Prophet Inequalities with Cancellation Costs

In Proceedings of the 56th Annual ACM Symposium on Theory of Computing (STOC '24)

Chicago Booth Research Paper No. 24-15

42 Pages Posted: 22 Apr 2024 Last revised: 29 Apr 2024

See all articles by Farbod Ekbatani

Farbod Ekbatani

University of Chicago - Booth School of Business

Rad Niazadeh

University of Chicago - Booth School of Business

Pranav Nuti

Stanford University

Jan Vondrák

Stanford University

Date Written: March 31, 2024

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 X_1, X_2, ... 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 X_i, we can still discard X_i later and accept another variable X_j, at a \textit{buyback cost} of fX_i. 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(f log(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.

Keywords: Online Algorithms, Prophet Inequalities, Buyback

Suggested Citation

Ekbatani, Farbod and Niazadeh, Rad and Nuti, Pranav and Vondrák, Jan, Prophet Inequalities with Cancellation Costs (March 31, 2024). In Proceedings of the 56th Annual ACM Symposium on Theory of Computing (STOC '24), Chicago Booth Research Paper No. 24-15, Available at SSRN: https://ssrn.com/abstract=4779633

Farbod Ekbatani (Contact Author)

University of Chicago - Booth School of Business ( email )

5807 S. Woodlawn Avenue
Chicago, IL 60637
United States

Rad Niazadeh

University of Chicago - Booth School of Business ( email )

5807 S Woodlawn Ave
Chicago, IL 60637

HOME PAGE: http://https://faculty.chicagobooth.edu/rad-niazadeh

Pranav Nuti

Stanford University ( email )

450 Jane Stanford Way
Stanford, CA 94305

Jan Vondrák

Stanford University ( email )

Stanford, CA 94305
United States

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