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Algorithmic Information Disclosure in Optimal Auctions

Yang Cai, Yingkai Li, Jinzhao Wu

TL;DR

A polynomial-time approximation scheme (PTAS) for computing the optimal joint design with at most an ϵ multiplicative loss in expected revenue and it is shown that in this joint design problem, the seller can significantly reduce the information rent of the agents by providing partial information, which ensures a revenue that is at least 1 - 1/e of the optimal welfare for all valuation distributions.

Abstract

This paper studies a joint design problem where a seller can design both the signal structures for the agents to learn their values, and the allocation and payment rules for selling the item. In his seminal work, Myerson (1981) shows how to design the optimal auction with exogenous signals. We show that the problem becomes NP-hard when the seller also has the ability to design the signal structures. Our main result is a polynomial-time approximation scheme (PTAS) for computing the optimal joint design with at most an $ε$ multiplicative loss in expected revenue. Moreover, we show that in our joint design problem, the seller can significantly reduce the information rent of the agents by providing partial information, which ensures a revenue that is at least $1 - \frac{1}{e}$ of the optimal welfare for all valuation distributions.

Algorithmic Information Disclosure in Optimal Auctions

TL;DR

A polynomial-time approximation scheme (PTAS) for computing the optimal joint design with at most an ϵ multiplicative loss in expected revenue and it is shown that in this joint design problem, the seller can significantly reduce the information rent of the agents by providing partial information, which ensures a revenue that is at least 1 - 1/e of the optimal welfare for all valuation distributions.

Abstract

This paper studies a joint design problem where a seller can design both the signal structures for the agents to learn their values, and the allocation and payment rules for selling the item. In his seminal work, Myerson (1981) shows how to design the optimal auction with exogenous signals. We show that the problem becomes NP-hard when the seller also has the ability to design the signal structures. Our main result is a polynomial-time approximation scheme (PTAS) for computing the optimal joint design with at most an multiplicative loss in expected revenue. Moreover, we show that in our joint design problem, the seller can significantly reduce the information rent of the agents by providing partial information, which ensures a revenue that is at least of the optimal welfare for all valuation distributions.
Paper Structure (23 sections, 16 theorems, 97 equations, 5 algorithms)

This paper contains 23 sections, 16 theorems, 97 equations, 5 algorithms.

Table of Contents

  1. Introduction
  2. Our Results and Techniques
  3. Related Work
  4. Preliminaries
  5. Signal Structures
  6. Mechanisms
  7. Discussion of Modeling Choices
  8. Optimal $k$-Signal Problem
  9. Illustrative Example
  10. NP-hardness
  11. Polynomial Time Algorithms
  12. Warm-up: A Simple Constant Approximation Algorithm
  13. PTAS
  14. Discretization
  15. Dynamic Programming Algorithm
  16. ...and 8 more sections

Key Result

Theorem 1

Both the Optimal Signal problem and the Optimal $k$-Signal problem for any $k\geq 2$ are NP-hard when $\max|V_i| \geq 3$.

Theorems & Definitions (32)

  • Definition 1: Partitional Signal Structures
  • Theorem 1: NP-hardness
  • Lemma 3.1
  • Lemma 3.2
  • Lemma 3.3
  • Theorem 2: Welfare Approximating Algorithm
  • Corollary 4.1
  • Lemma 4.1
  • proof
  • Theorem 3: PTAS
  • ...and 22 more