The Competition Complexity of Prophet Inequalities
Johannes Brustle, José Correa, Paul Dütting, Tomer Ezra, Michal Feldman, Victor Verdugo
TL;DR
This work analyzes the classic single-choice prophet inequality under resource augmentation, introducing the (1−ε)-competition complexity to quantify how many extra copies are needed for online algorithms to approach the offline optimum. It proves a tight separation between block and single-threshold strategies: block threshold algorithms achieve a tight $k=Θ(\log\ log(1/ε))$, while single-threshold policies require $k=Θ(\log(1/ε))$, implying a doubly-exponential convergence for the former and an exponential gap for the latter. The authors develop a novel approximate stochastic dominance framework and a max-min optimization to bound performance, complemented by explicit threshold constructions and a detailed analysis of different arrival orders, including a γ-displacement model. They further extend the framework to combinatorial settings, showing that block-consistent pricing in submodular and XOS auctions attains $O(\log(1/ε))$ competition, while static pricing yields $O(1/ε)$, highlighting practical implications for mechanism design and pricing strategies.
Abstract
We study the classic single-choice prophet inequality problem through a resource augmentation lens. Our goal is to bound the $(1-\varepsilon)$-competition complexity of different types of online algorithms. This metric asks for the smallest $k$ such that the expected value of the online algorithm on $k$ copies of the original instance, is at least a $(1-\varepsilon)$-approximation to the expected offline optimum on a single copy. We show that block threshold algorithms, which set one threshold per copy, are optimal and give a tight bound of $k = Θ(\log \log 1/\varepsilon)$. This shows that block threshold algorithms approach the offline optimum doubly-exponentially fast. For single threshold algorithms, we give a tight bound of $k = Θ(\log 1/\varepsilon)$, establishing an exponential gap between block threshold algorithms and single threshold algorithms. Our model and results pave the way for exploring resource-augmented prophet inequalities in combinatorial settings. In line with this, we present preliminary findings for bipartite matching with one-sided vertex arrivals, as well as in XOS combinatorial auctions. Our results have a natural competition complexity interpretation in mechanism design and pricing applications.