality in diverse auction environments.

Contribution

It introduces a unidimensional principal curve for feasibility testing, characterizes extreme points, and extends Myerson's virtual values to nonlinear settings for optimal auction design.

Findings

Feasibility of interim winning probabilities can be tested along a principal curve.

Explicit characterization of extreme points of the feasible set.

Optimal mechanisms use principal virtual values to allocate goods.

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.