Are Elo Ratings Zero-Sum? Are Elo ratings a zero-sum system in chess? In other words, if I gain +9 points from a game, does my opponent necessarily take a -9 point loss? If so, would it make sense to say that Elo points are a kind of &quot;currency&quot; which gets traded around the pool of chess players? Would players that are overrated because of playing in a weaker pool have less &quot;purchasing power&quot;?
Between most players, Elo ratings are zero-sum.
The formula for updating an Elo rating looks like this. Suppose player 1 and player 2 are playing a game. First, their ratings are used to generate a prediction: an expected score W that player 1 will get against player 2. Then, we compare this to the actual score, X. player 1 will get K*(X-W) points, where K is a &quot;development factor&quot; - more on this later on.
From player 2's point of view, the expected score is 1-W and the actual score is 1-X (a score of 1 for player 1 means a score of 0 for player 2, and vice versa). So player 2's rating changes by K*((1-X)-(1-W)) or K*(W-X) points, which is the negative of player 1's change. Zero-sum!
The development factor K is the only thing that can break this. In the FIDE version of the Elo rating system, this value is 20 for most players, but it's 40 for new players (to represent the uncertainty in their rating) and 10 for players that have ever had a rating over 2400 (for the opposite effect). If two players from different categories play, then the result will not be zero-sum.
Also, globally, the sum of ratings does change, because players are leaving and entering the pool. A new player starts with an Elo rating determined by their first few games, adding to the supply. That rating may become larger or smaller by the time they stop playing chess for whatever reason - removing another number from the supply.