# British Skat Association Ratings

## Criteria

I have tried to devise a system that satisfies the following criteria:
• Ratings should reflect playing strength as measured by results of BSkA Synchron tournaments.
• Recent results should have more weight than old results.
• Larger tournaments, in terms of number of players or number of deals played, should have more weight than smaller tournaments.
• Players should be rewarded to some extent for playing frequently and regularly. In particular, a player who achieves a high rank should not be able to keep it indefinitely without continuing to play well.
• A strong player who is new to BSkA tournaments should be able to achieve a higher rating than a well established average player within a reasonable time - say after achieving excellent results in two or three consecutive tournaments.

## Proposed system

A system that satisfies all the above requirements is inevitably rather complicated. Here is my suggestion.

A player's rating is a number from 0 to 100. It represents the number of match points that player typically scores in BSkA tournaments as a percentage of total match points available.

Each tournament is given a weight W depending on the number of players and the number of hands played.

If a player whose previous rating was R scores M match points out of a possible T in a tournament with weight W, the player's rating will change to

`	(1 - W)*R + 100*W*M/T`

The weight of a tournament is

`	W = 1 - exp(-sqrt(P*H)/100)`
where P is the number of players and H is the number of hands played by each player.

In a tournament where the numbers of players and hands varies from round to round, P*H in the above formula is simply replaced by the sum of the player-hands for each round:

`	P1*H1 + P2*H2 + P3*H3 + P4*H4.`

Ratings of players who play part of a tournament only are based on the percentage of available match points they score in the rounds that they play. For the rounds where they did not play they are credited with a percentage of match points equal to their previous rating. If the round(s) that they played in had p players and h hands, this is equivalent to counting their score only for the rounds in which they played and giving the tournament a weight of W*(p*h)/(P*H) for such players.

Once a year, on New Year's Eve, an imaginary tournament is played with weight 0.1 in which all players in the system score 20%. This moves everyone's score towards the starting rating of 20, and ensures that a high rating will slowly decay if the player does not sustain it by continuing to play tournaments.

## Example

A new player enters a typical BSkA tournament with 15 players playing 72 hands and scores 18 MP out of a possible 32. The tournament weight is 0.28, so the player's rating rises from 20 to
`	(1 - 0.28)*20 + 100*0.28*18/32 = 30.15.`

## Notes

1. If a player's result (percentage of maximum) in a particular tournament exactly reflects his or her rating before the tournament, the rating stays the same.
2. A player who achieves above average results in a series of tournaments will eventually have a rating above 50.
3. The starting value of 20 was chosen with the expectation that in practice ratings will range from around 20 to 80. With a lower starting value it would take longer for a new player to achieve a rating reflecting their playing strength. With a higher starting value, some new players would see their rating sink below the starting value, which might be discouraging.
4. The rating decay caused by the imaginary New Year tournament is fairly mild. A player with a respectable rating of 60 who did not play for 3 years would see it reduce to 49.2. Playing two tournaments with 0.3 weight and scoring 70% in each would restore it to 59.8.
5. For rounds with 4-player tables the number of hands played (H) represents the number of hands each player actually plays, so will be only 3/4 of the total number of deals in the tournament.
6. The tournament weight formula was chosen with the aim that a normal BSkA tournament would have a weight of around 0.25. The first idea considered was a weight of P*H/4000, but this makes too much difference between small and large tournaments: for example 9 players and 48 hands have weight 0.11; 15 players and 72 hands have weight 0.27; 20 players and 80 hands have weight 0.40. Sqrt(P*H)/100 is better: weights for the same three examples would be 0.21, 0.33 and 0.40, but both these formulae have the theoretical drawback that a very large tournament could have a rating greater than 1. The proposed exponential formula corrects this and gives weights of 0.19, 0.28 and 0.33 for the three example sizes.
7. When the tournament director plays in part or all of a tournament to make up the numbers, the director's result is included in the ratings, even though this player was not entitled to a prize.
8. It is unfortunately the case that a player who scores very well in the first round of a tournament might do well (in terms of rating) to retire from the tournament at that point, rather than risk a poorer result at a strong table in the next round. I can see no fair way to avoid this, but I hope that the incentive to win a tournament prize will be sufficient to discourage such behaviour.

John McLeod
March 2005

This page is part of the web site of the British Skat Association.