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Optimal single threshold stopping rules and sharp prophet inequalities

Alexander Goldenshluger, Yaakov Malinovsky, Assaf Zeevi

TL;DR

This paper addresses the finite-horizon iid optimal stopping problem of selecting the maximum value and benchmarks performance against a prophet with perfect foresight through sharp, non-asymptotic prophet inequalities. It introduces randomized single-threshold stopping rules and a unifying game-theoretic framework that recasts the problem as two-person zero-sum games on the unit square, yielding precise constants via saddlepoint solutions. A discretization-based computational paradigm using finite matrix games computes sharp ratio-type and difference-type constants, and the framework accommodates restricted distribution families (e.g., bounded variance and Pareto-like tails) with concrete numerical results. The work advances both theory and computation, providing explicit formulas, minimax representations, and practical algorithms to derive optimal simple stopping rules with finite-horizon guarantees and broad applicability in online decision-making. The results illuminate the structure of optimal stopping under iid assumptions and offer distribution-sensitive performance bounds with direct implications for mechanism design and sequential decision problems.

Abstract

This paper considers a finite horizon optimal stopping problem for a sequence of independent and identically distributed random variables. 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 characterize sharp prophet inequalities for single threshold stopping rules as solutions to infinite two person zero sum games on the unit square with special payoff kernels. The proposed game theoretic characterization allows one to derive sharp non-asymptotic prophet inequalities for different classes of distributions. This, in turn, gives rise to a simple and computationally tractable algorithmic paradigm for deriving optimal single threshold stopping rules.

Optimal single threshold stopping rules and sharp prophet inequalities

TL;DR

This paper addresses the finite-horizon iid optimal stopping problem of selecting the maximum value and benchmarks performance against a prophet with perfect foresight through sharp, non-asymptotic prophet inequalities. It introduces randomized single-threshold stopping rules and a unifying game-theoretic framework that recasts the problem as two-person zero-sum games on the unit square, yielding precise constants via saddlepoint solutions. A discretization-based computational paradigm using finite matrix games computes sharp ratio-type and difference-type constants, and the framework accommodates restricted distribution families (e.g., bounded variance and Pareto-like tails) with concrete numerical results. The work advances both theory and computation, providing explicit formulas, minimax representations, and practical algorithms to derive optimal simple stopping rules with finite-horizon guarantees and broad applicability in online decision-making. The results illuminate the structure of optimal stopping under iid assumptions and offer distribution-sensitive performance bounds with direct implications for mechanism design and sequential decision problems.

Abstract

This paper considers a finite horizon optimal stopping problem for a sequence of independent and identically distributed random variables. 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 characterize sharp prophet inequalities for single threshold stopping rules as solutions to infinite two person zero sum games on the unit square with special payoff kernels. The proposed game theoretic characterization allows one to derive sharp non-asymptotic prophet inequalities for different classes of distributions. This, in turn, gives rise to a simple and computationally tractable algorithmic paradigm for deriving optimal single threshold stopping rules.
Paper Structure (19 sections, 10 theorems, 137 equations, 3 tables)

This paper contains 19 sections, 10 theorems, 137 equations, 3 tables.

Table of Contents

  1. Introduction
  2. Optimal stopping.
  3. Single threshold stopping rules
  4. Main results
  5. Remarks on the sharpness of prophet inequality (\ref{['eq:cor-11']})
  6. Characterization of prophet inequalities via a game theoretic formulation
  7. Computation of sharp constants
  8. Ratio--type prophet inequalities
  9. Difference-type prophet inequalities
  10. Discussion and examples
  11. Difference--type prophet inequality for distributions with bounded variance.
  12. Ratio--type prophet inequality for Pareto--like distributions.
  13. Proofs of main results
  14. Proof of Theorem \ref{['th:prophet']}
  15. Proof of Theorem \ref{['th:R-A']}
  16. ...and 4 more sections

Key Result

Theorem 1

(Performance of optimal single threshold rules) Let $X_1, \ldots, X_n$ are iid random variables with common distribution $F$, and Then

Theorems & Definitions (19)

  • Theorem 1
  • Remark 1
  • Remark 2
  • Corollary 1
  • proof
  • Remark 3
  • Corollary 2
  • proof
  • Remark 4
  • Theorem 2
  • ...and 9 more