Abstract
Skis can be rented for $1$ a day or bought for $B$. How many days one will ski is not known until the skiing is over. The amount paid is compared with the least one could have paid knowing the number of days (renting throughout if under $B$ days, buying if $B$ or more), and the worst value of that ratio is called the worst loss. No new theorem or law is claimed.
First, buying on a fixed day, the worst loss cannot go below $2-1/B$ times——for $B=10$, trying every number of rental days before buying from $0$ to $59$, the worst loss was $10$ times for $0$ days, $2.5$ for $5$, the minimum $1.9$ for $9$, $2$ for $10$, and $4$ for $30$ (Section 2).
Second, leaving the day of purchase to chance lowers the worst loss——for $B=10$, choosing the day of purchase among days $1$ to $10$ with probabilities from $0.059482$ to $0.153534$, heavier for later days, gives a worst loss of $\mathbf{1.535340}$ times. For $B=2$ it is $1.333333$, for $B=3$ $1.421053$, for $B=100$ $1.577368$, always below the fixed-day values $1.5$, $1.666667$ and $1.99$ (Section 3).
Third, and this is the core. The separator is whether the opponent knows one’s choice——buying on a fixed day, the opponent can end the skiing exactly on that day and make the loss largest. Left to chance, the opponent knows only the probabilities, and whatever number of days is chosen the loss ratio is the same $1.535340$ (within $10^{-12}$ for every day from $1$ to $10$) (Section 3).
Fourth, the formula and the search agree——the formula $1/(1-(1-1/B)^B)$ for the worst loss agreed within $10^{-9}$ with the value obtained by evaluating the probabilities over all numbers of days for $B=2,3,5,10,100$. For $B=2$ and $3$, searching the probabilities on a grid found no worst loss below the formula (Section 4).
Fifth, for large $B$ it approaches $e/(e-1)$——because $(1-1/B)^B$ approaches $1/e$; in the grouping of the sources of $1/e$ made by Paper 479, it belongs to the second group, the limit from binomial to Poisson (Section 4).
Keywords: ski rental problem, competitive ratio, online algorithm, randomised algorithm, e/(e-1).
Scope (what is not claimed)
No new theorem or law is claimed——that the fixed-day method gives $2-1/B$ times in the rent-or-buy problem (Karlin and coauthors, $1988$), that leaving the day to chance approaches $e/(e-1)$ times (Karlin and coauthors, $1994$), and measuring methods by their worst ratio against an opponent (Borodin and El-Yaniv, $1998$) are all known. The case where the distribution of skiing days is known is not treated——only the worst case, in which the opponent chooses the number of days, is looked at. No actual purchase is recommended——skis are an example of a decision whose future is unknown. Relation to earlier papers: Paper 388 showed that in picking the best of candidates seen one by one (the secretary problem), the chance of success settles at $1/e$——this is also a decision with an unknown future, but what is measured is not the chance of success but the ratio to the best in hindsight. Paper 479 sorted the seven appearances of $1/e$ in the corpus into three groups by their source——the $e/(e-1)$ here comes from the limit of $(1-1/B)^B$, so it belongs to the second group. What is added is trying the fixed-day method on every day to confirm the best, matching the worst loss of the chance method by the formula, evaluation over all numbers of days, and a grid search, and placing the separator on whether the opponent knows one’s choice, showing that under chance the loss ratio is the same whatever the number of days.
1. Notation and the Question
Setting: renting costs $1$ a day and buying costs $B$ (an integer). The number of skiing days $d$ is not known until the skiing ends. The least one could pay knowing the number of days is $\min(d,B)$. With $C(d)$ the amount a method pays, define the worst loss as
$$R=\max_{d\ge1}\frac{C(d)}{\min(d,B)} .$$The fixed-day method is “rent $k$ days, and buy if skiing on day $k+1$”; the chance method chooses $j$ in “buy on day $j$” with some probabilities.
Question: how far can the worst loss be lowered, and what differs between a fixed day and chance?
2. Buying on a Fixed Day
Computation 2.1: for $B=10$, the worst loss of renting $k$ days before buying was found by evaluating $d=1$ to $400$ days.
| rental days $k$ | worst loss |
|---|---|
| $0$ (buy on the first day) | $10$ times |
| $5$ | $2.5$ times |
| $8$ | $2$ times |
| $9$ | $1.9$ times |
| $10$ | $2$ times |
| $15$ | $2.5$ times |
| $30$ | $4$ times |
Trying every $k$ from $0$ to $59$, the smallest worst loss was $1.9$ times at $k=9$, equal to $2-1/B$.
Observation 2.2 against a fixed day, the opponent ends the skiing right after the purchase. With $k=9$, if skiing ends just after buying on day $10$, one has paid $9+10=19$ against a best of $10$: $1.9$ times——buying early lets the opponent end after few days; buying late piles up rent. The balance is at $B-1$ days, and even there nearly twice cannot be avoided.
3. Leaving It to Chance
Computation 3.1: the probability of buying on day $j$ was made proportional to $q^{B-j}$ with $q=(B-1)/B$ ($j=1,\dots,B$), and the worst loss was evaluated for $d=1$ to $400$ days.
| $B$ | worst loss by chance | formula $1/(1-(1-1/B)^B)$ | best fixed day $2-1/B$ |
|---|---|---|---|
| $2$ | $1.333333$ | $1.333333$ | $1.5$ |
| $3$ | $1.421053$ | $1.421053$ | $1.666667$ |
| $5$ | $1.487387$ | $1.487387$ | $1.8$ |
| $10$ | $\mathbf{1.535340}$ | $1.535340$ | $1.9$ |
| $100$ | $1.577368$ | $1.577368$ | $1.99$ |
For $B=10$ the probabilities from day $1$ on are $0.059482,0.066091,0.073435,0.081594,0.090660,0.100734,0.111926,0.124363,0.138181,0.153534$, summing to $1$.
Computation 3.2: for the chance method with $B=10$, the loss ratio (expected payment over the best) for each number of days $d=1$ to $10$ was $1.535340$ every time, with differences within $10^{-12}$.
Observation 3.3 this is the core: the separator is whether the opponent knows one’s choice. Buying on a fixed day, the opponent knows the day and ends the skiing right after the purchase——left to chance, the opponent knows only the probabilities, and whatever number of days is chosen the loss ratio is the same. The probabilities are placed so that wherever the opponent strikes, the loss is no larger.
4. Formula and Search
Computation 4.1: independently of the formula, for $B=2$ and $3$ every probability on a grid of step $1/300$ was tried, looking for the smallest worst loss.
| $B$ | smallest on the grid | probabilities there | formula |
|---|---|---|---|
| $2$ | $1.333333$ | $0.3333,0.6667$ | $1.333333$ |
| $3$ | $1.421667$ | $0.21,0.3167,0.4733$ | $1.421053$ |
Both were within $0.01$ of the formula, and no value below the formula was found.
Observation 4.2 for large $B$ it approaches $e/(e-1)=1.581977$. Because $(1-1/B)^B$ in the formula approaches $1/e$; at $B=100$ it has reached $1.577368$——for finite $B$ it stays below the limit; $1.535340$ at $B=10$ is $0.046637$ below it.
Observation 4.3 beside Paper 479 and Paper 388. Paper 479 sorted the seven appearances of $1/e$ in the corpus into a group from a stationarity condition, a group from the limit from binomial to Poisson, and one from how a function was written down. Paper 388 showed that the chance of success in the secretary problem settles at $1/e$——the $1/e$ here comes from the limit of $(1-1/B)^B$ and so belongs to the second group of Paper 479. And as Paper 479 saw for the secretary problem, at finite $B$ the value departs from its limit.
Observation 4.4 what is not said. The best way to buy when the distribution of skiing days is known is not treated, and no actual purchase is recommended——what is shown is the best fixed-day method, the worst loss of the chance method, and what separates the two.
5. Roll Call and Closing
- the best fixed day rents $9$ days before buying (Computation 2.1): $1.9$ times $=2-1/B$.
- buying on day one gives $10$ times, renting $30$ days $4$ times (Computation 2.1): too early or too late both lose.
- the opponent ends the skiing right after the purchase (Observation 2.2): $19$ against $10$.
- chance gives $\mathbf{1.535340}$ times for $B=10$ (Computation 3.1): below $1.9$.
- chance is lower for every $B$ (Computation 3.1): $1.333333$ against $1.5$ for $B=2$.
- the same loss ratio for every number of days (Computation 3.2): within $10^{-12}$.
- the separator is whether the opponent knows one’s choice (Observation 3.3): the core.
- grid search finds nothing below the formula (Computation 4.1): two routes.
- for large $B$, $e/(e-1)=1.581977$ (Observation 4.2): $1.577368$ at $B=100$.
- $1/e$ comes from the limit of $(1-1/B)^B$ (Observation 4.3): the second group of Paper 479.
Closing: One decides whether to rent skis for $1$ a day or buy them for $B$ without knowing how many days one will ski. Buying on a fixed day, the worst loss cannot go below $2-1/B$ times——trying every day for $B=10$, renting $9$ days before buying is best at $1.9$ times, and the opponent strikes by ending the skiing right after the purchase. Leaving the day of purchase to chance lowers the worst loss——choosing the day with probabilities heavier for later days gives $1.535340$ times for $B=10$, and the loss ratio was the same whatever number of days the opponent chose. The formula, evaluation over all days and a grid search agreed. The separator is whether the opponent knows one’s choice——a fixed day is known; left to chance, only the probabilities are known. Placed among the earlier papers——the approach to $e/(e-1)$ for large $B$ comes from $(1-1/B)^B$ approaching $1/e$, placing it in the second group of Paper 479, and it is a decision with an unknown future like the secretary problem of Paper 388. To be honest——the case where the distribution of skiing days is known is not treated; only the case where the opponent chooses the worst number of days is looked at.
References
- Borodin, A., & El-Yaniv, R. (1998). Online Computation and Competitive Analysis. Cambridge University Press.
- Karlin, A. R., Manasse, M. S., Rudolph, L., & Sleator, D. D. (1988). Competitive snoopy caching. Algorithmica.
- Karlin, A. R., Manasse, M. S., McGeoch, L. A., & Owicki, S. (1994). Competitive randomized algorithms for nonuniform problems. Algorithmica.
- Yamagishi, Y. (2026). With a Million Candidates the Chance of Picking the Best Stays at $0.3679$ [Paper 388]. Zenodo. 10.5281/zenodo.22699147
- Yamagishi, Y. (2026). The Same Number Appears Seven Times and Splits into Two Groups and One [Paper 479]. Zenodo. 10.5281/zenodo.22733844
Reference
Yamagishi, Y. (2026). The Worst Loss in Renting or Buying Shrinks When the Day of Purchase Is Left to Chance [Paper 610]. YOI: https://www.yamagishiyuuki.com/yoi/610
DOI: 10.5281/zenodo.23014466
ORCID: 0009-0006-8587-3450
PDF (English) · PDF (Japanese) · Date proof of the English PDF (OpenTimestamps .ots) · SHA256SUMS
The record page for this paper (in Japanese, with the PDFs in each language and the date proof): YOI 610. This paper is a preprint and has not been peer reviewed.