Prophet Inequalities with Cancellation Costs

2024-03-31

Symposium on the Theory of Computing

AI Summary

This paper studies the prophet inequality problem with linear buyback costs: an online decision maker may revoke previously accepted random variables, incurring a cost equal to $f$ times the revoked value; the objective is to maximize net reward—the sum of finally accepted values minus total buyback costs. For this novel “reversible yet costly” model, we establish the first tight optimal competitive ratio: $1 + f/(1+2f)$ for $f geq 1$, and nearly matching upper and lower bounds asymptotically tight at $1 - Theta(f log(1/f))$ for $0 < f < 1$. Our analysis unifies classical probabilistic arguments, LP duality, network flow theory, and cut-based techniques. This reveals fundamental trade-offs between buyback cost magnitude and decision reversibility in online optimization. The work provides the first systematic theoretical framework for sequential decision making with costly backtracking.

Technology Category

• Reasoning under Uncertainty: Sequential Decision Making
• Machine Learning: Online Learning & Bandits
• Search and Optimization: Learning to Search

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

Abstract

Most of the literature on online algorithms and sequential decision-making focuses on settings with “irrevocable decisions” where the algorithm’s decision upon arrival of the new input is set in stone and can never change in the future. One canonical example is the classic prophet inequality problem, where realizations of a sequence of independent random variables X1, X2,… with known distributions are drawn one by one and a decision maker decides when to stop and accept the arriving random variable, with the goal of maximizing the expected value of their pick. We consider “prophet inequalities with recourse” in the linear buyback cost setting, where after accepting a variable Xi, we can still discard Xi later and accept another variable Xj, at a buyback cost of f × Xi. The goal is to maximize the expected net reward, which is the value of the final accepted variable minus the total buyback cost. Our first main result is an optimal prophet inequality in the regime of f ≥ 1, where we prove that we can achieve an expected reward 1+f/1+2f times the expected offline optimum. The problem is still open for 0<f<1 and we give some partial results in this regime. In particular, as our second main result, we characterize the asymptotic behavior of the competitive ratio for small f and provide almost matching upper and lower bounds that show a factor of 1−Θ(flog(1/f)). Our results are obtained by two fundamentally different approaches: One is inspired by various proofs of the classical prophet inequality, while the second is based on combinatorial optimization techniques involving LP duality, flows, and cuts.

Problem

• Prophet inequalities with linear cancellation costs (buyback model).
• Maximize expected net reward with cancellations and costs.
• Design competitive online policies against optimal offline benchmark.

Innovation

• Introduces prophet inequalities with linear cancellation costs
• Optimal online policy via differential equation solution
• Order-agnostic randomized adaptive policy achieves optimal ratio