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Procurement without Priors: A Simple Mechanism and its Notable Performance

Dirk Bergemann, Tibor Heumann, Stephen Morris

TL;DR

The paper tackles procurement when supplier costs are unknown and no prior over costs is available. It shows that simple constant utility share mechanisms, calibrated to demand elasticity, guarantee a positive fraction of the efficient social surplus and attain the competitive ratio across both linear and convex nonlinear cost environments. The authors provide two complementary justifications: a direct maximin analysis and a Bayes-optimal interpretation under appropriate priors, establishing a saddle-point structure. The framework extends naturally to regulation and nonlinear pricing, offering robust, scale-invariant design principles that rely on key elasticities rather than costly cost-distribution estimates.

Abstract

How should a buyer design procurement mechanisms when suppliers' costs are unknown, and the buyer does not have a prior belief? We demonstrate that simple mechanisms - that share a constant fraction of the buyer utility with the seller - allow the buyer to realize a guaranteed positive fraction of the efficient social surplus across all possible costs. Moreover, a judicious choice of the share based on the known demand maximizes the surplus ratio guarantee that can be attained across all possible (arbitrarily complex and nonlinear) mechanisms and cost functions. Similar results hold in related nonlinear pricing and optimal regulation problems.

Procurement without Priors: A Simple Mechanism and its Notable Performance

TL;DR

The paper tackles procurement when supplier costs are unknown and no prior over costs is available. It shows that simple constant utility share mechanisms, calibrated to demand elasticity, guarantee a positive fraction of the efficient social surplus and attain the competitive ratio across both linear and convex nonlinear cost environments. The authors provide two complementary justifications: a direct maximin analysis and a Bayes-optimal interpretation under appropriate priors, establishing a saddle-point structure. The framework extends naturally to regulation and nonlinear pricing, offering robust, scale-invariant design principles that rely on key elasticities rather than costly cost-distribution estimates.

Abstract

How should a buyer design procurement mechanisms when suppliers' costs are unknown, and the buyer does not have a prior belief? We demonstrate that simple mechanisms - that share a constant fraction of the buyer utility with the seller - allow the buyer to realize a guaranteed positive fraction of the efficient social surplus across all possible costs. Moreover, a judicious choice of the share based on the known demand maximizes the surplus ratio guarantee that can be attained across all possible (arbitrarily complex and nonlinear) mechanisms and cost functions. Similar results hold in related nonlinear pricing and optimal regulation problems.

Paper Structure

This paper contains 26 sections, 12 theorems, 191 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Motivation
  3. Related Literature
  4. Model
  5. Payoffs
  6. Private Information and Mechanism
  7. The Buyer Surplus Ratio Guarantee
  8. The Linear Cost Environment
  9. Constant Share Mechanisms
  10. The Ratio Guarantee with Constant Share Mechanisms
  11. The Competitive Ratio and General Mechanisms
  12. The Constant Share Mechanism is a Bayes Optimal Mechanism
  13. The Nonlinear Cost Environment
  14. The Environment with Convex Cost
  15. General Cost and Utility Functions
  16. ...and 11 more sections

Key Result

Proposition 1

$\quad$ For all constant share mechanisms, $t(q)=z\cdot u\left( q\right)$, we have that The inequality is an equality if and only if $z^{\ast }=\sigma$, in which case the bound is attained for all $c\in \mathbb{R}_{+}$.

Figures (8)

  • Figure 1: Buyer Surplus Guarantee as Share of Social Surplus
  • Figure 2: Buyer and Seller Surplus Guarantee as Share of Social Surplus
  • Figure 3: Quadratic Cost Function and Piecewise Linear Cost Function
  • Figure 4: Buyers' Surplus and Social Surplus
  • Figure 5: Surplus share $z^{\ast}(\sigma)$ as function of the elasticity $\sigma$.
  • ...and 3 more figures

Theorems & Definitions (12)

  • Proposition 1: Constant Share Mechanisms
  • Proposition 2: Competitive Ratio
  • Proposition 3: Bayes Optimal Mechanism
  • Theorem 1: Buyer Surplus Guarantee
  • Lemma 1: Convexity of the Transformed Cost
  • Theorem 2: Competitive Ratio with General Cost and Utility Functions
  • Theorem 3: Competitive Ratio of Regulation
  • Corollary 1: Comparative Statics of Competitive Ratio
  • Proposition 4: Competitive Ratio for Nonlinear Pricing
  • Lemma 2: Concavification of $t$
  • ...and 2 more