The rating of a perfect player

What would be the rating of a perfect chess player?

The definition of a perfect chess player is (rather boringly) someone who plays as if they had access to the complete tablebase of chess (a dataset so large some people think it could not be created within our universe). So they never fail to get at least the theoretical result from a position. If one imagines enough players of intermediate strength existing, such a player has a well-defined rating (if you believe that the statistical foundation of the rating system is solid).

Of course this question is as impractical to answer as it is to have a perfect chess player (human or computer), but computer chess makes it slightly easier to attack conceptually. We know that computer programs that can see a little further are significantly stronger, so depth of search influences rating. The relationship is believed to be not linear, with diminishing returns for greater search depth. So the graph of rating against search depth has a decreasing slope. It is eventually assymptotic to some fixed rating.

But how can one get a better estimate? One idea relies on games having a fairly limited length (based on existing examples). Typical high level games between opponents who make only small errors take the best part of 100 moves, but rarely get past 200 moves. The average number of moves may increase somewhat as the strength increases. If you think of the search tree for a position, as it gets deeper, the more it overlaps with the search tree for the previous position and for the one after. In a sense, games become made up of smaller numbers of overlapping trees as the search depth increases, until for hypothetical enormous depth, there might be a position in the middle of the game which is in the initial search tree, but whose tree reaches the final position. For smaller (but still enormous) search depth, the game might take 3 steps rather than 2 (in this loose sense) and so on. One reason for a decrease in the effect of increased search depth is this decrease in the number of steps from the start to the end because small errors get less chance to add up to a critical one.

I'd like to hear other people's opinions. Based on your judgement, intuition or more quantitative analysis, what would be your estimate of the answer? Is perfect chess Elo 3500, 4000, 5000, 10000 or what? (extrapolated from your choice of currently used scales)

Whilst I understand the intent off your question. The elo off a 'perfect' chessplayer is determined largely by the strength off the #2 player.

As elo calculations go I believe you hit your roof at around a rating difference off 800. Meaning that defeating someone rated +/- 800 below you will not be able to boost your rating anymore.  So if we assume that the second highest rated player has a rating off lets say 2800 and the perfect chessplayer defeats the #2 every time, then his/her rating will become no more then 3600

There's something missing in the definition of the perfect player : can he (it) adapt its perfect play to the weaknesses of his opponent ?

What would be the score of a game between the best engines and top players only trying to draw ? 75% for the engine maybe ? That would put them around 3000 elo.

Nothing missing, hicetnunc. This theoretical best player always assumes his opponent will play perfectly. Of course if the opponent makes a mistake, playing perfectly may lose after that, and the perfect player simply makes sure of this result from then on.

Interesting question about human computer matches. I am not sure: top players have not been playing the top engines much in recent years. The computer-computer rating lists are distorted by various rules, especially strong restrictions on opening books.

My original question is more about computers much more powerful than today's. How strong would Houdini be on a processor 1,000,000 times faster? 10^20 times faster? And so on. Pretty strong by today's standards, I reckon. I would expect near 100% results against today's engines. If so, that would add another 600 points or more.

One thing that is noticeable about today's matches is there is quite a spread of results. More than 50% decisive games with the weaker player getting a fair number of wins.

Interestingly, I think that if the "perfect" player always played the (same) "best" move in a given position, then, unless the game is won for white on move one, this player would, after a few years, obtain nothing but draws on either side of the pieces.  The primary reason for this is that the play from this player would be extremely consistent, and over time the top GMs would figure out the single line they needed to play in order to get a draw against this player, and just play that specific line every time.  Only by mixing it up a bit (i.e. selecting a move that might be 0.001 "worse" but cause more difficulties or just be different) would the perfect player be able to keep winning.

Very interesting discussion. Thank you.

Just an idea however... do you think an individual would neccesarily have to have a mega brain to retain every and all possibility or rather a heightened and super in-depth problem solving / pattern recognition?

I know that GM's have memorized a ton of info but what is the ration between applying pattern recogniztion to already known knowledge and coming up with or at least seeing the combination that others do not?

If the game is theoretically a draw, the perfect player would evaluate every move that keeps the position theoretically drawn as 0.00 and can play any one of these moves. In a given position there are probably several. For example, 1. e4, 1. d4, 1. Nf3, 1. c4, probably none of these moves are 'bad' enough to change the game from a draw to a loss.

The perfect player can play quite a variety of games without playing any non-perfect moves.

Some time ago someone one this site argued that a rating of 10 000 would be possible.

I still think that today's top guys would get an occasional draw against the perfect player, putting the maximum rating around 3300-3500. I have nothing to base this on, it's just what I think....

Loomis - I don't think the evaluation would necessarily work that way, or else the player would also be able to play all sorts of random nonsense (i.e 1. a4) as long as there was a least a single drawing line still available.  Rather, I would think that the evaluation would take into account the number of winning lines following a moves and play the move that, while still drawing, offered the greatest chance of entering a winning line.  Otherwise, you might see the effect that you sometimes get with computers in which they just give away material in the endgame once they see that the game is completely drawn (with perfect play) and so they just pick any random move that still leads to a draw rather than trying to press for advantage and allow the opponent maximum opportunity to make a mistake.  If the perfect player followed this method of picking the move with the most winning changes, then the line of play would be quite predictable.

nola, if you believe the perfect player is allowed to make subjective judgements such as which move provides the best winning chances instead of considering all moves that maintain the draw to be equal, then this provides another way out of playing predictable lines.

Playing the same lines repeatedly would quickly provide less winning chances than playing a variety of lines. So, the perfect player would naturally vary it's play among several moves that 1) preserve the draw 2) provide reasonable winning chances against an imperfect (but strong) player.

I would agree with hicetnunc's point:

A perfect player in the sense defined above (assuming a complete tablebase for chess) will score worse in real play than a player who adapts his moves to make life difficult for his weaker opponents.

Doing endgame training in chess can be much tougher versus a human than versus a perfect tablebase engine.

In checkers, there are tablebases covering large parts of the entire game. Yet, the human checkers champion found it relatively easy to draw most games versus such a machine, as it did not give any additional value to moves that are harder to defend against.

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So the rating of a "perfect" player will depend to a large degree on how successfully he uses his superiority to create difficulties for his opponents. In particular, the percentage of draws he allows his opponents will probably have the biggest influence on the rating margin he will be able to lay in between himself and his highest-rated opponents.

Also, the achievable rating will depend on the highest ratings of the opponents he can play. A computer engine playing only human grandmasters (rating up to about 2800) would achieve a rating of not more than something like 2900-3100 ELO or so. In engine vs engine competitions, and therefore including opponents rated higher than 2900 or 3000, higher ratings can be achieved (such as 3200 or more, as can be seen from the engine rating lists).

Because, as is known, a rough rule of thumb concerning the rating formulae is that even if you win 100% of your games against an opponent about 400 points weaker than yourself, this will not really increase your ELO by any significant amount.

Well said!

Thing is, if you just said pick the move with the most paths leading to a win, that could lead to pointless exchanges of pieces. I.e. "If I capture this knight with my knight, all replies lead to a win for me except one (the obvious recapture)".

An idea I've had for a computer which has solved the game, to stop it playing rubbish moves as you describe above, would be to stick with the current method of evaluating moves, but before making any move, check it against a database of all lost positions just to make sure it doesn't allow any winning chances for the opponent.

perfect player is an artist..

I see what you're saying, and it's a very good argument.  I wonder though if it would be as easy to draw as you say.  I think of something as "simple" as QvR or the RvR with f+h pawns draw and even against an EGTB I have trouble (no doubt harder vs a human but still).  The point being how many "inaccuracies" can the average 15 man position tolerate?

Because we're ultimately talking about computer vs computer ratings I wonder how much this even applies?  Meaning what would the rating difference be between the imaginary ultimate program when designed to try to exploit certain failings in other programs to calculate or evaluate vs when it doesn't do this.

For what it's wroth I'd guess we're still at least two 400-600 point leaps away from when the rating vs error rate graph starts to asymptote.  This might be a good question for some of those on engine teams such as at the rybka forums who may have some interesting engine data and results to back up their ideas.

 Rating is based on a curve.  At some point, the area under the curve is less than the size of a single player's rating change, and then there can be no movement to the right.  This depends solely on the population size, so the greater the number of players, the greater the highest rating.  Also, it means there is a mathematical limit to highest rating, so it can't just increase willy-nilly.

It's true. A computer that purely does something like that will probably lose a lot (just think about what would happen if such a line started with a queen moving into harm's way)

Generally, we don't look for "most winning lines," but the line with the best "worst-case scenario." I'd say the perfect player would be able to find this line immediately every single time.

Allthough I have no opinion to add to this debate, I would just like to say how interesting this thread is.

Thank you for the post. 

I fully agree that it is not easy to defend such endgames versus a human. But when defending these endgames against a pure tablebase engine, it can be surprisingly simple to defend them, because the tablebase does not see which lines are difficult to defend.

In a Q vs R endgame, a human will of course try to cause the defender as many difficulties as possible, for example by driving the king towards the edge. To a tablebase, lots of continuations just have the same evaluation, because they are all a draw. So the tablebase may just as well only move around the pieces in the middle of the board or even exchange the queen for the rook, because that does not change the evaluation of the position in its eyes.

So  in order to make a tablebase a tough attacking endgame training partner, one would need to combine it with some engine that adds a different kind of evaluation to choose between all the different kinds of drawn positions and try those continuations that may be most difficult for the defender.

(If the tablebase has the task of holding a drawn endgame, it is a perfect training partner, of course. And if its task is to try to defend a losing position, it will often be an excellent training partner, because it will choose the variation with the longest distance to mate, which may in many cases really be one of the toughest defenses possible.)