Optimal Single Threshold Stopping Rules and Sharp Prophet Inequalities

Core Concepts

The paper characterizes 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. This game-theoretic formulation allows for the derivation of sharp non-asymptotic prophet inequalities for different classes of distributions and provides a simple and computationally tractable algorithmic paradigm for determining optimal single threshold stopping rules.

Abstract

The paper considers an optimal stopping problem where the objective is to design stopping rules that attempt to select the random variable with the highest value in a sequence of independent and identically distributed (iid) random variables. The performance of any stopping rule is benchmarked relative to the selection of a "prophet" that has perfect foreknowledge of the largest value in the sequence. Such comparisons are typically stated in the form of "prophet inequalities".
The key contributions of the paper are:

Characterization of 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. This game-theoretic formulation allows for the derivation of sharp non-asymptotic prophet inequalities for different classes of distributions.

Demonstration that several classical observations in the literature regarding the sharpness of prophet inequalities for single threshold stopping rules are either incorrect or incomplete.

Development of a simple and computationally tractable algorithmic paradigm for deriving optimal single threshold stopping rules based on the game-theoretic formulation.

Establishment of two-sided bounds on the worst-case competitive ratio and regret of optimal single threshold stopping rules in terms of the optimal values of the associated finite matrix games.

The paper provides a unified and rigorous treatment of prophet inequalities for single threshold stopping rules, correcting and extending previous results in the literature.

Stats

Mn(F) = E max1≤t≤n Xt
V^*_n(Tsingle; F) = sup_θ≥0 [θ P(Mn(F) > θ) + (1 - F^(n-1)(θ)) ∫_θ^∞ (1 - F(x)) dx]
R_n(Tsingle; F) = V_n(Tsingle; F) / Mn(F)
A_n(Tsingle; F) = Mn(F) - V_n(Tsingle; F)

Quotes

"The 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."
"Prophet inequalities compare the optimal value of the stopping problem with the expected value of the maximal observation Mn(F), which is the performance of the 'prophet' with complete foresight."

Key Insights Distilled From

Optimal single threshold stopping rules and sharp prophet inequalities

by Alexander Go... at arxiv.org 04-22-2024

https://arxiv.org/pdf/2404.12949.pdf

Deeper Inquiries

How can the game-theoretic formulation be extended to more general classes of stopping rules beyond single threshold rules

The game-theoretic formulation developed in the paper can be extended to more general classes of stopping rules by considering a broader range of decision-making strategies. While the paper focuses on single threshold stopping rules, one can incorporate more complex stopping rules that involve multiple thresholds, randomized decisions, or other decision-making mechanisms. By expanding the set of possible stopping rules, the game-theoretic approach can provide insights into the optimal strategies for a wider range of scenarios. This extension may involve defining new pay-off functions, considering different constraints, or exploring alternative solution concepts in game theory to analyze the interactions between decision makers and nature in the stopping problem.

What are the implications of the results in this paper for the design of online auction mechanisms and other practical applications of optimal stopping theory

The results presented in the paper have significant implications for the design of online auction mechanisms and other practical applications of optimal stopping theory. By characterizing sharp prophet inequalities for single threshold stopping rules, the paper offers a systematic approach to evaluating the performance of stopping rules in sequential decision-making problems. This can be particularly valuable in online auctions where sellers need to determine optimal pricing strategies to maximize revenue while facing uncertainty about buyers' valuations. The game-theoretic framework developed in the paper can help in designing efficient auction mechanisms, setting appropriate reserve prices, and improving overall auction outcomes. Additionally, the insights from the paper can be applied to various other domains such as inventory management, resource allocation, and strategic decision-making in dynamic environments.

Are there connections between the game-theoretic approach developed here and other optimization techniques used in the analysis of stochastic processes, such as dynamic programming or Markov decision processes

There are indeed connections between the game-theoretic approach developed in the paper and other optimization techniques used in the analysis of stochastic processes. Dynamic programming, for example, is a powerful optimization method commonly used in sequential decision-making problems. The game-theoretic formulation can be seen as a specialized form of dynamic programming where the interactions between decision makers and nature are modeled as a two-person zero-sum game. By framing the stopping problem in this way, the optimal stopping rules can be derived by solving the game and finding the equilibrium strategies for both players. Similarly, Markov decision processes (MDPs) can also be related to the game-theoretic approach, as both involve modeling decision-making in a stochastic environment and finding optimal strategies to maximize rewards or minimize costs over time. By drawing parallels between these optimization techniques, researchers can leverage insights from game theory to enhance the analysis of stochastic processes and improve decision-making strategies in dynamic settings.