Our club's rating system
This is a description of how the NASPA Club #620 rating system works. It is primarily a mathematical discussion, so if that kind of thing makes your brain melt, you may prefer to study some words instead.
This system is a long-term experiment. Its primary purpose is to determine the relative playing strengths of our club members compared with each other. As such, it has absolutely no relevance at other clubs or at sanctioned tournaments.
Theory: The bell curve and you
Ratings in our club are derived from the old NSA rating system, which in turn is derived from Elo ratings used in chess, which in turn are derived from the cumulative distribution function. The cumulative distribution function, related to the bell curve, is a graph representing probability across a domain ranging from positive to negative infinity. As x increases without bound, y comes closer and closer to 1 - and as x grows more and more negative, y approaches zero. If x is zero, the function is exactly 0.5.
The primary assumptions in place are:
- Two equally-rated players competing against each other will each have a 50% chance of winning.
- Higher-rated players have a higher probability of winning against lower-rated players.
- Likewise, lower-rated players have a lower probability of winning against higher-rated players.
Given these assumptions, we can use the difference between the two players' ratings as a factor in the x value, and the y value can be mapped via the cumulative distribution function to the probability of one player winning over the other.
The devil, of course, is in the details.
Setting probability parameters
The equation for the cumulative distribution function is well-known and governed by two parameters: the mean and the variance. The mean determines the value of x where y becomes 0.5, and the variance determines how "wide" the curve is. By setting these parameters appropriately, the x values can be restricted to most often fall within a certain range. This is a desirable characteristic of this system because it can be used to prevent players' club ratings from being confused with official NSA ratings.
The mean is easiest to determine. Since a 50% probability of winning is indicative of no difference between two players' ratings, the mean needs to be zero.
As previously stated, it is desirable to keep this rating system in a different range than the one currently used by NASPA. An unrated player who participates in a tournament and loses every game will receive a rating of 500. Thus, the club rating system should try to stay below 500. To accomplish this, we can examine values of x which yield certain probabilities and then map them into a range of zero to 500. At x=1, y=0.8413; at x=2, y=0.9772; and at x=3, y=0.9987. Using whole integers above 3 won't add much more to y, so 3 will suffice. And finally, to map this into a range of zero to 500, we can take this 3 and multiply it by the fraction made up of the ratings difference between you and your opponent in the numerator and 500 in the denominator.
Thus, your probability of winning a game at our club is assumed to be the cumulative distribution function of three times the difference between your club rating and your opponent's club rating divided by 500. It's important to note this is an assumption, and a somewhat arbitrary one. At some point the actual probability will be determined and this calculation will be revised as necessary. For now, since the rating system is designed to be a rough indicator of relative performance, it's close enough.
Final touches
With win and loss probabilities in place, club rating points can now be awarded to players who win when they are expected to lose (or deducted in the opposite case). We must also consider the special case of a player who has no club rating playing one who does (or the very special one of two unrated players against each other, like at our very first meeting). Finally, phenoms who storm into our club and start winning every game should face tougher competition sooner rather than later. All of these are addressed below.
Awarding and deducting points
If you are expected to lose and instead win, your rating is too low and should be increased. This is accomplished by subtracting your win probability from one and multiplying that by a rating point multiplier. Thus, if your chances of winning are extremely low and you are still triumphant, your reward is greater than if your chances of winning are high.
If you lose your game, your win probability is subtracted from zero (i.e., made negative) and multiplied instead.
The multiplier
Your first ten games at our club use a rating multiplier of 30, and all subsequent games use a multiplier of 20. Using a higher multiplier for the first ten games allows exceptional players to more quickly move up to the top tier and face tougher competition. The period when the larger multiplier is used is called the acceleration period and is indicated with a + symbol next to the rating.
Unrated players
If you have no club rating, you are awarded points based on the fraction of the cumulative score that was yours in the game. This also takes care of cases where two unrated players play each other.
Corrections to the system
There have been two corrections to this rating system, both involving initial ratings. In the first case, players were accidentally given a large probability of winning their first game, and in the second, the fraction of new players' contributions toward the cumulative score in their first game was calculated incorrectly.
Implementation
As the system is described above, since we use a mean value of zero, a player could theoretically obtain a negative club rating. To mitigate this, in addition to the above calculations, each player automatically receives 31 points per game for the first ten games played. After the first ten games this no longer applies. Thus, new players at our club can expect to see ratings which seem much lower than everyone else's. Rest assured as you get closer to the ten-game mark, your rating will rise accordingly.
All club ratings are expressed to two decimal places to further separate them from NSA ratings.
Ratings are recalculated after each game and tallied after each meeting. This is done using a large and convoluted spreadsheet in OpenOffice. The ratings are published at least twice a month both on this site and in our club newsletter.