Controlling Tail Risk in Two-Slope Ski Rental

Qiming Cui, Michael Dinitz

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 specific value \(\gamma \) is bounded by a constant \(\delta \). This extends the recent study of tail bounds for ski rental initiated 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, but the rental cost 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.

External IDs:dblp:conf/waoa/CuiD25