## Ratings Explained

### Introduction

The rating system is a measure of the overall relative strength of the members of the club. Rating points are awarded after match outcomes in a manner that provides additional reward to players who beat an apparently stonger player. The nature of the system is such that more recent wins have a greater effect than older ones, so as your game improves, your rating will increase to appropriate levels, even if you were thrashed as a novice! Ratings are persistent and the table is never reset.

All matches played at any club count towards a player's rating. The ratings table for a particular club shows the global rating for players who have played at that club. Their detailed match history shows matches played elsewhere that have contributed to their ratings history, but these matches can be hidden if desired.

### Experience Points

New players on the ratings system start with a rating of 1,500. If they are especially strong or weak, this figure will not be an accurate representation of their true skill level, so we use experience – defined as the sum of the match lengths of all recorded matches – as an additional data point.

Experience is used in two ways:

- The ratings system itself boosts ratings changes arising from matches for low-rated players, in an effort to speed them to their correct rating more quickly. Ratings movements are multiplied by a factor (known as the K-factor) which starts at 5 for a completely new player, descending smoothly to 1 (i.e. no multiplier) when their experience reaches 400 and beyond.
- By displaying experience alongside rating, viewers of the ratings table can make a personal assessment of the reliability of that player's advertised rating. Opinion is divided on what counts as an experience level low enough to consider someone's rating provisional, but the design of the ratings calculation clearly implies that ratings less than 400 fall into this category, and some people are cautious about accepting ratings for even higher experience levels.

It should be borne in mind that ratings always fluctuate as match results come in, and players have lucky and unlucky streaks, so even a 'mature' rating could easily be an over- or under-estimate of a player's skill.

### The Calculation

At the heart of the rating system is an estimate of the probability of an "upset" – that is, a weaker player beating a stronger one. The rating system, presuming that the rating already established is a reasonable estimate of playing strength, calculates this estimate as follows:

P_{upset}= 1 / (1 + 10^{D√M/2000})

where D is the difference in rating of the two players and M is the match length. (The square root is taken due to mathematical results involving "random walks" along a line.)

P_{upset} should always less than or equal to ½ – if it were more likely than an evens chance it would hardly count as an upset! – and the formula ensures this. The rating change will be multiplied by P_{upset} if the higher-rated player wins, or by 1 – P_{upset}, which is larger, if the lower-rated player wins.

We would also like players new to the system to have their ratings move more quickly to their appropriate level. We do this by multiplying their rating change by a number K for each player which starts at 5 and descends gradually to 1 as experience (the number of points played) approaches 400:

K = max(1, 5 - experience/100)

This is the only way in which experience comes into the calculation.

The final calculation is: for the winner, the rating goes up by

4 * K_{winner}* √M * P

and for the loser, the rating goes down by

4 * K_{loser}* √M * P

where P is P_{upset} if the higher-rated player won, and 1 – P_{upset} if the lower-rated player won.

### Effects

As the difference between the ratings grows, the effect of using P_{upset} in this way becomes more marked, and this puts a brake on runaway rating changes because of the huge penalty associated with an upset, which, even for a strong player, is bound to happen from time to time. The strong player can avoid this by only playing highly-rated players, but then they are more likely to lose! The converse also takes place at the low end of the scale. Ratings start at 1,500 (a conventional, but arbitrary choice), and are hard to sustain in normal play outside the 1,200-1,800 range.

### Practical Considerations

As always, a "match" of a given length is won by the person who reaches the required number of points first, with no consideration given to final scoreline or margin of victory. The doubling cube and Crawford Rule should be used.

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