Minimization I.I.D. Prophet Inequality via Extreme Value Theory: A Unified Approach

TL;DR

This paper introduces a unified framework using Extreme Value Theory to analyze and optimize threshold algorithms for the I.I.D. Prophet Inequality, covering both maximization and minimization settings with new theoretical insights.

Contribution

It develops a universal analysis framework based on Extreme Value Theory, deriving closed-form competitive ratios and extending results to multi-unit cases for the minimization problem.

Findings

Competitive ratio for minimization depends on the extreme value index $oldsymbol{\gamma}$.

The same function $oldsymbol{\Lambda(oldsymbol{\gamma})}$ describes maximization and minimization ratios for different $oldsymbol{\gamma}$.

Poly-logarithmic guarantees for single-threshold algorithms in minimization.

Abstract

The I.I.D. Prophet Inequality is a fundamental problem where, given $n$ independent random variables $X_1,\dots,X_n$ drawn from a known distribution $\mathcal{D}$, one has to decide at every step $i$ whether to stop and accept $X_i$ or discard it forever and continue. The goal is to maximize or minimize the selected value and compete against the all-knowing prophet. For maximization, a tight constant-competitive guarantee of $\approx 0.745$ is well-known (Correa et al, 2019), whereas minimization is qualitatively different: the optimal constant is distribution-dependent and can be arbitrarily large (Livanos and Mehta, 2024). In this paper, we provide a novel framework via the lens of Extreme Value Theory to analyze optimal threshold algorithms. We show that the competitive ratio for the minimization setting has a closed form described by a function $\Lambda$, which depends only on the extreme value index $\gamma$; in particular, it corresponds to $\Lambda(\gamma)$ for $\gamma \leq 0$. Despite the contrast of maximization and minimization, our framework turns out to be universal and we recover the results of (Kennedy and Kertz, 1991) for maximization as well. Surprisingly, the optimal competitive ratio for maximization is given by the same function $\Lambda(\gamma)$, but for $\gamma \geq 0$. Along the way, we obtain several results on the algorithm and the prophet's objectives from the perspective of extreme value theory, which might be of independent interest. We next study single-threshold algorithms for minimization. Using extreme value theory, we generalize the results of (Livanos and Mehta, 2024) which hold only for special classes of distributions, and obtain poly-logarithmic in $n$ guarantees. Finally, we consider the $k$-multi-unit prophet inequality for minimization and show that there exist constant-competitive single-threshold algorithms when $k \geq \log{n}$.