Ratings Explained

by Glenda Foster

As current custodian of the New Zealand Scrabble ratings system, a number of people have asked me for an explanation of why their ratings go up or down so much after a tournament. This article provides a relatively simple explanation of how the NZ ratings system works.

In the Beginning

Ratings calculations are based on the American system. Former NZASP president Roy Vannini established the database that I currently hold, which contains rating records as far back as July 1988 (New Plymouth tournament).

In 1998 Nigel Richards proposed amendments to the existing system. These were accepted by the NZASP executive, and Nigel’s computer program was introduced on 1 January 1999.

Expected Wins

The system is based on probability, and the notion that a player who plays enough games against an opponent of equal strength will win about half of the games. So, in a 7-game round robin tournament where all the players started on the same rating, they would all be expected to win half their games (ie 3.5 wins). Of course, in real life there are rating differences between the players in a grade. Expected wins are therefore adjusted to take into account the rating differential in each game played. Table 1 shows the expected wins for Joyce (rated at 1500) in a hypothetical grade where the ratings of the opponents range from 1650 down to 1400.

As you can see, when Joyce is playing a player with a higher rating than hers she is expected to win less than half her games, but her win expectancy is higher than 50% when she is playing a lower rated player. So in this hypothetical 7-game round robin, Joyce is expected to win a total of 3.3 games.

So the first part of the ratings calculation is to establish the win expectancy for each player in a grade. If the draw is not a round robin it doesn’t matter because the formulas work off the actual opponents for each player rather than the average for the grade (as used to happen).

Reality Sets In

A player’s final rating after a tournament is based on the number of wins they actually achieved compared to their win expectancy. Each player’s placing in the grade is also recorded, but it is not used in the rating calculation. Wins that result from byes are not counted, and the number of games played is also adjusted to reflect actual games played.

Table 2 shows the new ratings for players in Joyce’s grade. Because Joyce achieved more wins than her expectancy, her rating goes up. Conversely, players who achieved fewer wins than expected sustain a drop in rating. How much your rating goes up or drops depends on the rating you started with and the size of the difference between your actual wins and your win expectancy. In the example above, David drops about 27 points for each win less than expected, but Robin drops 33 points per win differential. The higher your rating, the lower is the rating change. At the 1400-1650 level you drop or gain around 25-30 points per win differential, but at the 2050-1800 level the rating change is about 20-25 points and at the 700-950 level it is 40-45 points. So higher rated players tend to have more stable ratings than lower rated players.

Unrated Players

You might be thinking that the foregoing explanation is fine, but what happens when new players enter a tournament and they don’t have a current NZ rating? What about overseas players? How is a player’s initial rating established? Good questions!

Our rules allow tournament organisers to place players without a NZ rating in any grade that they deem appropriate. This may be based on performance in their club or on overseas ratings. Once a player obtains a NZ rating then they must be graded in rating order.

An unrated player does not have a win expectancy prior to the tournament, because you can’t work out a rating differential for each game played. So after the tournament when the actual performance of the player is known, a provisional rating is calculated for the player. This is done by applying a formula based on the percentage of wins achieved against rated players and the average rating of those players. And, because at least one player in the grade has an unknown starting rating, you can’t work out win expectancies for the remaining players in the grade prior to the tournament.

This is best illustrated by an example. Suppose that our friend Joyce did not have a NZ rating and was placed in this grade by the tournament organiser. Table 1 cannot be calculated prior to the tournament, so all win expectancies are unknown at this stage. The next step is calculating Joyce’s provisional rating. There are 7 opponents who have an average rating of 1521.78. Joyce wins 4/7 (57.1%) of her games. The formula then looks up the point on the probability curve where a player would win 57% of their games against a player with a 1521 rating. This becomes the new provisional rating — in this case it is 1572. Now the remaining players’ new ratings can be calculated as shown in Table 3.

Provisionally Rated Players

A player’s rating is treated as provisional until they have played 35 tournament games. This is because there needs to be a reasonable number of games for probability theory to work properly. So for the first 35 games, all players are treated like players without a rating and a new provisional rating is calculated at the end of each tournament in the manner described above. This allows new players to find their ‘true’ rating level more quickly, but it also means that there can be very large rating swings when players win significantly more or less than 50% of their games. It also means that win expectancies cannot be truly calculated prior to tournaments in grades where there are any provisionally rated players. This is often worked around by calculating win expectancies prior to a tournament as if all players had established ratings. In such a case, if provisionally rated players perform significantly above or below expectations, it will affect the rating gain or drop of other players in the grade.