In the Multislope Ski Rental problem, the user needs a certain

resource for some unknown period of time. To use the resource, the

user must subscribe to one of several options, each of which

consists of a one-time setup cost (``buying price''), and cost

proportional to the duration of the usage (``rental rate''). The

larger the price, the smaller the rent. The actual usage time is

determined by an adversary, and the goal of an algorithm is to

minimize the cost by choosing the best option at any point in time.

Multislope Ski Rental is a natural generalization of the classical

Ski Rental problem (where the only options are pure rent and pure

buy), which is one of the fundamental problems of online

computation. The Multislope Ski Rental problem is an abstraction

of many problems where online decisions cannot be modeled by just

two options, e.g., power management in systems which can be shut

down in parts. In this paper we study randomized algorithms for

Multislope Ski Rental. Our results include the best possible

online randomized strategy for any additive instance, where the

cost of switching from one option to another is the difference in

their buying prices; and an algorithm that produces an

$e$-competitive randomized strategy for any (non-additive)

instance.