*This is a guest post by tournament director Chris Schaffner.*

*For td-tuesdays articles: http://leaguevine.com/blog/tags/td-tuesdays/*

Posted on October 8th, 2012 by Mark "Spike" Liu

There are a number of advantages in using the innovative Swissdraw format compared to the more common pool format. All teams can potentially play each other. The Swissdraw format is designed so that teams of similar strength match up quickly and within a few rounds, the ranking of the teams represents their level of play. This guarantees attractive games against different opponents of comparable strength. While I am personally a big fan of the Swissdraw format, I will argue that the currently used system to rank teams has the problem that teams are awarded the same amount of Swiss points for a certain point differential, independent of the strength of the opponent. This drawback has particularly bad consequences in big divisions with a widespread level of play, where in later rounds of Swiss draw, teams can still make big jumps in the ranking by winning/losing by big margins. In this post, I would like to suggest another method of ranking the teams which will make the Swissdraw format work even better.

The basic assumption is that every team can be assigned a numerical value representing its

If there are more teams with many games played among them, it will become more difficult to assign strengths to the teams, but we can nevertheless try to

In more mathematical terms, we assume that the game outcome

There exist efficient methods (such as the least-square method) to compute strength vectors

The results of the first round are as follows:

resulting in the following strength values and swiss points:

PowerRank denotes the team's rank according to their strength, whereas SwissRank is the team's rank according to the amount of Swiss points earned so far.

All game outcomes can be perfectly explained with those strengths. After the first round (and assuming no prior knowledge of the strength of these teams), it is impossible to compare Team Alice with Team Bob, because there is no connection between them yet.

After a second round with the following results:

We can compute the following strength values, Swiss points and according ranks:

Notice that Team Charlie would be ranked first if sorted according to swiss points, as it had the largest marginal of

The seventh column (entitled "predicted margin") is the difference in current strength of the teams involved in a particular game which can be interpreted as the margin predicted by the strength. The values in the last columns are the squared differences of the actually observed and predicted margins. If such a value is high, the model could not predict this game outcome well. Hence, big values stand for surprising game outcomes.

The least-square procedure tries to find strength values that minimize the sum of the surprise values in the last column.

Playing one more round with results:

gives:

By now the teams are clearly separated in strength. However, notice that the Swiss points still do not reflect the strengths of the teams correctly. Sorting according to number of wins as first criterion (and Swiss points as second) would put Alice on first place (she is the only one with three wins), but it would still place Team Eve ahead of Team Danny (both have one win and two losses).

Let us examine the example graphically in the following chart. Clicking on series in the legend will toggle its visibility. Clicking on particular points in the chart will show detailed explanations how that strength was obtained.

For comparison, let us consider the graph of average Swiss points:

The Swiss score does not reflect the
correct order of the teams, neither after round 2 nor after round
3. In contrast, the power ranking "gets it right" already after two
rounds. The frequent crossings of the lines indicates that teams make
jumps in their placement as illustrated here: (click on series in the
legend to toggle visibility of the power ranks)

Here are the final evaluations of the games, based on the team's strength after Round 3, sorted by the upset/surprise value.

The first line can be read as follows: based on the strengths computed based on all game results, the biggest surprise of all games happened in the round-2 game, where Team Charlie won against Team Danny with a margin of

Equipped with all this knowledge you can dive into the power scores for Windmill Windup and Wisconsin Swiss provided in the Further Analysis section at the end.

Here is a list of pros and cons compared to the currently used swiss-point system:

- Your strength depends on the performance of your opponents. A large win against a strong opponent counts more than against a weak one.
- Converges faster to the "real" ranking, smaller jumps in rankings from one round to the next.
- Strengths say more about teams than swiss points (e.g. the difference in strength of two teams directly predicts the game outcome and point differential. This also allows us to get a sense of "surprising" results. This information might be of interest for spectators live at a tournament or when reporting about it afterwards.)

- Difficult to understand
- Games of previous rounds are "re-evaluated", you never have a certain amount of points for sure.
- Your strength depends on the performance of your opponents.

I am very curious to hear what you think about the suggested power-ratings in Ultimate. Do you see more advantages and disadvantages?

- Windmill Windup 2012 Open Division
- Windmill Windup 2012 Mixed Division
- Windmill Windup 2012 Women's Division
- Wisconsin Swiss 2011
- Wisconsin Swiss 2012

- Leake, R. J. (1976), "A Method for Ranking Teams with an Application
to 1974 College Football," Management Science in Sports, North Holland.

- Massey, Kenneth (1997), "Statistical Models Applied to the Rating of
Sports Teams", Honors Project: Mathematics, Bluefield College. http://masseyratings.com/theory/massey97.pdf

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