The Ski-Rental Problem

You're going on a ski trip of unknown length. Renting skis costs $1 per day; buying them outright costs $B once, and after buying you ski for free. If you knew in advance how many days you'd ski, the answer would be trivial: rent if the trip is shorter than $B$ days, buy if it's longer. The ski-rental problem asks what to do when you don't know the trip length in advance and must decide each morning, using only what has happened so far.

The standard solution: rent every day up until you've paid a cumulative $B$ in rental fees, then buy on that day. Compare this rule's cost against the best decision you could have made in hindsight, for every possible trip length. If the trip turns out shorter than $B$ days, you paid exactly what the hindsight-optimal renter would have paid — no loss. If the trip runs long, you paid $B$ in rental fees before buying, plus the purchase price $B$, for a total of $2B$ — exactly double what a hindsight buyer would have paid for the same trip, and this ratio of 2 is provably the best any strategy can guarantee against every possible trip length. This worst-case-to-best-case ratio is called the competitive ratio, and 2 is the minimum achievable competitive ratio for this problem with a deterministic strategy.

Interviewers use this puzzle to test comfort with decision-making under total uncertainty about the future, and it generalizes directly to real trading problems like deciding when to switch from a flexible short-term hedge into a fixed long-term one.