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Bounds on the revenue gap of linear posted pricing for selling a divisible item

Ioannis Caragiannis, Zhile Jiang, Apostolis Kerentzis

TL;DR

The paper investigates revenue gaps for linear posted pricing when selling a perfectly divisible item to $n$ agents with private, concave valuation functions drawn from known distributions. It introduces an ex-ante relaxation and a revenue-gap program parameterized by the natural IMOTV measure $κ$, and proves three results: a tight $O(\log κ)$ bound for a single agent, a black-box $O(κ^2)$ bound via reduction to anonymous pricing, and a direct $O(\log κ+\log n)$ bound for the multi-agent case. The strongest bound combines a constructive near-feasible solution with a careful decomposition into cases, yielding a logarithmic dependence on both $κ$ and $n$. The findings advance understanding of simple versus optimal pricing for divisible goods and point to future work on tight bounds and extensions to other mechanisms.

Abstract

Selling a perfectly divisible item to potential buyers is a fundamental task with apparent applications to pricing communication bandwidth and cloud computing services. Surprisingly, despite the rich literature on single-item auctions, revenue maximization when selling a divisible item is a much less understood objective. We introduce a Bayesian setting, in which the potential buyers have concave valuation functions (defined for each possible item fraction) that are randomly chosen according to known probability distributions. Extending the sequential posted pricing paradigm, we focus on mechanisms that use linear pricing, charging a fixed price for the whole item and proportional prices for fractions of it. Our goal is to understand the power of such mechanisms by bounding the gap between the expected revenue that can be achieved by the best among these mechanisms and the maximum expected revenue that can be achieved by any mechanism assuming mild restrictions on the behavior of the buyers. Under regularity assumptions for the probability distributions, we show that this revenue gap depends only logarithmically on a natural parameter characterizing the valuation functions and the number of agents. Our results follow by bounding the objective value of a mathematical program that maximizes the ex-ante relaxation of optimal revenue under linear pricing revenue constraints.

Bounds on the revenue gap of linear posted pricing for selling a divisible item

TL;DR

The paper investigates revenue gaps for linear posted pricing when selling a perfectly divisible item to agents with private, concave valuation functions drawn from known distributions. It introduces an ex-ante relaxation and a revenue-gap program parameterized by the natural IMOTV measure , and proves three results: a tight bound for a single agent, a black-box bound via reduction to anonymous pricing, and a direct bound for the multi-agent case. The strongest bound combines a constructive near-feasible solution with a careful decomposition into cases, yielding a logarithmic dependence on both and . The findings advance understanding of simple versus optimal pricing for divisible goods and point to future work on tight bounds and extensions to other mechanisms.

Abstract

Selling a perfectly divisible item to potential buyers is a fundamental task with apparent applications to pricing communication bandwidth and cloud computing services. Surprisingly, despite the rich literature on single-item auctions, revenue maximization when selling a divisible item is a much less understood objective. We introduce a Bayesian setting, in which the potential buyers have concave valuation functions (defined for each possible item fraction) that are randomly chosen according to known probability distributions. Extending the sequential posted pricing paradigm, we focus on mechanisms that use linear pricing, charging a fixed price for the whole item and proportional prices for fractions of it. Our goal is to understand the power of such mechanisms by bounding the gap between the expected revenue that can be achieved by the best among these mechanisms and the maximum expected revenue that can be achieved by any mechanism assuming mild restrictions on the behavior of the buyers. Under regularity assumptions for the probability distributions, we show that this revenue gap depends only logarithmically on a natural parameter characterizing the valuation functions and the number of agents. Our results follow by bounding the objective value of a mathematical program that maximizes the ex-ante relaxation of optimal revenue under linear pricing revenue constraints.

Paper Structure

This paper contains 17 sections, 21 theorems, 62 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Our contribution and techniques
  3. Related work
  4. Roadmap
  5. Concepts, techniques, and notation
  6. Selling an indivisible item with an anonymous price
  7. Shortsighted agents
  8. A mathematical program for the revenue gap
  9. Warming up: the single-agent case
  10. A lower bound
  11. The multi-agent case: a black-box reduction
  12. The multi-agent case: a direct approach
  13. Case 1: $q_i(x)<1/n$.
  14. Case 2: $q_i(x)>1/n$.
  15. Case 1: $\ell\cdot (1-F(\ell)) \geq z\cdot F^{-1}(1-z)$.
  16. ...and 2 more sections

Key Result

Theorem 1

For every $R\geq 1$, the objective value of the mathematical program (eq:mp-indivisible) is at most $e\cdot R$.

Figures (1)

  • Figure 1: An illustration of the argument in the proof of Lemma \ref{['lem:triangular']} (Case 2) using the revenue-quantile curve $q\cdot F^{-1}(1-q)$. To compare the two rectangles with thick sides $A$ and $B$, we show that due to concavity of the revenue curve $q\cdot F^{-1}(1-q)$, rectangle $A$ contains the lower-left shaded region corresponding to rectangle $\widetilde{A}$ and the upper-right shaded region corresponding to rectangle $\widetilde{B}$ contains rectangle $B$. To complete the proof, it suffices to observe that the two shaded rectangles have equal areas.

Theorems & Definitions (40)

  • Theorem 1: Alaei et al. AHNPY19
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • Theorem 5
  • proof
  • Corollary 6
  • ...and 30 more