Prophet Inequalities with Cancellation Costs

TL;DR

This work extends the prophet inequality framework to online decisions with cancellations and linear buyback costs. The authors derive the optimal online performance for all buyback factors $f\ge 0$ by solving a nonlinear differential equation linked to a continuum flow representation, showing the optimal competitive ratio is $\alpha(f)=\frac{1}{2-y_f(1)}$. They also design a simple, polynomial-time, order-agnostic randomized policy that attains this optimum, using a scale-invariant Poisson point process to simulate quantile acceptance and carefully calibrated thresholds. The analysis hinges on a chain of reductions to monotone two-point distributions, factor-revealing LPs, a generalized flow with leakage, and a differential-equation embedding; worst-case instances are constructed from the same $y_f$ solution to demonstrate tightness. Practically, cancellations improve online revenue management across cloud markets, ads, and hotel bookings, and the order-agnostic policy provides a robust, implementable approach that leverages distributional knowledge without requiring arrival order.

Abstract

Most of the literature on online algorithms in revenue management focuses on settings with irrevocable decisions, where once a decision is made upon the arrival of a new input, it cannot be canceled later. Motivated by modern applications -- such as cloud spot markets, selling banner ads, or online hotel booking -- we introduce and study "prophet inequalities with cancellations" under linear cancellation costs (known as the buyback model). In the classic prophet inequality problem, a sequence of independent random variables $X_1, X_2, \ldots$ with known distributions is revealed one by one, and a decision maker must decide when to stop and accept the current variable in order to maximize the expected value of their choice. In our model, after accepting $X_j$, one may later discard $X_j$ and accept another $X_i$ at a cost of $f \times X_j$, where $f\geq 0$ is a parameter. The goal is to maximize the expected net reward: the value of the final accepted variable minus the total cancellation cost. We aim to design online policies that are competitive against the optimal offline benchmark. Our first main result is an optimal prophet inequality for all parameters $f \ge 0$. We fully characterize the worst-case competitive ratio of the optimal online policy against the optimal offline via the solution to a certain differential equation (for which we provide a constructive solution). Our second main result is to design and analyze a simple and polynomial-time randomized adaptive policy that achieves this optimal competitive ratio. Importantly, our policy is order-agnostic (à la [Samuel-Cahn, 1984]), as it only needs the set of distributions and not their arrival order. These results are obtained by novel techniques related to factor-revealing LPs and generalized flow, reductions to a differential equation, and embedding of problem instances into specific Poisson point processes.

Figures (5)

• Figure 1: Comparison of different competitive ratio bounds. The blue curve is the optimal competitive ratio (this paper), the red curve is the classic 0.5 bound samuel1984comparison and the black curves are the optimal bounds in the adversarial setting for deterministic policies babaioff2008selling and for general randomized policies ashwinkumar2009randomized.
• Figure 2: Interpreting \ref{['eq:discrete-LP']} as a generalized flow; the red arrow indicates "lost" flow or the leakage (due to the buyback).
• Figure :
• Figure EC.1: Different possible outcomes of the construction, for different values of $\theta$
• Figure EC.2: The results of the numerical experiments for different values of $f\in\{0.1,0.2,0.3,0.5,1,2,3,5,7,10\}$. The orange lines specify the median while the green triangles identify the mean performance ratio of the policy across 100 sampled instances.

Theorems & Definitions (39)

• Proposition 1
• Lemma 1
• Lemma 2
• Lemma 3
• Lemma 4
• Lemma 5
• Proposition 2
• Definition 1: Y-Function Differential Equation
• Theorem 1: Existence of Proper Y-Function
• Theorem 2: Optimal Competitive Ratio -- Lower bound