This article is part of a series on designing a tournament:
Part 1 - Designing a Tournament
Part 2 - A grading system
Part 3 - Calculation of Win Percentages (this article)
One holdup in determining a grade for different tournament system is to settle on an algorithm to give consistent win percentages that scale with the difference in true skill of the competitors. Initially I had used a 100% win ratio to the higher ranked player for determining the inherent bias in a tournament system so that the result could be clear of as many other biases as possible, however this precludes tournaments that attempted to add more games to a match for bias reduction. Without some chance for the lower ranked player to overcome a higher ranked player (even a slim chance) there would be no need to play best of 3 matches, best of 5 matches and the like for the better player to prove their worth.
Ideally I'd like to come up with a system that gives a win chance similar to a standard deviation bell curve or some sort of sigmoid function that granted ever higher win chances the greater the skill difference was between players. As it turns out, the bell curve presented as a cumulative chance for success also appears as a sigmoid. Also, the idea of having ranks of importance to also determine the relative skill differences appeals as the difference between 2nd & 5th should be far more significant than the difference between 32nd and 35th. Even though there may not be a bell curve of skill displayed at the tournament, it would be prudent to assume that we have the top end of the bell curve in attendance.
After a couple of attempts at mapping the ranks of importance onto units of standard deviation, another mathematical nicety presented itself: The inverse ratio of each player's true skill produces the same ratio between ranks: 2:1. Eg: 2nd in true skill play 4th in true skill (1 rank apart) gives 4:2 => 2:1. 8th plays 16th (1 rank apart) gives 16:8 => 2:1. 2nd plays against 8th (2 ranks apart) gives 8:2 => 4:1. 4th plays 16th (2 ranks apart) gives 16:4 => 4:1.
This inverse ratio means a difference of one rank has 66% chance to win over their opponent, a difference of 2 ranks gives 75% chance to win, 3 ranks gives 88.5% chance, etc. Not quite as steep as the standard deviation steps, but follows along the same shape as cumulative chance for success. This should fit quite nicely for now.