Competition Versus Complexity in Multiple-Selection Prophet Inequalities

2026-02-23

AI Summary

This work investigates how introducing minimal competition—by slightly increasing the number of candidate values—enables simple mechanisms to closely approximate the prophet benchmark in multi-choice sequential decision-making, thereby circumventing the need for complex algorithmic designs. Focusing on welfare maximization in multi-unit pricing, the study analyzes the performance gap of a single-threshold strategy when the observation sequence exceeds the prophet’s horizon. The key contributions include uncovering a phase transition phenomenon wherein an infinitesimal increase in competition dramatically improves the approximation ratio; fully resolving the open problem of the competitive complexity required for a (1−ε)-approximation when k=1, establishing it precisely as ln(1/ε); and proving that without competition, the single-threshold algorithm’s performance is upper-bounded by 1−1/√(2kπ), whereas merely 1% additional observations suffice to achieve a (1−exp(−Θ(k)))-approximation. The analysis leverages infinite-dimensional linear programming and duality theory.

Technology Category

• Game Theory and Economic Paradigms: Mechanism Design
• Reasoning under Uncertainty: Sequential Decision Making
• Multiagent Systems: Mechanism Design

Application Category

• Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environments
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements

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.

Problem

• prophet inequalities
• competition complexity
• multi-unit pricing
• single-threshold algorithms
• welfare maximization

Innovation

• competition complexity
• prophet inequalities
• single-threshold algorithms
• phase transition
• infinite-dimensional linear programming