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Optimal Auction Design with Contingent Payments and Costly Verification

Ian Ball, Teemu Pekkarinen

Abstract

We study the design of an auction for an income-generating asset such as an intellectual property license. Each bidder has a signal about his future income from acquiring the asset. After the asset is allocated, the winner's income from the asset is realized privately. The principal can audit the winner, at a cost, and then charge a payment contingent on the winner's realized income. We solve for an auction that maximizes the principal's revenue, net of auditing costs. The winning bidder is charged linear royalties up to a cap, beyond which there is no auditing. A higher bidder pays more in cash upfront and faces a lower royalty cap.

Optimal Auction Design with Contingent Payments and Costly Verification

Abstract

We study the design of an auction for an income-generating asset such as an intellectual property license. Each bidder has a signal about his future income from acquiring the asset. After the asset is allocated, the winner's income from the asset is realized privately. The principal can audit the winner, at a cost, and then charge a payment contingent on the winner's realized income. We solve for an auction that maximizes the principal's revenue, net of auditing costs. The winning bidder is charged linear royalties up to a cap, beyond which there is no auditing. A higher bidder pays more in cash upfront and faces a lower royalty cap.
Paper Structure (29 sections, 5 theorems, 74 equations, 1 figure)

This paper contains 29 sections, 5 theorems, 74 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Model
  3. Setting
  4. Discussion
  5. Protocol
  6. Mechanisms
  7. Principal's program
  8. Benchmarks
  9. Non-contingent payments
  10. Fully contingent payments
  11. Solving for an optimal mechanism
  12. Myersonian approach
  13. Optimal mechanism
  14. Special cases
  15. Comparison with other contingent-payment auctions
  16. ...and 14 more sections

Key Result

Lemma 1

Let $(q,t,r,a,p)$ be a mechanism that satisfies Condition L:punishment_constraint and the constraints IC2, IC1, and IR for each agent $i$. The principal's expected payoff from $(q,t,r,a,p)$ is at most where for each agent $i$, the function $\Psi_i \colon \Theta \to \mathbf{R}$ is given by

Figures (1)

  • Figure 1: Winner's total equilibrium payment as a function of income for two type reports.

Theorems & Definitions (6)

  • Lemma 1: Payoff bound
  • Theorem 1: Optimal mechanism
  • Remark 1: Noisy auditing
  • Corollary 1: Free auditing
  • Corollary 2: Binary menu
  • Theorem 2: Comparative statics