Controlling Tail Risk in Online Ski-Rental

TL;DR

This paper investigates how imposing tail risk constraints affects the optimal strategies in the classic ski-rental problem, revealing significant structural changes in the solution when limiting the probability of high competitive ratios.

Contribution

It introduces a new perspective on ski-rental strategies by incorporating tail risk constraints, showing how these constraints alter the optimal solution's structure.

Findings

Tail risk constraints lead to non-monotone, discontinuous strategies.

The probability of purchasing skis can become arbitrarily large under small tail risk.

Structural properties of optimal solutions are significantly affected by tail risk bounds.

Abstract

The classical ski-rental problem admits a textbook 2-competitive deterministic algorithm, and a simple randomized algorithm that is $\frac{e}{e-1}$-competitive in expectation. The randomized algorithm, while optimal in expectation, has a large variance in its performance: it has more than a 37% chance of competitive ratio exceeding 2, and a $\Theta(1/n)$ chance of the competitive ratio exceeding $n$! We ask what happens to the optimal solution if we insist that the tail risk, i.e., the chance of the competitive ratio exceeding a specific value, is bounded by some constant $\delta$. We find that this additional modification significantly changes the structure of the optimal solution. The probability of purchasing skis on a given day becomes non-monotone, discontinuous, and arbitrarily large (for sufficiently small tail risk $\delta$ and large purchase cost $n$).