Bouldering shoes

Question: You go bouldering for the first time. At the bouldering hall is a sign that says: “Bouldering shoes: €5 to rent and €80 to purchase.”

You have no idea how often you will go bouldering. When should you switch from renting shoes to purchasing shoes to minimize the maximum regret?

Note: you may ignore shoe-depreciation and opportunity costs.

Answer: On the 16th visit.

Maximum regret is determined by the maximum amount you paid versus the minimum amount you could have paid. Maximum regret is often considered as price difference or price ratio. E.g. if you buy shoes the first time, and never go again your regret is €80/€5 = 16x or €80-€5=€75. If you switch from renting to buying on your 16th visit the maximum regret is at most 2x $(\frac{80+15\cdot5}{16\cdot5})$, and €75 $(15\cdot5)$.