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Dynamic Decoupling in Multidimensional Screening

Eric Gao

Abstract

I study multidimensional sequential screening. A monopolist contracts with a buyer who privately observes information about the distribution of their eventual valuations for multiple goods. After initial private information is reported and the contract is signed, the buyer learns and reports realized valuations. In these settings, the monopolist frontloads surplus extraction: Any information rents given to the buyer to elicit their true valuations can be extracted in expectation before those valuations are drawn, transforming the multidimensional screening problem by distorting buyer information rents compared to static screening. If the buyer's distributions over valuations are commonly FOSD ordered and regular for each good; and satisfy invariant dependencies (valuations can be dependent across goods, but how valuations are coupled cannot vary), the optimal mechanism coincides with independently offering the optimal sequential screening mechanism for each good. This rationalizes membership payments followed by separate sales schemes seen across multiple industries.

Dynamic Decoupling in Multidimensional Screening

Abstract

I study multidimensional sequential screening. A monopolist contracts with a buyer who privately observes information about the distribution of their eventual valuations for multiple goods. After initial private information is reported and the contract is signed, the buyer learns and reports realized valuations. In these settings, the monopolist frontloads surplus extraction: Any information rents given to the buyer to elicit their true valuations can be extracted in expectation before those valuations are drawn, transforming the multidimensional screening problem by distorting buyer information rents compared to static screening. If the buyer's distributions over valuations are commonly FOSD ordered and regular for each good; and satisfy invariant dependencies (valuations can be dependent across goods, but how valuations are coupled cannot vary), the optimal mechanism coincides with independently offering the optimal sequential screening mechanism for each good. This rationalizes membership payments followed by separate sales schemes seen across multiple industries.
Paper Structure (13 sections, 8 theorems, 103 equations, 3 figures)

This paper contains 13 sections, 8 theorems, 103 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Illustrative Example
  3. Related Literature
  4. Model
  5. Sufficient Assumptions
  6. Dynamic Decoupling
  7. Relaxing A3
  8. Relaxing A2 or A1
  9. Extensions
  10. Additional Periods
  11. Observed Orthogonalization
  12. Downstream Auction
  13. Discussion

Key Result

Lemma 1

The monopolist's problem can be written as where $u(\gamma, \theta) = \theta \cdot q(\gamma, \theta) - t_2(\gamma, \theta)$ is the buyer's utility net of $t_1$ and $\nabla_\theta u(\gamma, \theta)$ is the gradient of $u(\gamma, \theta)$ in $\theta$ at some fixed $\gamma$. Furthermore, a necessary condition for IC1 to hold is which is used to arrive at the objective function.

Figures (3)

  • Figure :
  • Figure :
  • Figure :

Theorems & Definitions (16)

  • Lemma 1
  • Lemma 2
  • Theorem 1: Optimality of Separate Sales
  • Proposition 1: Separate Ironing
  • Proposition 2: Irrelevance of Additional Periods
  • Proposition 3: No Post-Contractual Information Rents
  • Proposition 4: Relaxed Solution is Implementable
  • proof
  • proof
  • proof
  • ...and 6 more