What are Elo ratings ~ Elo Explained

Background

Elo ratings attempt to mathematically assign a number by which the relative skill of two players in a game can be judged.  The system was first developed by the mathematician Arpad Elo as an improved method for computing chess ratings. For more information on the history of Elo ratings (it is chess-centric), see http://en.wikipedia.org/wiki/Elo_rating_system.  A rating system based on the same principals has been applied to backgammon. Many online backgammon sites maintain ratings of their players based on match results.  This was first done on FIBS (First Internet Backammon Server), and backgammon ratings based on the FIBS Elo ratings formula are standard. NYC Backgammon uses the FIBS Elo rating formula.

The Elo rating is a number ranging from about 1000 to about 2000.  Players joining NYC Backgammon are initially assigned a 1500 rating. The rating changes each time you complete a match.

When calculating the change in ratings by winning or losing a match, there are three factors taken into account:

1. The length of the match.
2. The experience of the player.
3. The difference in ratings between the two players.

A one-point match played between two "experienced" players [see below] with identical ratings is worth 2 points; that is, the winner's rating will go up exactly 2.00 points, while the loser's rating will go down exactly 2.00 points.

The three factors -- match length, player experience, and player ratings -- affect the ratings calculations when a match is played as follow:

I.  Match Length

The rating change is multiplied by the square root of the length of the match. So, for example, a 4- point match would be worth twice as much as a 1-point match. Why twice? Because the square root of 4 is 2, so the match is worth 2 times what a 1-point match would be worth. A 9-point match is worth 3 times as much; a 16-point match is worth 4 times as much. Most NYC Backgammon matches are 3, 5, or 7 points long, so it's not as easy to figure those in your head, but hopefully you get the idea.

The match length is the length agreed upon when the invitation was accepted, not the final score. In other words, a 5-point match will always count for 5 in both rating and experience calculations, regardless of whether the final score was 12-0 or 5-4.

II.  Player Experience

A player's experience is taken into account when determining the rating change after a match is completed. Only a player's own experience level is used in this calculation; not the opponent's experience. A player is considered to be "experienced" when the player has an experience level of 400 or more. This number is simply the running total of the length of all matches completed. In other words, a "newbie" starts with experience 0; after a 5-point match the experience is 5, after another match for 3 points the experience would be 8; and so on. The length of the match is added to the player's experience before performing the ratings calculation. If your rating is 400 or higher when the match is over, experience does not affect the ratings calculation as described above; i.e., if you win a 1-point match against someone with an identical rating, and your experience after completing the 1-point match is 400 or more, your rating will go up exactly 2 points. If experience level is less than 400, the rating change for that player will be more: If experience is 300, the rating change is doubled. If experience is 200, rating change is tripled. Experience of 100 means rating change is quadrupled.

When your experience is 400 or more, your experience level isn't a factor in your rating change calculations. When you're very new on our Elo ratings, and for your first several hundred games, your rating is very volatile and will go up and down a lot.

III.  Player Ratings

Our Elo ratings take player ratings into account when calculating ratings changes. If you defeat the best player in NYC Backgammon, your rating should go up a lot more than if you beat the worst player; similarly, if you lose to the best player, your rating shouldn't suffer nearly as much as if you had lost to the worst player.

Elo calculates the probability of winning a match based on the difference in ratings between the two players and the length of the match. The larger the difference in ratings, the more "mismatched" the two opponents are, and the higher the probability of the favorite winning any given game of the match. The longer the match, the more likely the best player will win the match. (Usually, the longer the match, the more likely it is that the luck of the dice will even out and the more likely it is that the better player's skill and knowledge will prevail).  Examples: Two players of identical rating are each 50% favorites to win the match, whatever its length. A player with a rating 100 points higher than the opponent is a 52.9% favorite to win a 1-point match. Not a huge difference. However, that same 100-point rating differential results in a different prediction when the match is longer. For example, in a 13-point match, the 100-point higher rated player is a 60.2% favorite to win the match.

Let's look at another example. This time a 1700-rated player plays a 1- point match with a 1400-rated player. The 300-point difference in their ratings results in the higher-rated player being considered to be a 58.5% favorite. When the match length increases, the higher-rated player becomes even more favored. For a 3-point match, the 1700-player is considered a 64.5% favorite; for a 5-point match, 68.4%; for a 7-point match, 71.4%; 9-point: 73.8%; etc.

If you play a higher-rated player calculated to be an overwhelming 75% favorite to win the match, and if you played that player 100 matches, you would be assumed to win 25 of those matches, and that you'll lose the other 75. If you do, in fact, win 25 and lose 75, your rating won't have changed after those 100 matches. Neither will your opponent's.  Whenever you win against such a higher-rated opponent, your rating will go up by 3 times as many points as it will go down when you lose. Since you'll lose 3 times as many of these matches as you'll win, the net result will be no change.

IV.  The formula:

How the Variables are calculated:

How the rating change is calculated: