### Zermelo Incomplete Team Tournament Scoring

Thank you for showing interest in my incomplete team tournaments scoring method.

#### NEW! Paper published in The Mathematical Scientist!

On this page you can find the program I use to calculate team tournaments in my local club (in German only so far). It can be downloaded here. A detailed description of the input file used if input from file is chosen can be found here

This method is a way to calculate fair rankings in a team tournament when not all matches have been played. Basically teams are compensated for the strength of opposition they received.

This method is especially useful for time-limited tournament like for example a night at the club. If you have between 6 and 10 teams, as many club tournaments have, you have to play very short matches in an all-play-all, and if you play a 4-round Swiss you will have a huge emphasis on who plays who when, and what happens in the last round.

### Example tournament

At a club night we had 8 teams and we played 4 rounds of 7 boards. The final VP table like this:
```           4    8    7    1    2    9    6    3    5      VP
______________________________________________
Team  4 |  *        16   25                  25   21   |  87
Team  8 |       *   15        17        16        25   |  73
Team  7 | 14   15    *   14        22                  |  65
Team  1 |  4        16    *   20        23             |  63
Team  2 |                10    *             22   13   |  58
Team  6 |            8              *   16    9   20   |  53
Team  9 |      14         7        14   *    17        |  52
Team  3 |  5                   8   21   13    *        |  47
Team  5 |  9    2             17   10              *   |  38
```
It is clear that not all teams had equally strong opposition. The Zermelo valuation compensates for this and calculates the most likely results for the matches that did not take place and then adjusts the final VP table according to these results. Although there is no certainty that those matches would actually have these results one should see it as compensation for playing against strong or weak opponents.

The corrected VP table looks like this:

```
VP
4    7    1    8    2    9    6    3    5     before  after
______________________________________________
Team  4 |  *   16   25   19.3 21.5 21.7 22.1 25   21   |   87    85.8
Team  7 | 14    *   14   15   19.2 19.5 22   20.2 20.7 |   65    72.3
Team  1 |  4   16    *   16.4 20   23   20.2 20.3 20.8 |   63    70.3
Team  8 | 10.4 15   13.4  *   17   16   19.0 19.1 25   |   73    67.5
Team  2 |  7.8 10.4 10   13    *   15.2 16.0 22   13   |   58    53.7
Team  9 |  7.6 10.2  7   14   14.6  *   14   17   16.6 |   52    50.5
Team  6 |  7.1  8    9.4 10.7 13.9 16    *    9   20   |   53    47.0
Team  3 |  5    9.4  9.3 10.6  8   13   21    *   15.6 |   47    46.0
Team  5 |  9    8.8  8.7  2   17   13.3 10   14.2  *   |   38    41.5
```
Team 8 did not win any match against reasonable opponents with more than 17 VP and got to second place by beating last-place Team 5 with 25 VP. After the compensation teams 1 and 7, who had to work harder for their VPs, have moved up.

### Mathematical strategy

From the set of played matches, each team is assigned a strength si which is determined by an iterative process and yields one unique solution as proven in the paper above. From these strengths all the expected results for the missing matches are calculated using the statistical observations by John Manning (described in this mirror of his site. The final result the sum of the VP over all matches, real and virtual, adjusted for the actual number of played matches.

### Other applications

#### Large team tournaments

In longer tournaments with a large number of teams, a Swiss tournament is usually decided in the last round. This is because the possible deviation from the average result for a team (win with 25 VP) is much more than the average deviation from the average win. This means that even if you win your last match 17 - 13, you may get thrown off a vital qualification spot by teams who happen to be lucky in some way or another in the last round. An example of this is the Transnational Teams as was played in Estoril in 2005.
##### Swissing it in Estoril
In Estoril there were of course some examples of teams profiting and suffering because of the Swiss system. First the two teams that lost out: Team Lantaron was 6th in the Swiss before the last round and won their last match 16 - 14. However because three lower teams won by larger margins, they dropped 3 spots and finished 9th where 8 qualifying spots were available. Team Gotard was another last round victim. Having been in the top 8 since round 5 they played all the strong teams, but finally lost in the last round, dropping from 3rd to 10th.

Which teams profited from this? Team China Open finished 7th after being 84th after round 8 of 15. From that point it went upwards but with a similar VP score as the two teams I mentioned previously, they had much easier opponents. The same is true for the team that finished the Swiss in 8th, Team Spector. They were 42nd after 11 rounds of 15, and won the last four matches after that.

Of course these teams deserved their spot in the playoffs under the actual scoring method, but it does feel unfair. In the Swiss movement each next match is more important than the previous one. On the other hand, using the Zermelo method you will get the correct compensation for the strength of your opponents. To see the final results from the Transnational Round Robin, compensated for the opponent's playing strength, have a look at the Zermelo adjusted Round Robin results. Note that in these adjusted rankings, the Swiss pairings ensures that strong teams play more often against strong teams and weak teams play more often against weak teams. As such the distance in VPs over the whole field is larger than the "real" VPs.

#### Zermelo Swiss Teams

Let's define a new kind of movement: Swiss teams based on Zermelo strengths. The next pairings are not decided by the current VP standings of the teams but by the current Zermelo playing strengths. Playing 15 rounds of a combined Zermelo Swiss will ensure that every team gets enough chances to play against teams of similar strength and this method is therefore better suited to find the best EIGHT teams in the field, rather than Swiss that is best suited in finding the best ONE team in the field (and will often fail to do that). More details on the Zermelo Swiss page.

#### Teams refusing to play

In the 2004 European Team Championship, the Ladies' team of Lebanon refused to play against Israel. Regardless of your personal opinion on this, the problem is how to score this. The normal way would be to score it Israel 18 - Lebanon 0, but during the tournament it became clear that this would not be fair for Israel, given that many teams of similar strength beat Lebanon by a larger margin. However to make this precise is something else.

It was decided (see Daily Bulletin 11, page 16) that Israel get the average result against Lebanon of the eight teams nearest to them in the ranking or 18 VP, whichever is the greater. Why eight teams? With the Zermelo incomplete tournament scoring method you will not choose some random information to calculate the expected result of the missing match, but include all matches played in the tournament, surely the fairer method. I also agree that Israel should not have gotten less than 18.0 VP, regardless of the expected match result.

The Zermelo method finds an expected match result of 20.6 VP for Israel, which were a full VP away from the actual results used by the committee (21.6 VP, but surprisingly this was then scored as 21 VP for some reason). Given the close results, this decision could have been decisive for the fifth qualifying spot, although it turned out it wasn't.

(c) Gerben Dirksen, 2005 - 2006