## Philosophy

The algorithm that we have implemented is derived from the Elo rating system. The Elo system was originally developed to provide a relative ranking system for two-player games such as chess, but it has been generalized and applied to many other sports including college football, basketball and Major League Baseball.

The basic assumption of the Elo system is that the performance of any individual (or team) is a normally distributed (Gaussian) random variable. And, while any individual performance might differ significantly from one game to the next, the mean value of these performances would be relatively constant. Thus, the true skill would be best represented by that mean value. Further statistical studies have shown that rather than using the normal distribution, individual performances are better represented by the closely related logistic distribution. As such, we have chosen to use the logistic distribution for the FTS algorithm.

In creating the power ratings, we had a few arbitrary choices that relate to how the power ratings would be presented. We chose the range such that most teams would have a power rating between 500 – 1000. While there is nothing inherently restricting the possible values of the power ratings from going higher or lower, the system feeds back on itself in such a way that it becomes increasingly difficult for a team to stray much beyond this range.

## The Scoring Metric

It is necessary to provide a measurable way of judging the outcome of a bout. Simply using a win or loss does not give enough fidelity to accurately gauge the strength of the performance. We instinctively use the score, more specifically the difference in the score, to measure a team’s strength (i.e. if Team A beat Team B by 250 points while Team C only beat Team B by 100 points, then Team A must be stronger than Team C). However, this overly rewards teams with a high-output offense while negatively valuing teams with a good defense. Consider, for example, Team A beats Team B by the score of 320 to 205 while Team C beats Team B by the score of 95 to 3. By score difference alone, Team A would be considered the stronger team. However, it could well be argued that Team C had more dominant control of their game, scoring at will and keeping Team B from scoring points. To account for this, we use a Difference-over-Sum (DoS) method to provide a normalized ratio of the scores that is independent of the overall scoring:

The DoS spans the range between +1 (a dominant win by Team A) and -1 (a dominant win by Team B), with 0 representing a tie between the two. In the previous example, the DoS for Teams A and B would be 0.22 while the DoS for Teams C and B would be a much more dominant 0.94. Throughout our testing, we have found that the DoS is a very reliable gauge of a team’s strength and we have adopted it as the primary metric through which predictions and bout results are compared.

## The Prediction Algorithm

The standard Logistic Cumulative Distribution Function (CDF) provides a value between zero and 1 that is symmetric around zero. To match the values of the DoS, which range from -1 to +1, and to allow flexibility in the implementation of the power rating values, we have generalized the Logistic CDF as follows:

where RB and RA refer to the power ratings of Teams B and A respectively and provides a mechanism to adjust for any biases which may exist between the two teams. The CDF thus provides a prediction of the DoS between any two teams based on their relative rankings as shown in the graph below.