Short Explanation

The rating system used is based on the same logic used for KRACH (Ken's Ratings for American College Hockey), a rating system that dates back to mathematical research done by Zermelo in 1929 and Bradley and Terry in the 1950s. See the References section for mathematical research papers on which this method is based.

In short, each team has a rating based on two factors: win ratio (NOT winning percentage) and strength of schedule. The two are multiplied by each other to determine a rating.

Basic Goals

The foundation of the system is based on these basic principles:

• A win, no matter the opponent, should always improve a team's rating. (This is not the case in RPI, for example.)
• A loss, no matter the opponent, should always cause a team's rating to go down.
• Teams that play tougher schedules should be rewarded for that appropriately. Thus, an undefeated team should not necessarily be the best team in the country.

What the Ratings Mean

The ratings are designed in such a way that they are on an odds scale. A perfectly average (mean, not median) team will have a rating of 100--teams that are better than that will be higher while teams worse than that will be lower. The ratings are multiplicative in nature rather than additive. So, for example, in comparing teams with ratings of 600 and 400, if the teams were to compete head-to-head, it would be expected that the team with the higher rating would win 1.5 times as often as the team with the lower rating. This is because 600/400 = 1.5. Another way to translate this is that the probability that the team with the higher rating would win 600/(600+400) = 60% of the time.

Again, please see some of the References for a better explanation.

How It Works

Several of the websites in the References section explain this more thoroughly than I will here--please feel free to refer to them. Each team has a win ratio equal to the number of wins divided by the number of losses (ties are counted as half a win and half a loss). This win ratio is multiplied by the strength of schedule to determine a rating.

The strength of schedule is a weighted average of the ratings of a team's opponents. This is done in the following way:

1. Choose an opponent that you played a game against.
2. Find the number of times you competed against that opponent.
3. Divide that number by the sum of your rating and their rating to get a weighting factor.
4. Multiply that weighting factor by that opponent's rating.
5. Repeat steps 1 through 4 for all of your opponents.
6. Add all of these results together and divide it by the sum of the weighting factors.

Now you might be thinking that you need to know everyone's rating in order to calculate everyone's rating, and in a way, you're right. This process is recursive, but it has been shown to converge to a final, stable result. In our calculations, we start by giving every team a rating of 100, performing the calculations to determine each team's new rating, and repeating the process until the ratings stabilize. For our current system, we deem the ratings stable when the difference in each team's rating between consecutive iterations is less than 0.00001.

Special Circumstances

If you understand everything up to this point, you'll quickly see that undefeated and winless teams pose problems. Although it is not ideal, this is our current resolution, which appears to cause minimal effect on the ratings (though this has not yet been proven). In addition to all of the opponents a team actually faces, each team's rating is calculated by taking into account a tie game with a fictitious team with a rating of 100. This has the effect of interconnecting all of the teams, a requirement for this system. Although it is ideal for all of the teams to be fully interconnected naturally, considering that this is a national rating system, this is highly unlikely.

Final Comments

This system is in development, and I am very interested in your comments and recommendations. Please feel free to e-mail me (dbykowski@aiquizbowl.com) if you're interested in discussing this system in greater depth.

Further refinements of this page may be added in the future depending on demand.

References

Frequently Asked Questions about KRACH

John Whelan's deeper analysis of KRACH

One final explanation of KRACH

HTML version of Generalized Bradley-Terry Models and Multi-Class Probability Estimates

There are several other papers that are not online, such as:

E. Zermelo. Die berechnung der turnier-ergebnisse als ein maximumproblem der wahrscheinlichkeitsrechnung. Mathematische Zeitschrift, 29:436-460, 1929.

R. A. Bradley and M. Terry. The rank analysis of incomplete block designs: I. the method of paired comparisons. Biometrika, 39:324-345, 1952.

Jech, Thomas. 1983. The ranking of incomplete tournaments: A mathematician's guide to popular sports. American Mathematical Monthly 90(4) 246-266

Stob, Michael. 1984. A supplement to "A mathematician's guide to popular sports." American Mathematical Monthly 91(5) 277-282