How to determine who improved the most considering diminishing returns

Question

My friends and I are competing in a fitness challenge against each other. We will determine the winner based on who has the greatest percentage improvement after three months. Here is the problem: if someone can do $1$ pushup and they improve to $10$, that is a huge percent difference compared to someone that goes from $21$ to $30$, even though going from $21$ to $30$ is much harder. So is there a way to compare the percent improvements while taking into account the diminishing returns problem?

Answer 1

A percentage improvement is a percentage improvement.

Let's say, for argument's sake, that you're the fittest one. You can do 21 pushups, and your most out-of-shape friend can only do one.

For every extra pushup he can do at the end of the challenge period, he's experienced a 100% increase. In order to keep pace with that, you have to be able to do 21 more pushups for every one more that he does.

I wouldn't have agreed to this challenge if I were you, but, "the destructor has been chosen."

In essence, you're looking for a way to explain why you don't have to improve as much on a percentage basis as he does. The simplest way, in my book, would have been to mutually agree on a goal of $n_{later}$ pushups to each participant, where $n_{later} > n_{now}$, and then choose the winner based on how far they exceed their goal (in number of pushups).

But, that should have been negotiated before the challenge began, not during. :)