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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 "currency" which gets traded around the pool of chess players? Would players that are overrated because of playing in a weaker pool have less "purchasing power"?

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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 "development factor" - 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.

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    Good post! Just a small addition, when you start out with your FIDE rating your first rating is in fact the performance you played in your first few games. (so for instance if you score 50% against 1700 opposition then your entry rating is 1700) Doesn't change your point of course. – koedem Nov 21 at 6:28
  • Assuming that roughly as many people leave the rating system as are entering it, that's also roughly zero-sum. – Peter - Reinstate Monica Nov 23 at 4:35
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    @Peter-ReinstateMonica That isn't the safest assumption. In the 1970s, Arpad Elo computed every single FIDE player's rating himself, because there were fewer than 2000 of them. (According to Wikipedia, anyway.) – Misha Lavrov Nov 23 at 4:41
  • @Misha Yes, I wondered whether there is an imbalance. Today it probably depends on whether young players pick up the game or whether it becomes an old-man sport. – Peter - Reinstate Monica Nov 23 at 4:55
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Are Elo ratings a zero-sum system in chess?

No, they are not zero sum. There are two reasons for this. First of all, when a player first gets a rating then that is a pure plus. There is no corresponding minus. The player gains a rating of 1000+ (the lower bound of the rating system) by virtue of wins and draws against rated players. Since the player was unrated at the time of the games there is no corresponding effect on the already rated players' ratings.

The second reason is more complicated reason. There is an inbuilt throttling in the rating calculations to limit or exaggerate changes and this is different for different players. This is called the "k factor" or "development coefficient".

Here is the section of the FIDE Rating Regulations effective from 1 July 2017 which describes this:

8.55 (a) Use table 8.1(b) to determine the player’s score probability PD
(b) ΔR = score – PD. For each game, the score is 1, 0.5 or 0.
(c) ΣΔR x K = the Rating Change for a given tournament, or Rating period.

8.56 K is the development coefficient.
K = 40 for a player new to the rating list until he has completed events with at least 30 games.
K = 20 as long as a player's rating remains under 2400.
K = 10 once a player's published rating has reached 2400 and remains at that level subsequently, even if the rating drops below 2400.
K = 40 for all players until their 18th birthday, as long as their rating remains under 2300.
If the number of games (n) for a player on any list for a rating period multiplied by K (as defined above) exceeds 700, then K shall be the largest whole number such that K x n does not exceed 700.

This means that, in the extreme case (say a 2290 junior player beating an ageing 2400 rated GM) the winner can gain as much as 4x what the loser loses.

If so, would it make sense to say that Elo points are a kind of "currency" which gets traded around the pool of chess players?

This can apply when a pool of players never (or almost never) play outside their pool. When the elite players all restrict their play to closed, elite tournaments where only players in the world top 20 then this can happen.

However, in recent years the prize money in large open tournaments like Gibraltar and Isle of Man have reached such attractive levels that more players from the elite have started competing and risking / sharing their Elo points. In particular the 2019 and 2021 Isle of Man tournaments have automatic qualifying places for the Candidates tournament (1 in 2019 and 2 in 2021) which makes it doubly attractive in 2021.

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  • Hm. If you play 700 games in the rating period (k = 1), and loose one of them against a new guy (who is already rated, but k = 40), (probably due to exhaustion), wouldn't it be 40 to 1? – Deduplicator Nov 23 at 0:55
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In addition to the other answers: it would depend on the organization running the Elo system as to whether there are additional modifiers.

e.g. back when I was playing chess, the Chess Federation of Canada awarded "participation points" basically points just for playing. Obviously not zero-sum.

The North American Scrabble Player's Association awards both "acceleration points" for performance in a tournament well above expectation, and "feedback points" to the opponents. These are also clearly not zero-sum.

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Rating floors make it non-zero sum as other have pointed out. Many years ago, I was talking to a major tournament organizer who supported this. I think one reason (in chess) that they did not want people to be able to lose a lot of rating points was cheating by losing in cheap events and then playing in big-money events against much weaker players. This indeed happened but is really not much of way to earn a living given travel expenses, etc. Nonetheless, "sandbagging" occurred as did player playing under fake names which probably is impossible today due simply to them requiring ID for tax purposes.

But the tournament organizer felt that what caused some people not to play was fear of losing rating points. I thought this absurd -- I think people want ratings to accurately reflect true ability as much as possible but last I checked, ratings have become not only wildly inflated (maybe by 200 or more points over the 1970s) but also in the case of people who have lost skill due to lack of practice etc. simply not reflective of ability and of course this is contagious as they play and other players gain points they would not have if the first player's rating had been allowed to realistically drop.

99.9% or chess players will not make a living at chess but they can try to learn the game and improve (not to mention the social aspect) and I am sure the vast majority of people want a rating that is accurate and meaningful, not fake points that I have seen also obtained by a small club running many rated events but only against each other -- Claude Bloodgood did something like that in prison and became iirc the highest rated player in the USA but he really probably was not even much above master, far below the IM level that would be minimum needed to qualify usually for the US championship that he insisted, based on his fake rating, he be invited to.

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