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Competition Versus Complexity in Multiple-Selection Prophet Inequalities

Eugenio Cruz-Ossa, Sebastian Perez-Salazar, Victor Verdugo

Abstract

Competition complexity formalizes a compelling intuition: rather than refining the mechanism, how much additional competition is sufficient for a simple mechanism to compete with an optimal one? We begin the study of this question in multi-unit pricing for welfare maximization using prophet inequalities. An online decision-maker observes $m \geq k$ nonnegative values drawn independently from a known distribution, may select up to $k$ of them, and aims to maximize the expected sum of selected values. The benchmark is a prophet who observes a sequence of length $n \geq k$ and selects the $k$ largest values. We focus on the widely adopted class of single-threshold algorithms and fully characterize their $(1-\varepsilon)$-competition complexity. Notably, our results reveal a sharp competition-induced phase transition: in the absence of competition, single-threshold algorithms are fundamentally limited to a $1-1/\sqrt{2kπ}$ fraction of the prophet value, whereas even a $1\%$ multiplicative increase beyond $n$ observations suffices to achieve a $1-\exp(-Θ(k))$ fraction. Another notable result happens when $k=1$: we show that the $(1-\varepsilon)$-competition complexity is exactly $\ln(1/\varepsilon)$, fully resolving an open question by Brustle et al. [Math. Oper. Res. 2024]. Our analysis is based on infinite-dimensional linear programming and duality arguments.

Competition Versus Complexity in Multiple-Selection Prophet Inequalities

Abstract

Competition complexity formalizes a compelling intuition: rather than refining the mechanism, how much additional competition is sufficient for a simple mechanism to compete with an optimal one? We begin the study of this question in multi-unit pricing for welfare maximization using prophet inequalities. An online decision-maker observes nonnegative values drawn independently from a known distribution, may select up to of them, and aims to maximize the expected sum of selected values. The benchmark is a prophet who observes a sequence of length and selects the largest values. We focus on the widely adopted class of single-threshold algorithms and fully characterize their -competition complexity. Notably, our results reveal a sharp competition-induced phase transition: in the absence of competition, single-threshold algorithms are fundamentally limited to a fraction of the prophet value, whereas even a multiplicative increase beyond observations suffices to achieve a fraction. Another notable result happens when : we show that the -competition complexity is exactly , fully resolving an open question by Brustle et al. [Math. Oper. Res. 2024]. Our analysis is based on infinite-dimensional linear programming and duality arguments.
Paper Structure (11 sections, 11 theorems, 51 equations, 1 table)

This paper contains 11 sections, 11 theorems, 51 equations, 1 table.

Table of Contents

  1. Introduction
  2. Our contribution.
  3. Problem Formulation
  4. Technical Overview of Our Results
  5. Related Work
  6. A Linear Programming Formulation
  7. The LP formulation.
  8. Solving the Linear Program via Duality
  9. Proof of Lemma \ref{['prop:supremum-q-d']}
  10. Competition Complexity Bounds
  11. Concluding Remarks

Key Result

Proposition 1

Given $F \in \mathcal{F}$, $q \in (0,1)$, $k \in \mathbb{N}$, and $n,m \in \mathbb{N}$ such that $n,m \geq k$, the following holds with $f(u) = F^{-1}(1-u)$:

Theorems & Definitions (22)

  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • Theorem 1
  • Proposition 3
  • proof
  • Lemma 1
  • proof : Proof of Theorem \ref{['thm:optimal-solution-PNK']}
  • Proposition 4
  • ...and 12 more