Competition Versus Complexity in Multiple-Selection Prophet Inequalities

TL;DR

This paper investigates how increased competition allows simple online algorithms to approximate the optimal prophet benchmark in multi-unit welfare maximization, revealing a phase transition and exact competition complexity for the single-unit case.

Contribution

It characterizes the competition complexity for single-threshold algorithms in multi-unit prophet inequalities, revealing a phase transition and resolving an open question for the case k=1.

Findings

Single-threshold algorithms are limited to a 1-1/√(2kπ) fraction of the prophet value without competition.

Slight increases in the number of observations dramatically improve the approximation ratio.

Exact competition complexity for k=1 is ln(1/ε), resolving prior open questions.

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 $1-1/\sqrt{2k\pi}$ fraction of the prophet value, whereas even a $1\%$ multiplicative increase beyond $n$ observations suffices to achieve a $1-\exp(-\Theta(k))$ 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.