Controlling tail risk in two-slope ski rental

2025-08-08

AI Summary

This paper studies the two-slope ski-rental problem with tail-risk control: optimizing the rent-or-buy decision under the constraint that the probability of the competitive ratio exceeding threshold γ is at most δ. The model generalizes the classical ski-rental problem to capture realistic “rent-then-buy” scenarios where a one-time purchase reduces—but does not eliminate—the per-unit-time cost thereafter. Methodologically, we propose a greedy + binary search-based efficient approximation algorithm and an exact algorithm leveraging linear programming and structural characterization. Theoretically, we reveal nonstandard optimal policy structures under tail-risk constraints: existence of pure strategies that indefinitely defer purchase, and mixed strategies requiring randomized purchase at finite time points with positive probability; we prove non-uniqueness of optimal solutions across multiple regions and establish a complete structural characterization theorem. Experiments confirm near-optimal performance across diverse parameter regimes and expose risk-sensitive behavioral patterns overlooked by classical models.

Technology Category

• Search and Optimization: Mixed Discrete/Continuous Search
• Reasoning under Uncertainty: Stochastic Optimization
• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization

Application Category

• Economics, Online Markets and Human Computation: The sharing economy
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

Abstract

We study the optimal solution to a general two-slope ski rental problem with a tail risk, i.e., the chance of the competitive ratio exceeding a value $γ$ is bounded by $δ$. This extends the recent study of tail bounds for ski rental by [Dinitz et al. SODA 2024] to the two-slope version defined by [Lotker et al. IPL 2008]. In this version, even after "buying," we must still pay a rental cost at each time step, though it is lower after buying. This models many real-world "rent-or-buy" scenarios where a one-time investment decreases (but does not eliminate) the per-time cost. Despite this being a simple extension of the classical problem, we find that adding tail risk bounds creates a fundamentally different solution structure. For example, in our setting there is a possibility that we never buy in an optimal solution (which can also occur without tail bounds), but more strangely (and unlike the case without tail bounds or the classical case with tail bounds) we also show that the optimal solution might need to have nontrivial probabilities of buying even at finite points beyond the time corresponding to the buying cost. Moreover, in many regimes there does not exist a unique optimal solution. As our first contribution, we develop a series of structure theorems to characterize some features of optimal solutions. The complex structure of optimal solutions makes it more difficult to develop an algorithm to compute such a solution. As our second contribution, we utilize our structure theorems to design two algorithms: one based on a greedy algorithm combined with binary search that is fast but yields arbitrarily close to optimal solutions, and a slower algorithm based on linear programming which computes exact optimal solutions.

Problem

• Optimizing two-slope ski rental with tail risk constraints
• Characterizing complex solution structures under risk bounds
• Designing algorithms for near-optimal and exact solutions

Innovation

• Extends two-slope ski rental with tail risk bounds
• Develops structure theorems for optimal solutions
• Designs greedy and LP-based solution algorithms