Optimal Auction Design with Contingent Payments and Costly Verification

This paper characterizes the revenue-maximizing auction for an income-generating asset with costly post-allocation verification, demonstrating that the optimal mechanism combines upfront cash payments with linear royalty contracts that feature bidder-specific caps beyond which auditing ceases.

The Gist

Imagine you are a scientist who has invented a magical new battery. You know it could power a city, but you don't know exactly how much energy it will produce until you actually hook it up and turn it on. You want to sell the rights to use this battery to the highest bidder, but you have a problem: you can't see the results until after the sale.

Once a buyer takes the battery, they will see exactly how much energy it generates. But they might lie to you and say, "Oh, it only produced a little bit of power," so they can pay you less. You could hire an inspector to check their records, but that costs money.

This paper solves a very tricky puzzle: How do you design an auction to sell this battery so you make the most money, while accounting for the cost of checking if the buyer is lying?

The Core Idea: The "Royalty Cap"

In the real world, companies often use royalty agreements. Instead of paying a huge lump sum upfront, the buyer pays a small fee now and then pays you a percentage of their sales later.

The authors of this paper discovered the perfect way to structure these deals. Their solution has two main parts:

1. The "Capped" Royalty: You tell the buyer, "You will pay me 10% of your sales, but only up to a certain limit (the cap). Once you hit that limit, you pay me nothing more, and I won't check your books."
2. The "Audit" Threat: If the buyer claims their sales are below that cap, you have the right to hire an inspector. If the inspector finds they lied (under-reported sales), they have to pay a heavy penalty.

Why This Works: The "Debt" Analogy

Think of this mechanism like a loan with a twist.

• The Low-End (The "Debt" part): If the battery produces low energy, the buyer pays you a percentage of what they made. Because they are paying a percentage, they have an incentive to tell the truth. If they lie and say they made less, they save money now, but if you audit them, they lose a lot later. So, they tell the truth.
• The High-End (The "Cap" part): If the battery is a massive success and the buyer makes a fortune, they hit the "cap." Once they hit that cap, they stop paying you extra, and you stop checking their books.

Why stop checking?
Because checking costs money. If the buyer is already paying you a huge amount (hitting the cap), the cost of hiring an inspector to find out if they made even more money isn't worth it. You save on inspection fees, and the buyer is happy because they keep the extra profit.

The "Secret Sauce": Who Pays More?

The paper also explains how to treat different bidders. Imagine two people want to buy your battery:

• Person A: A small startup with shaky finances.
• Person B: A giant tech corporation with deep pockets and a reputation for honesty.

The authors' model says:

• Person B (The "High Type"): They pay a higher upfront cash fee to get the battery. In exchange, they get a lower royalty cap. This means they can keep more of their massive profits, and you won't bother auditing them as often because they are likely to hit that cap quickly.
• Person A (The "Low Type"): They pay less upfront, but they face a higher royalty cap. This means you will audit them more often to make sure they aren't hiding their small profits.

The Metaphor:
Think of it like a tax system.

• The "High Type" is like a wealthy person who pays a large entry fee to get a "fast pass" (lower taxes/royalties) and less scrutiny.
• The "Low Type" is like a small business that pays a small entry fee but faces strict audits and higher taxes on their earnings.

Why This Matters in the Real World

You see this exact structure in the real world, often in patent licensing or franchise deals (like a McDonald's franchise).

• The Problem: If you charge a flat fee, the buyer might not have enough cash. If you charge a flat percentage forever, you spend a fortune auditing them every single year.
• The Solution: The "Royalty Cap" is the sweet spot. It encourages the buyer to be honest when they are doing poorly (because of the audit threat), but it stops you from wasting money auditing them when they are doing incredibly well.

Summary in One Sentence

The best way to sell a future income stream is to charge a percentage of the income up to a limit, and then stop checking once that limit is reached; this saves you money on inspections while still keeping the buyer honest.

Technical Summary

Here is a detailed technical summary of the paper "Optimal Auction Design with Contingent Payments and Costly Verification" by Ian Ball and Teemu Pekkarinen.

1. Problem Statement

The paper addresses the design of an optimal auction for an income-generating asset (e.g., an intellectual property license, a casino license, or an oil lease) where the future income generated by the asset is privately observed by the winner and can only be verified at a cost.

• The Setting: A principal (seller) auctions an asset to $N$ risk-neutral agents.
• Information Structure:
  • Ex-ante: Agents observe a private signal (type $\theta_i$) regarding their expected future income.
  • Ex-post: After the asset is allocated, the winner's realized income ($\pi_i$) is drawn from a distribution conditional on $\theta_i$. This income is private to the winner.
• The Mechanism: The principal can charge upfront transfers, contingent royalties (based on reported income), and penalties. Crucially, the principal can conduct a costly audit to reveal the true income. If an audit occurs, the principal can charge a penalty based on the discrepancy between reported and realized income.
• Constraints:
  • Costly Verification: Auditing incurs a cost $c_i$.
  • Limited Liability/Moral Hazard (Condition A): The principal cannot extract more than a fraction $\phi_i$ of the income in penalties. Specifically, the penalty function $p(\pi)$ must satisfy $0 \le p(\hat{\pi}) - p(\pi) \le (\hat{\pi} - \pi)\phi_i$. This ensures the winner retains at least a fraction$(1-\phi_i)$ of any additional dollar earned, preventing the winner from "burning" cash or diverting funds to avoid payments.

Objective: Maximize the principal's expected revenue (payments minus auditing costs) subject to incentive compatibility (truthful reporting of types and income) and individual rationality.

2. Methodology

The authors employ a dynamic mechanism design approach using the Myersonian framework extended to sequential screening and costly verification.

• Relaxation and Envelope Theorem: The authors formulate the principal's problem as a constrained optimization. They use the Envelope Theorem to derive an upper bound on the principal's payoff.
• Virtual Value Derivation: A key technical contribution is the derivation of an endogenous virtual value ($\Psi_i$). Unlike standard Myerson auctions where the virtual value depends only on the type distribution, here it depends on the allocation rule and the auditing rule.
  • The virtual value is defined as:
    $$\Psi_i(\theta) = \theta_i - \frac{1-F_i(\theta_i)}{f_i(\theta_i)} + E_{\pi_i|\theta_i} [a_i(\theta, \pi_i)(\mu_i(\theta_i, \pi_i)\phi_i - c_i)]$$
    Where $\mu_i$ represents the sensitivity of the information rent to the type, and the term in the expectation captures the trade-off between the cost of auditing ($c_i$) and the benefit of reducing information rents via steeper penalties ($\phi_i$).
• Pointwise Maximization: The authors solve for the optimal mechanism by maximizing the virtual surplus pointwise. They identify conditions (Assumption 1) under which the optimal allocation and auditing rules are monotonic, ensuring global incentive compatibility.
• Regularity Conditions: They introduce a class of distributions called "transformed additive noise" to satisfy the necessary regularity conditions (single-crossing properties) for the optimal solution to be implementable.

3. Key Contributions and Results

A. The Optimal Mechanism (Theorem 1)

The optimal mechanism features a specific structure that rationalizes common industry practices like royalty caps:

1. Allocation Rule: The asset is allocated to the agent with the highest virtual value ($\psi_i$), provided it is non-negative. The virtual value is strictly higher than the standard Myerson virtual value because contingent payments reduce information rents.
2. Auditing Rule: The winner is audited if and only if their reported income is below a specific threshold (the "royalty cap" level).
   • If reported income $\pi' \ge \pi^*$, the agent pays a flat royalty cap and is not audited.
   • If reported income $\pi' < \pi^*$, the agent pays a linear royalty (slope $\phi_i$) and is audited with probability 1.
3. Payment Structure:
   • Linear Royalty + Cap: The winner pays a royalty of $\phi_i \times \pi'$ up to a cap.
   • Penalty: If audited and found to under-report, the agent pays the underpaid royalties as a penalty.
   • Upfront Transfer: Determined by the envelope theorem to ensure individual rationality.
4. Shape of Payments: The equilibrium payment curve is linear up to a cap and then flat. This is the inverse of the "call option" structure found in standard security-bid auctions (DeMarzo et al., 2005) where payments are flat then linear.

B. Economic Intuition

• Trade-off: The mechanism balances the information rent extraction (via auditing and steep penalties) against the cost of auditing.
• Royalty Caps: Auditing is costly. To save on auditing costs, the principal stops auditing once the reported income is high enough (above the cap). At this point, the marginal benefit of extracting more information is outweighed by the cost of verification.
• Type Dependence: Higher types (agents with higher expected income) face lower royalty caps and are less likely to be audited. This is counter-intuitive from a statistical perspective (high types with low reports are suspicious), but optimal because high types are more likely to generate high incomes naturally; auditing them would be wasteful if they are truthful.

C. Comparative Statics (Theorem 2)

• Auditing Costs ($c_i$): As auditing becomes cheaper, the royalty cap increases, and the principal extracts more surplus.
• Penalty Sensitivity ($\phi_i$): If the principal can impose steeper penalties (higher $\phi_i$), the royalty cap increases, and the asset is allocated more frequently.
• Hazard Rate: If an agent's type distribution shifts to the "hazard-rate order" (making high types relatively more likely), the agent wins less often but faces a higher royalty cap.

D. Special Cases

• Free Auditing ($c_i=0$): The principal always audits. The mechanism reduces to a linear royalty at rate $\phi_i$ with no cap (or an infinite cap).
• Single Agent: The mechanism can be implemented as a menu of contracts (lump-sum vs. linear royalty with cap). High types choose the lump-sum (no royalties), while middle types choose the royalty contract.

4. Significance and Implications

1. Rationalizing Royalty Caps: The paper provides a rigorous theoretical justification for the widespread use of royalty caps in licensing agreements (e.g., patents, franchises). It shows that caps are not merely contractual limitations but optimal responses to the trade-off between information extraction and verification costs.
2. Extension of Security Design: It bridges the gap between auction theory and security design. While previous literature (e.g., DeMarzo et al.) assumed public cash flows, this paper incorporates private information and costly verification, showing that the optimal security structure changes from a call option to a debt-like contract (linear then flat).
3. Policy and Business Applications:
   • Cross-Border Licensing: The model predicts that royalty caps will be lower when auditing is more difficult (e.g., the licensee is in a different country), as the cost $c_i$ rises.
   • Technological Advances: If technology reduces auditing costs (e.g., better data tracking), the model predicts royalty caps will increase, allowing sellers to extract more revenue.
4. Methodological Advancement: The paper successfully applies the Myersonian approach to a dynamic setting with sequential information revelation and costly state verification, deriving a closed-form solution for the optimal mechanism under general conditions.

In summary, Ball and Pekkarinen demonstrate that when income is private and verification is costly, the optimal auction design naturally evolves into a capped royalty scheme where high-income realizations are not audited to save costs, while low-income realizations are scrutinized to prevent under-reporting.