Reduced Forms: Feasibility, Extremality, Optimality

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.

💡 Research Summary

The paper tackles the design of optimal auctions in independent private values (IPV) environments where the seller’s expected revenue is a nonlinear function of bidders’ interim winning probabilities. This non‑linearity arises in several economically important settings, such as endogenous‑valuation models (where bidders invest before the auction) and models with non‑expected‑utility preferences (e.g., constant relative risk aversion). The authors develop a unified framework that (i) characterizes feasibility of any reduced form (the vector of interim winning probabilities), (ii) describes the extreme points of the feasible set, and (iii) solves the seller’s optimal‑auction problem under a natural regularity condition.

Feasibility. Classical Border’s theorem provides a high‑dimensional family of linear inequalities that a reduced form must satisfy. In asymmetric settings with many bidders these constraints are intractable. The authors introduce the principal curve, a canonical one‑dimensional path inside the unit hyper‑cube. They prove (Theorem 1) that a reduced form is feasible if and only if Border’s inequalities hold along this single curve. This reduces the verification problem to a univariate test, even for fully asymmetric environments with many bidders.

Extremality. Because the feasible set of reduced forms is convex and compact, any optimum of a (quasi‑convex) objective must lie at an extreme point. Building on the principal‑curve representation, the paper shows (Theorems 2 and 3) that a reduced form is extreme precisely when Border’s constraints bind everywhere on its principal curve. Moreover, every extreme reduced form can be implemented by a score allocation: each bidder’s type is mapped to a real‑valued score, the good is awarded to the highest non‑negative score, and ties occur with probability zero. Score allocations are exactly the mechanisms that generate extreme reduced forms, providing a clean Bayesian‑dominant‑strategy equivalence.

Optimality via optimal control. The seller’s problem is reformulated as a continuous‑time optimal control problem in which “time” runs from high to low types along the principal curve. The control variables are the rates at which interim winning probabilities are allocated to each bidder, subject to a dynamic feasibility constraint (the Border condition along the curve). Applying Pontryagin’s maximum principle yields a marginal‑revenue equalization condition: on any interval where monotonicity does not bind, the optimal principal curve equalizes the bidders’ marginal revenues. This leads to the definition of principal virtual values, a nonlinear extension of Myerson’s virtual values that incorporates the shadow cost of feasibility.

Applications.

• In the classic linear IPV model, principal virtual values coincide with Myerson’s virtual values, reproducing the known optimal auction.
• In endogenous‑valuation models, interim winning probabilities affect the return on pre‑auction investment. The optimal mechanism shifts probability toward bidders with higher investment returns, reducing duplicated investment costs and raising revenue.
• In CRRA risk‑averse settings, the optimal mechanism may allocate a fractional unit to low‑type, risk‑averse bidders (fractional score allocations) to lower informational rents while incurring modest efficiency loss. The analysis predicts a reversal of preference: risk‑averse bidders are favored at low allocations (low types) but disfavored at high allocations (high types), a pattern that follows directly from marginal‑revenue equalization.

Overall, the paper provides a powerful three‑step methodology—principal‑curve feasibility, extreme‑point characterization via score allocations, and marginal‑revenue equalization—to solve optimal auction design problems that were previously tractable only in symmetric or two‑bidder settings. By extending Myerson’s virtual‑value intuition to nonlinear, asymmetric environments, it opens the door to practical optimal‑auction design in many modern markets where revenue depends on allocation probabilities in a complex way.