The core of my formula is n/r, the number of contestants divided by a contestant's rank. For various reasons, a fraction need to subtracted from r. The logical choice is 1/2, but you can play with other fractions. So we have n/(r-.5).
This needs to be run through a nonlinear function. I use the log formula. So we now have log(n/(r-.5)).
This is multiplied by some fraction, to control how large the awards are. We have k * log(n/(r-.5)). The value of k is decided last, to make the awards conform to the size of previous awards.
There is one other choice to make. (r-.5) can be multiplied by a number. This influences how quickly the awards drop off after the first-place award. (r - .5) looks fine to me for club games, where 40% of the participants win awards. For tournaments, roughly 10% of the contestants win awards, and the actual awards look better if (r - .5) is multiplied by 2.
I propose using
where j is 1 for club games and 2 for tournaments.