the highest rating I have ever seen on chess.com is 3006.
Imagine this player plays 1000000 rated games with you. Can you guess what will be his rating after the games?
3006
Could be. It depends on the rating formula adopted by chess.com.
If both ratings are accurate, then it will stay 3006.
That is it! Rating is a RELATIVE measure of STRENGTH. If you are invincible, your rating will be infinite. Thats why it is important to assume the strength of the opposition. In my case, I proposed the real world situation. Anand is below 2900. What is the average of top 20? Nobody is 3600 or 4000, right?
The problem is that the top players might draw it. Therefore the perfect player might only have a 95% score so the ratings won't be infinite. There will be a point there a draw vs a much lower rated player lowers its rating by the same amount as the many wins in between will raise it.
That is it! Rating is a RELATIVE measure of STRENGTH. If you are invincible, your rating will be infinite. Thats why it is important to assume the strength of the opposition. In my case, I proposed the real world situation. Anand is below 2900. What is the average of top 20? Nobody is 3600 or 4000, right?
The problem is that the top players might draw it. Therefore the perfect player might only have a 95% score so the ratings won't be infinite. There will be a point there a draw vs a much lower rated player lowers its rating by the same amount as the many wins in between will raise it.
That's the issue: to generate the winning probability of a perfect player against his opposition to find out what the number could be.
A win will give the perfect player at least +0.8 while a lost -4.2
We can model the opposition with creating a tournament where the top 20 FIDE become the participants.
Now how can we model the perfect player? Isn't it logical to say that the perfect player will have at least the strength of the strongest machine? Now imagine the fastest computer that can be built (it is not yet built!) to run the best chess software. Some of the engine already have a similar rating. That is based on only a few games. What if we double the number of the games!
After we have the result now assume a few more fictitious games where win/draw/loss is a direct copy of the first tournament, but with players rating adjusted accordingly.
Just because in the practical sense it would be hard to achieve a given rating, even as a perfect player, depending on level of competition, does not mean we cannot talk about it notwithstanding! The entire discussion is based on speculation anyway, isn't it?
I will repeat something I said earlier here:
But again, what we, or I, really mean when it comes to strength, is the strength at which the person would consistently perform, which could be determined by many long game matches. For example, scoring xx against carlsen out of 100 games.
In other words, if it's hard to actually achieve ratings in say the 4000s, it may be more conclusive to take a look at the performance rating over the course of many matches against others, to determine the general strength of this player.
The problem I am seeing here is that we are looking at two situations.
One, a perfect player's rating when playing versus today's GMs.
Two, the theoretical assesment of chess.
Analysing the first situation, the perfect player's rating would be lower against today's GMs than a theoretically highest possible rating based on the pool of players. A perfect player's rating would show a dominance, which has been shown in this thread to peak at around 800 points maximum difference, where no decernable increases would occur. The assumption is that all the games would be won by this player. So in effect, the highest rating a perfect player could achieve in this pool would be 3600. Now if draws are included, then this maximum number will be lower.
Now, what if there were two great players dominating the field. Lets say player #2 dominates the field of players and player #1 dominates player #2 and the rest of the pool. Player #2 would approach the 3600 peak, but player #1 would eclipse 4000 showing the dominance of the pool. I am not sure if you can postulate a perfect rating as ratings reflect the strength of a player against the pool of players. For that to be evaluated, there will have to be perfect statistical results based on a 32 piece database, in effect, the solving of chess. If all players play perfect based on this database, then all the ratings should centralize around one number.
Two, with increasing rating, white tends to show an increase in the percentage of wins and draws. So in effect, a perfect white game would not end in a loss according to the statistics. But can the statistics show that with even further increasing rating, is there an increased tendency towards a draw or win? At some point, one of these statistics will hit a peak and will experience an decrease while the other will show a continual increase. This will show the theoretical result of chess.
Good post. I think Elroch tried to get the theoretical number by integrating search depth to rating or something (it's back there somewhere).
And I think that's the main question... such as building a computer 200 elo above today's top computer (you can even call that one standard deviation) and so how many standard deviations (of these 200 point increases) until you approach the 32 man table base.
For what it's worth, I think (completely guessing of course) that today's top human players would score much less than 1% vs a theoretical perfect player if this perfect player played for a win by choosing lines that required the most technique / most chance to blunder.
Browse this thread and you will find most high rated players have the same opinion that this perfect player is a lot stronger than a computer. How much stronger? Waffllemaster stated at least 99% stronger than top human player!! (a rough guess)
The fact is, once you have this probability number assumed, you can calculate the rating number.
There is a percentage where by probability theory (and current FIDE rule), the perfect player will always increase his rating when defeating other players. It is just how many games allowed that will prevent this increasing number to become infinite.
By definition of the original ELO curve (which is not adopted 100% by FIDE), once this perfect player rating difference with his opponent's is above 735, the perfect player is expected to win by 100%, so that if he wins he doesn't get point but if he loses he loses point.
But FIDE has a rule that once the difference is very high, it is asumed that the maximum difference is 400 with winning expectation of 92% (not 100%). If this perfect player is expected to score 92% while in actual he scores 100% there will be rating increase of K*(100%-92%)=K*8%. With K=10 (as in current practice) this is +0.8.
This is a small number, but if we assume that the number of games is infinite (which is of course unreal!), the total rating will also be infinite.
Lets say the the number 2 player is "Perfect misus 0.000000000001%". Then at that level of play, one might expect that games played between "Perfect Player" and "Perfect Player misus 0.00000000001%" would with almost certainity would always be a draw. Therefore one would never know which one is better player. Therefore ratings would be the same.
Those two players (even with much smaller different in strength) when play against each other will even out the rating. But he who plays the other players more will have the higher rating. It can be the #2 player who has the highest rating if he refuses to play the #1 and play the rest more often.
Murphy's law, you seem to have not thought hard enough about the core issues. You claim that the perfect player is not an absolute, but in theoretical terms it is an absolute - a perfect player is one who always gets at least the theoretical result of a position, and more if the opponent makes a serious error. i.e. he/she/it plays according to a 32-piece tablebase.
Interesting is that that against imperfect players, playing perfectly is not necessarily enough to get the best results! Suppose player A is a program that plays according to the 32-piece tablebase, but just picks any winning move, but player B also plays according to the 32-piece tablebase but chooses moves that it determines will create more difficulties for weaker players. As a result, although player A and player B always get 50% against each other, player B will get a better score against imperfect players. This shows an imperfection in the rating system because player B could achieve a higher rating than player A if the player pool included imperfect players, but always get 50% against player A.
Some people have talked about how strong "a computer" is, thinking of current computers. It is theoretically trivial that a sufficiently fast computer (utterly impractical as this is) would be a perfect player, so it would be more appropriate to refer to "current computers".
Oh, and FIDE's 400 rule is pretty stupid if applied to players with very large differences in rating. Eg if Kramnik played 1000 players with ratings of 2000, he would win such a large proportion of the games that his rating would go up several hundred points. Obviously this would be misleading if he then started playing super GMs again. When Elo designed the rating system on statistical grounds, he certainly did not include such an unjustifiable condition, and we should ignore this condition too.
~3800
No, I think the rating would be not even on the magnitude of thousands.
I'm thinking the perfect chess player would be rated at least in the billions--chess is an extremely complicated puzzle involving some 10^10000 possible games (Questioning this number would be interesting).
In fact, the perfect chess player could increase its rating arbitrarily high if desired.
Suppose two "perfect" chess players played (with arbitrarily high ratings). The game would end in a draw.
Interesting. I do not think it's infinity.
Well, I'm pretty sure there are nowhere near that amount of possible games; it's probably under 10^100.
The struggle with getting a rating of a billion, even with perfect play, is that in chess you can calculate all you want, but you cannot force a strong line of play to be there. With this in mind, non-perfect players who are very strong can try to defend for a long time, and since many things don't depend on 50 move variations, I don't think the ability to play perfectly is quite as significant as you think it is.
As said I think the perfect rating is somewhat higher than the strongest computers now, but to talk in anything more than thousands seems quite extreme.
Especially given that a draw with a lower-rated player decreases your rating; and the rating increase from beating a lower-rated player decreases as the difference in rating increases. Unless it turns out that chess is solved to be a forced win for one side or the other, I don't doubt that the current top GMs will be able to draw a non-negligible fraction of their games even against a perfect player (since such a large fraction of high-level games are drawn today, even in GM vs computer games). If that is the case, sooner or later the perfect player's rating will reach an equilibrium point where the points lost from draws match the points gained from wins.
EDIT: Hmmm, most of this stuff has been said before. I only skimmed some of the pages in this thread though.
The OP question is nonsense from several perspectives:
1) Mr. Arpad Elo, the inventor of chess rating, stated that such ratings can't and shouldn't be used to measure a player's strenght.
2) Perfect player in the OP meaning is nonsense too. If one side had access to whole information, it would be no longer player. Playing a game is based on the fact, that both sides have limited possibilities to dig informations and choose strategy.
I use term "perfect player" in another meaning. Perfect player has master level in understanding the game, has iron nerves, isn't lazy and enjoys the game he plays.
"1) Mr. Arpad Elo, the inventor of chess rating, stated that such ratings can't and shouldn't be used to measure a player's strenght."
I disagree with this slightly however, and the reason is that by now, there are many strong players with very well established ratings -- they usually stay within 25 points or so of a certain rating.
With this in mind, we could just have matches between say Carlsen, and the perfect player, and judging by how the perfect player performs against this >2800 player we can get a performance rating. We could make it out of as many games as we feel necessary -- 20, 60, 100, 500, etc.
By getting an accurate performance rating, I think we can attempt to measure that player's strength, from the perspective of elo only of course. This perspective is fine as most chess players are familiar with its implications.
No, I think the rating would be not even on the magnitude of thousands.
I'm thinking the perfect chess player would be rated at least in the billions--chess is an extremely complicated puzzle involving some 10^10000 possible games (Questioning this number would be interesting).
In fact, the perfect chess player could increase its rating arbitrarily high if desired.
Suppose two "perfect" chess players played (with arbitrarily high ratings). The game would end in a draw.
Interesting. I do not think it's infinity.
Some wild guesses and some statements that I would argue are not correct in this post.
For quantitative reasons I gave early in this discussion, the rating is very likely bounded by a very much smaller number than this poster thinks. Of course the perfect player cannot increase his rating indefinitely: because chess is a finite game, there are a finite number of steps in strength between you and the perfect player (where a step can be defined say as being good enough to score 75%). These steps of course correspond to equal Elo differences. Most people agree with the statement about a game between 2 perfect players. Being a pedantic mathematician who has studied game theory, I would say that we know a game between two perfect players has a certain result, but I cannot be sure what that result is (there is a slight chance that white can force a win, and a tiny chance that black can because white is in zugzwang in the initial position. Yes, this seems very unlikely, even to me, but no-one can prove it is not the case!). The statistical evidence from imperfect players coupled with intuition is not adequate to be sure. Mathematicians are fussy about the difference between something that is strongly believed and something that is proved to be true.
And yes, the answer is not infinity. 
I get 10^10000 from the fact that a chess game can last over 5000 moves, and that most positions have some 100 possibilities for the next move (for both halves of the move). Of course many positions have much more or much less than 100 possibilities, but I use this as an estimate which results in 100^5000 = 10^10000 possible games. True, the actual number is probably much less, but I didn't pull the number from nowhere.
It shouldn't be hard to find an upper bound on the number of possible games, but it might be harder to prove the existence of an upper bound that is less than 10^10000. (BTW, I am another one of those pedantic math people)
Frrixz, your reasoning is slightly flawed in that many of what you are counting as different games are just going to end up as transpositions. You can play e4 followed by Nf3 or Nf3 followed by e4... those would be different games, but if Black plays the same move in between, it has no effect on the outcome.
Also, the "perfect" side is constrained in their possible moves, since they can never make a move that would turn a win into a draw or a draw into a loss. They've probably still got multiple possible moves most of the time, but it does restrict them a lot. And the poor human playing that "perfect" player isn't a random number generator, either. They're going to strive to make the best move. So you can throw out all the trillions of theoretical games starting with 1.g4 e5 2.f4. (and 1.f3 e6 2.g4, and any of the other variations.) The perfect player isn't going to play those moves as White, and even if the opponent was moronic enough to do this (and the average player is not), the perfect player would take the win right away. 1.e4 g6 2.Qh5? Lots of possible moves due to the queen being "active"... but not something we need to worry about in any "perfect player" analysis.
Those 5000+ move games we were discussing in that other thread? I'm fairly sure those are not possible unless both sides cooperate. When talking about a perfect player, we assume that both sides are trying to win, not to extend the game as far as possible.
That is it! Rating is a RELATIVE measure of STRENGTH. If you are invincible, your rating will be infinite. Thats why it is important to assume the strength of the opposition. In my case, I proposed the real world situation. Anand is below 2900. What is the average of top 20? Nobody is 3600 or 4000, right?