Does elo rating have a limit?

Question

The highest Elo rating to date is 2882, which belongs to Carlsen, while the last Elo rating of Kasparov is 2851. Could I have an Elo rating of 2950 or maybe 3000? Does the Elo rating have a limit?

Answer 1

No, Elo ratings have no theoretical limit. If Alice scored 76% against Carlsen (rating 2853) consistently, she would stabilize at a rating of 3053*, and if Bob scored 76% against Alice, he would stabilize at a rating of 3253**, etc. There is no theoretical end to this sequence. However, consensus seems to be that in practice a perfect player would have a rating somewhere in the 3000s.

*Assuming Carlsen performed well enough against other people that his rating stayed at 2853

**Assuming Alice performed well enough against other people that her rating stayed at 3053

Answer 2

Yes ratings have a limit for chess, go, etc; as ratings go higher , the proportion of perfect games will increase and therefore if chess is a theoretical draw, the rate of draws will also increase (assuming most games are played against players of similar rating.) Once 100% draws are reached there is no better rating. This is borne out by the graph here which if extrapolated indicates that 100% draws would occur for engines at a rating of about 5200. Similar reasoning obtains if chess is a win or loss for white, namely that once perfect play is reached in 100% of games played, all white games are won and all black are lost or vice versa, again assuming play against similar beings. Even if one plays against dissimilar beings, an asymptote is reached in the calculation of elo points won per game; as ones rating rises , it is harder and harder to win points against lower rated rivals, again enforcing an upper bound. It is also clear that a perfect player will not have an inifinitely high rating since once in a while an imperfect player will play a perfect game, reducing the perfect players expectation from 1.0 to 1-epsilon thus enforcing a finite rating. (Look at the ratings difference curve (expectation as function of ratings difference) and you will see an asymptote to 1.0).

Answer 3

Elo is a relative scale that measures the winning probability between two players. The scale itself is arbitrary and there are many variations but here is no upper limit as such if the game is complicated enough and has enough room for continued improvements in game play.

In the game of Go a top professional has Elo rating around 6800 (KGS system), but a good AI engine can reach far beyond that. For example the Leela Zero Go engine has a current rating of 11,000+ (May 2018, after training on 7.5 million self-played games), and it is still improving. The game itself has enough complexities so how much further we can push up the Elo rating will be limited by how much computer resources were used to train the AI.

Answer 4

There is no theoretical maximum for the Elo rating.

There is, though, a theoretical maximum gap between the top rated player and the second which, in the first implementation, was 700+ point (when the difference of ratings between players was that or more the expected result was 1.0 so that winning doesn't earn you points).

Another factor to take in consideration is that the Elo rating is NOT a measure of absolute strength but a measure of performance relative to the ratings of the players. Since the strength of the players follows what in statistics is called a normal distribution, this was Elo's basic assumption, the probability to have a player rated at a rating R is never zero no matter how big R is, which means that if the total pool of rated players increases you automatically have higher top ratings simply by the law of large numbers. This is the basic reason (but there are are factors concurring) why the average rating of the top 10 players has increased over time: simply because the number of rated players has increased manifold.