Minimization I.I.D. Prophet Inequality via Extreme Value Theory: A Unified Approach
Vasilis Livanos, Ruta Mehta
TL;DR
This work introduces a unified Extreme Value Theory (EVT) framework to analyze the I.I.D. prophet inequality in the large-market regime, covering both reward maximization and cost minimization. A single function $Λ(γ) = \frac{(1-γ)^{-γ}}{Γ(1-γ)}$ governs the asymptotic competitive ratios for both settings, with $λ_M = \min\{Λ(γ),1\}$ and $λ_m = \max\{Λ(γ),1\}$ across the appropriate domains of $γ$, unifying prior results for the max case and extending to min. The analysis leverages regularly varying tail behavior, hazard-rate techniques, and Karamata's theorem to relate prophet-values and algorithm-threshold quantiles, yielding poly-logarithmic guarantees for single-threshold minimization and constant-competitive outcomes when $k \ge \log n$ in the multi-unit variant. The results bridge maximization and minimization under EVT, recover Kennedy–Kertz for max, and generalize Livanos–Mehta's min-prophet insights to the broader EVT domain, with implications for mechanism design and sequential decision-making under uncertainty.
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 $Λ$, which depends only on the extreme value index $γ$; in particular, it corresponds to $Λ(γ)$ for $γ\leq 0$. 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 $Λ(γ)$, but for $γ\geq 0$. 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 $k \geq \log{n}$.