Exponential Curved Graph Function

This is probably more of a math question than Excel, but can someone figure out a mathematical equation that will give me a curved graph based on the following requirements (see attached sample file).

Nutshell: A team of 20 people are ranked (X-axis) based on their individual performance. They each get a percentage share of a cash prize (Y-axis). How much each member wins compared to the others is not linear but curved. The highest ranked members get a bigger piece of the pie, proportionally, than the lower ones.

I need to be able to experiment with different degrees of curve and different numbers of contestants. The sum of all members' percent shares must equal 100%.