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Monotonic Mechanisms for Selling Multiple Goods

Ran Ben-Moshe, Sergiu Hart, Noam Nisan

TL;DR

This paper studies revenue-maximizing mechanisms for selling multiple goods to a single buyer in a Bayesian setting where valuations may be correlated. It proves that restricting to monotonic mechanisms yields at most a factor of $k$ relative to simple benchmarks like separate or bundled sale, while the true optimum Rev$(X)$ can be arbitrarily larger or even infinite; this motivates a deeper analysis of monotone subclasses. It then develops sharp characterizations and revenue bounds for allocation-monotone and related monotone mechanism classes, including a detailed study of quadratic mechanisms and symmetric deterministic mechanisms, showing $AMonRev(X)\leq O(\log k)\cdot SRev(X)$ and $SymDRev(X)\leq O(\log^2 k)\cdot SRev(X)$, among others. The paper also connects monotonicity to submodularity/supermodularity via the buyer payoff $b$ and the canonical pricing function $p_0$, and provides open problems that highlight gaps between monotone and nonmonotone mechanisms and potential refinements for symmetric cases.

Abstract

Maximizing the revenue from selling two or more goods has been shown to require the use of $nonmonotonic$ mechanisms, where a higher-valuation buyer may pay less than a lower-valuation one. Here we show that the restriction to $monotonic$ mechanisms may not just lower the revenue, but may in fact yield only a $negligible$ $fraction$ of the maximal revenue; more precisely, the revenue from monotonic mechanisms is no more than k times the simple revenue obtainable by selling the goods separately, or bundled (where k is the number of goods), whereas the maximal revenue may be arbitrarily larger. We then study the class of monotonic mechanisms and its subclass of allocation-monotonic mechanisms, and obtain useful characterizations and revenue bounds.

Monotonic Mechanisms for Selling Multiple Goods

TL;DR

This paper studies revenue-maximizing mechanisms for selling multiple goods to a single buyer in a Bayesian setting where valuations may be correlated. It proves that restricting to monotonic mechanisms yields at most a factor of relative to simple benchmarks like separate or bundled sale, while the true optimum Rev can be arbitrarily larger or even infinite; this motivates a deeper analysis of monotone subclasses. It then develops sharp characterizations and revenue bounds for allocation-monotone and related monotone mechanism classes, including a detailed study of quadratic mechanisms and symmetric deterministic mechanisms, showing and , among others. The paper also connects monotonicity to submodularity/supermodularity via the buyer payoff and the canonical pricing function , and provides open problems that highlight gaps between monotone and nonmonotone mechanisms and potential refinements for symmetric cases.

Abstract

Maximizing the revenue from selling two or more goods has been shown to require the use of mechanisms, where a higher-valuation buyer may pay less than a lower-valuation one. Here we show that the restriction to mechanisms may not just lower the revenue, but may in fact yield only a of the maximal revenue; more precisely, the revenue from monotonic mechanisms is no more than k times the simple revenue obtainable by selling the goods separately, or bundled (where k is the number of goods), whereas the maximal revenue may be arbitrarily larger. We then study the class of monotonic mechanisms and its subclass of allocation-monotonic mechanisms, and obtain useful characterizations and revenue bounds.
Paper Structure (31 sections, 34 theorems, 131 equations, 1 figure)

This paper contains 31 sections, 34 theorems, 131 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The Model
  4. Pricing Functions
  5. The Canonical Pricing Function
  6. Favorable Tie Breaking
  7. Submodularity and Supermodularity
  8. Monotonicity and Allocation Monotonicity
  9. Revenue of Monotonic Mechanisms
  10. Allocation-Monotonic Mechanisms
  11. Characterization of Allocation-Monotonic Mechanisms
  12. Revenue of Allocation-Monotonic Mechanisms
  13. Quadratic Mechanisms
  14. Allocation Monotonicity without Submodularity
  15. Revenue of Symmetric Deterministic Mechanisms
  16. ...and 16 more sections

Key Result

Proposition 2.1

A function $b:\mathbb{R}_{+}^{k}\rightarrow \mathbb{R}_{+}$ is a buyer payoff function of some mechanism if and only if $b$ is continuous, convex, nondecreasing, and nonexpansive. In this case, $b$ is obtained from the mechanism $\mu =(q,s)$ if and only if $q(x)\in \partial b(x)^{+}:=\{g\in \partial

Figures (1)

  • Figure 1: Deterministic mechanisms for two goods ($q$ denotes the allocation of the two goods, and $s$ denotes the payment; see Section \ref{['sus:model']})

Theorems & Definitions (42)

  • Example 1.1
  • Example 1.2
  • Example 1.3
  • Proposition 2.1
  • Proposition 2.2
  • Proposition 2.3
  • Proposition 2.4
  • Theorem A
  • Theorem B
  • Lemma 3.1
  • ...and 32 more