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Probabilistic Verification in Mechanism Design

Ian Ball, Deniz Kattwinkel

TL;DR

This paper introduces a tractable framework for probabilistic verification in mechanism design by modeling verification via tests with type-dependent passage probabilities. It develops a discernment-order structure on tests to determine when a single most-discerning test suffices and shows that, in that case, the mechanism design problem collapses to a reduced-form authentication rate $\alpha(\theta'|\theta)$. Using a Myerson-style envelope and a modified virtual value $\varphi(\theta)$ that incorporates verification, the authors characterize optimal revenue-maximizing mechanisms under exponential verification, including nonlinear pricing and the sale of a single good, with outcomes interpolating between no verification and full surplus extraction as verification strengthens. They further generalize to nonbinary tests and relate the approach to existing verification literatures, furnishing conditions for when the revelation principle and reduced-form results hold. The framework provides a foundation for evaluating the value of verification technologies and sets the stage for optimizing verification investment in dynamic mechanisms.

Abstract

We introduce a model of probabilistic verification in mechanism design. The principal elicits a message from the agent and then selects a test to give the agent. The agent's true type determines the probability with which he can pass each test. We characterize whether each type has an associated test that best screens out all other types. If this condition holds, then the testing technology can be represented in a tractable reduced form. We use this reduced form to solve for profit-maximizing mechanisms with verification. As the verification technology varies, the solution continuously interpolates between the no-verification solution and full surplus extraction.

Probabilistic Verification in Mechanism Design

TL;DR

This paper introduces a tractable framework for probabilistic verification in mechanism design by modeling verification via tests with type-dependent passage probabilities. It develops a discernment-order structure on tests to determine when a single most-discerning test suffices and shows that, in that case, the mechanism design problem collapses to a reduced-form authentication rate . Using a Myerson-style envelope and a modified virtual value that incorporates verification, the authors characterize optimal revenue-maximizing mechanisms under exponential verification, including nonlinear pricing and the sale of a single good, with outcomes interpolating between no verification and full surplus extraction as verification strengthens. They further generalize to nonbinary tests and relate the approach to existing verification literatures, furnishing conditions for when the revelation principle and reduced-form results hold. The framework provides a foundation for evaluating the value of verification technologies and sets the stage for optimizing verification investment in dynamic mechanisms.

Abstract

We introduce a model of probabilistic verification in mechanism design. The principal elicits a message from the agent and then selects a test to give the agent. The agent's true type determines the probability with which he can pass each test. We characterize whether each type has an associated test that best screens out all other types. If this condition holds, then the testing technology can be represented in a tractable reduced form. We use this reduced form to solve for profit-maximizing mechanisms with verification. As the verification technology varies, the solution continuously interpolates between the no-verification solution and full surplus extraction.

Paper Structure

This paper contains 46 sections, 13 theorems, 70 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Model
  3. Setting
  4. Principal--agent environment
  5. Verification
  6. Mechanisms and strategies
  7. Implementation
  8. Discussion of the model
  9. Trying on the test
  10. Test choice
  11. Ordering tests
  12. Discernment orders
  13. Implementation with most-discerning testing
  14. Sufficient conditions for discernment orders
  15. Testing in reduced form
  16. ...and 31 more sections

Key Result

Proposition 1

Every implementable (extended) social choice function is canonically implementable.

Figures (3)

  • Figure 1: Timing
  • Figure 2: Directed graph representing a $\{0,1\}$-valued authentication rate
  • Figure 3: Exponential authentication rate

Theorems & Definitions (28)

  • Proposition 1: Revelation principle
  • Example 1: Passing v. failing
  • Example 2: Exogenous authentication rate
  • Definition 1: $\theta$-discernment
  • Theorem 1: Test replacement
  • Proposition 2: $\theta$-discernment equivalence
  • Definition 2: Most-discerning
  • Theorem 2: Most-discerning implementation
  • Remark 3: Discernment orders under alternative specifications
  • Proposition 3: Relative performance
  • ...and 18 more