Maybe the relatable concept is just a stepladder to the general ongoing scenario, eg. modeling all consumers from a retailer’s perspective. Otherwise, the continuous to discrete assumption reads as a hand-wavy fiat.

Could someone who groks this math tell me why not buy the skis once you’ve paid half their price on rentals?

Another aspect of the solution that makes it rather abstract is it effectively assumes we know nothing about the distribution of the number of days.

Paying at 1/2 will be optimal if it ends before you buy, very bad (3x optimal) if it ends right after you buy, and slightly better than the solution in the post if it lasts at least twice that long (1.5x optimal vs e/(e-1)).

The metric in the post is just the worst of those ratios. Assuming the unproven statement in the post (that the solution which is a constant factor worse than optimal is best), any solution of the form you suggest is going to have similar tradeoffs. If we had a distribution, we could choose.

Why stop there and instead buy when 33% of the equipment cost is spent on renting?