Optimal single threshold stopping rules and sharp prophet inequalities

TL;DR

This paper develops a game-theoretic framework to derive sharp prophet inequalities for optimal stopping rules, enabling precise performance benchmarks for sequential decision-making problems.

Contribution

It introduces a novel game-theoretic approach to compute sharp constants in prophet inequalities for single threshold stopping rules.

Findings

Sharp constants are characterized by solutions to an infinite zero-sum game.

The framework provides a systematic way to handle restricted distribution classes.

An efficient algorithm computes sharp prophet inequality constants with high accuracy.

Abstract

This paper considers a finite horizon optimal stopping problem for a sequence of independent and identically distributed random variables, where the objective is to design stopping rules that attempt to select the random variable with the highest value in the sequence. The performance of any stopping rule may be benchmarked relative to the selection of a ``prophet" that has perfect foreknowledge of the largest value. Such comparisons are typically stated in the form of ``prophet inequalities." In this paper we develop a game-theoretic characterization that supports a principled approach for deriving sharp non-asymptotic prophet inequalities for single threshold stopping rules. We demonstrate that sharp constants in the ratio- and difference-type prophet inequalities are determined by the optimal values of infinite two-person zero-sum game on the unit square with particular payoff kernels, while the the solutions to the game provide optimal stopping rules and least favorable distributions. Among other things, this formulation also allows a systematic way to tackle restricted classes of distributions. The proposed framework leads to a numerically efficient algorithmic paradigm that allows computing sharp constants in prophet inequalities with any prescribed level of accuracy.