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omar
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Rating of a perfect chess player
« on: Oct 21st, 2004, 11:43pm » |
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I've often wondered what the rating of a perfect chess player would be. I know it has to be some finite number, although it would probably be an appoximate number with an error bound. No one has ever crossed a rating of 3000; so are we almost there; are we half way there; or are we not even close. Im not too concerened about having an exact number, I would be happy to have an approximate number that would give me some idea of where we are compared to a perfect player.
Is there any way to find this number; even approximately?
The other day I thought of a way to approximate this number. But I don't if it will actually work and it requires a lot of data. So Im posting it to see what others think.
First lets assume that a perfect game of chess is a draw (kind of like a complex version of tic-tac-toe). This assumption could be wrong, but for the moment lets assume it is true. Now if two perfect players were playing each other the result would always be a draw. The draw percentage would be 100%. Suppose we make a graph of chess rating (x-axis) vs percentage of draws (y-axis). The games used for this graph should be games between two players that have a fairly close rating (maybe 50 points of one another). We know that the percentage of draws among beginners is about 3% and among the top players it is above 50%. I wonder what this line or curve looks like. Maybe we can extrapolate the line to 100% draws and see what rating it corresponds to. Would this be the appoximate rating of a perfect chess player. Has anyone seen a graph like this?
Now there is the problem that in games between humans mutual draws increase the percentage quite a bit more than it really needs to be. So we have to keep in mind that the rating number may not really be the rating of a perfect chess player, but rather an approximation of what rating we can expect 100% draws among the human players. But if we use game between computers where draws were not so easily offered or accepted then it might give is a better ideas of the real chess max rating.
Jeff Sonus of www.chessmetrics.com has a database of lots of human games. I've contacted him to ask about this and waiting to hear from him.
Does anyone know if there is a big data base of computer games? That might give a better approximation.
Omar
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MrBrain
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Re: Rating of a perfect chess player
« Reply #1 on: Oct 22nd, 2004, 10:56am » |
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As with a previous discussion, I think "perfect chessplayer" is ill-defined. What does this mean? There may be many moves in a given position that lead to a draw, while other moves are a loss. But which of the drawing moves is the "perfect" move? This may depend highly on the opponent. Against another "perfect player", the distinction is moot. But against someone with say a random strategy, a C player, an Expert, etc, a different "optimal" move may result.
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Fritzlein
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Re: Rating of a perfect chess player
« Reply #2 on: Oct 22nd, 2004, 12:06pm » |
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I think MrBrain is quite right about the difficulty of defining perfect play. This problem has been much-discussed over the years, at least as far back at the late 1890's when the "modern" players thought they could basically codify all the principles of good chess play, which would eventually lead to equalization and a draw. Lasker was somewhat of an anomaly in his day because he wanted to provoke a fight, even if doing so required making an objectively bad move. Capablanca was sure that chess was heading for a death of draws, but years after his grim predictions, Tal thrashed his opponents with "unsound" moves.
To put MrBrain's question in a different way, would a perfect player playing against me choose a "good" move (i.e. one which can't lose with further perfect play) but which I know how to counter in preference to a "bad" move (i.e. one which will necessary lose with perfect play by me) but which I will likely blunder in response to? Is 10% chance of winning and 90% chance of drawing more perfect than 80% chance of winning and 20% chance of losing? In practice it is rather limiting to define perfect play as excluding any chance of losing.
The widespread (and unjustified) faith in the objective meaningfulness of chess ratings has given a new spin to an old discussion. Somehow it seems less ill-defined to ask "What would the rating of a perfect player be?" than it does to ask "What is perfect play?". In truth, however, assigning a rating to perfect play adds a new layer of ill-definedness on top of the previously unresolved questions of perfect play.
I would argue that a high percentage of draws, while correlated with good play, is sufficiently distinct to scuttle the project of extrapolating the percentage of draws to a point of 100% draws and considering that perfect play. One excellent counter-example is Ulf Andersson, who in his prime drew a huge percentage of his games due to incredible technique at saving theoretically lost endgame positions. Yet even when he was drawing a higher percentage of games than everyone else at the top of the chess world, he still wasn't the best, because he didn't win as many as some others.
Now, having made it perfectly clear that I think Omar's proposal is doomed and will produce a result that is not meaningful, let me say that it holds some mathematical interest for me. I don't have time right now, but I will try at some point to post a mathematical model which, if true, would allow the desired calculation to be made. Of course the model doesn't correspond to reality, but I am interested in the result of the calculation nonetheless.
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99of9
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Re: Rating of a perfect chess player
« Reply #3 on: Oct 22nd, 2004, 1:12pm » |
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Here's my definition of a perfect player that I'd be interested in knowing the rating of:
1) when in a theoretically winning position, randomly chooses from the moves which will keep it in a winning position.
2) when in a theoretically drawn position, randomly chooses from the moves which will keep it in a drawn position.
3) when in a theoretically lost position, randomly chooses from all legal moves.
Of course, as we discussed for tic-tac-toe, this is not the smartest player, and nor is it the player which will obtain the highest rating possible when playing in a human tournament. But it is a well defined, and traditionally "perfect" player.
I do not agree with Fritz that it is limiting to talk about perfect play as not having any chance of losing. His "winning probability" argument is not good when you don't know in advance what your opponent is. The opponent may well be perfect too. You are really talking about psychological exploitation of imperfect humans. Although I agree this is certainly possible, I don't think an alien (who had solved chess) would call this anything near perfect. Call it "exceptional used car salesman", able to sell junk.
However, as Mr Brain points out that we talked about in regard to tic-tac-toe, there are even different types of the traditional perfect. Some deliberately lay traps for humans when playing from a lost/drawn position. These are also psychology-dependent, so for simplicity this model player does not include this factor.
I think it is appropriate to ask the question Omar asked if this model player was placed in a pool of non-learning entities. (Perhaps different pools with incrementally increasing ratings as the rating of the player goes up, and wait til it reaches equilibrium). I don't know how well this draw % method would approximate the exact result, but I don't know of any better methods either. I'll be interested in seeing your model Fritz.
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Fritzlein
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Re: Rating of a perfect chess player
« Reply #4 on: Oct 22nd, 2004, 3:18pm » |
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OK, here's a (dubious) mathematical model for chess performance, based on the idea that you don't win with good moves, but you lose with bad moves, and if both players make good moves (or offsetting bad moves) it's a draw.
Imagine two riflemen shooting at a target. If both hit the target, they keep playing. Also if both miss the target they keep playing. But if one hits and the other misses, the one who hits wins the game.
If you assume that each player has a fixed percentage chance to hit the target on each round, independent of the other player and independent of previous shots, and you assume that the game goes on indefinitely until one or the other is victorious (i.e. draws are not possible), then it turns out that the Elo ratings model is a perfect match for this scenario. That is to say, from knowing how often A beats B, and how often B beats C, you can predict precisely how often A beats C. The prediction is independent of the players being good shots or poor shots; you can't deduce from wins and losses alone what anyone's percentage chance to hit the target is. The relative gaps are all that matter, and Elo ratings make the right transitive predictions. (A somewhat remarkable mathematical fact, actually.)
Now this is a terrible model for chess precisely because your move does influence the chance of the other player making a blunder. You aren't each trying to make good moves in a vacuum, you are trying to make good moves against each other. But let's forget that for a while and run with this model.
Suppose that instead of allowing the target shooting to go on indefinitely, we cut off the game after a finite number of shots, say 40, and call it a draw if it wasn't decisive before then. In chess this might correspond to the game trading down equally to a point that no more mistakes can be made after move 40. Now some more complicated equations emerge, with rather different behavior.
First, if we count a draw as half a win and half a loss, Elo's model no longer holds true. The ratings at the top end start getting squeezed together by the draws. In a manner of speaking, the game isn't long enough any more for the most excellent players to demonstrate their full superiority: A 99.99% chance of hitting usually draws versus a 99.999% chance of hitting, even though the latter would win about ten times as often if the game were indefinitely long.
Second, when draws are allowed we lose the property that the scale doesn't matter. That is to say, from knowing how often A beats B, and how often B beats C, you can't predict precisely how often A beats C.
Third, and most relevant for this discussion, is that now if you know for two players A and B the chance that A wins, the chance that B wins, and the chance that A and B draw, you can infer exactly where they are on the absolute scale. That is to say, you can deduce the probability with which each player hits the target on any shot. Solving the question analagous to Omar's chess question, you can deduce how close each of them is to hitting 100% of the time, and calculate from that how often each would draw against a perfect (100% accurate) player.
(Actually, there is still a problem because draws also get more common as the players both get close to zero percent accuracy. To eliminate the problem one could assume that no one is intentionally trying to miss, and that both players have at least a 50% chance of hitting on every shot, so that increasing the skill of equal players always increases the chance of a draw.)
Anyway, I emphasize again how bogus it would be to apply this model to chess. There could be two aggressive players who seldom draw against each other, and we would infer from this that they aren't very good, whereas two conservative players who often draw would be assumed (by this model) to be excellent players. Still, even knowing how bogus the model is, we could take any given set of chess game result data, fit it as closely as possible to the model, and see what results fall out.
Just for fun.
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| « Last Edit: Oct 22nd, 2004, 3:48pm by Fritzlein » |
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RonWeasley
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Re: Rating of a perfect chess player
« Reply #5 on: Oct 22nd, 2004, 3:59pm » |
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I wasn't part of the previous discussions about perfect play, but I would like to propose a variation on 99's definition.
1) when in a theoretically winning position, chooses the variation forcing the win in the fewest number of moves. If there is more than one with equal number, chooses the variation where the opponent is most likely to make a mistake resulting in a win in fewer than the original number of moves.
2) when in a theoretically drawn position, chooses the variation maintaining the draw where the opponent is most likely to make a mistake resulting in a winning position.
3) when in a losing position, chooses the variation where the opponent is most likely to make a mistake resulting in a non-losing position.
First, I think fewest moves to a win is most perfect.
So what do I mean about the opponent being most likely to make a mistake? It depends on the type of opponent, but relates to maximizing the likelihood of the result. For a computer opponent, this may mean the largest number of plys before its value function detects the change in state. For a human opponent, based on perceived experience level, this could mean seeking non-intuitive variations that you think you can analyze better. I'm falling short of a quantitative description of this quality, but I think it is an essential part of the intelligence question Omar is posing. Comments?
The play of this perfect player would produce draws against other perfect players, presumably, and wins almost every time against non-perfect players. The rating would depend on the presence of perfect, near-perfect, and not-close-enough-to-perfect players.
So what's the rating if you always win? What if half the games are draws and the other half wins? If a fixed fraction, f, of opponents are perfect, what is the rating as a function of f, assuming an equal number of games with all opponents?
Sorry in advance if this is obsolete thinking.
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Fritzlein
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Re: Rating of a perfect chess player
« Reply #6 on: Oct 22nd, 2004, 4:06pm » |
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on Oct 22nd, 2004, 1:12pm, 99of9 wrote:Here's my definition of a perfect player that I'd be interested in knowing the rating of:
1) when in a theoretically winning position, randomly chooses from the moves which will keep it in a winning position.
2) when in a theoretically drawn position, randomly chooses from the moves which will keep it in a drawn position.
3) when in a theoretically lost position, randomly chooses from all legal moves.
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That is a reasonable definition of a perfect player. I would predict, however, that such a player, active on the FIDE tour, would have a lower rating than Kasparov. I suspect that most grandmasters would have little trouble preserving the draw (assuming the initial position in chess is a draw) against aimless play like the above, whereas most grandmasters have considerable trouble preserving a draw against Kasparov.
For example, I expect that in the opening position in chess, 1.a3 preserves the draw for white, so your perfect player is as likely to play that as 1.e4. Even later in the game, whenever your perfect player builds up any pressure that isn't enough for the game to be "winning", it will be happy to give away that advantage, and even make a downright disadvantageous move as long as such a move doesn't make the position "theoretically lost".
Just on a hunch I'll peg your perfect player at a FIDE rating of 2600. Yes, it will gain rating points from playing anyone higher-rated, but it will lose rating points to anyone lower-rated who can hold the draw. Playing against the whole field, I'd expect its rating to average out somewhere well below world-championship level.
It's an interesting question the way you pose it, but I think most people have something more like Weasley's definition in mind, so as to insure that the perfect player will have a rating higher than any human.
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| « Last Edit: Oct 22nd, 2004, 4:10pm by Fritzlein » |
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Fritzlein
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Re: Rating of a perfect chess player
« Reply #7 on: Oct 22nd, 2004, 4:23pm » |
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on Oct 21st, 2004, 11:43pm, omar wrote:
First lets assume that a perfect game of chess is a draw (kind of like a complex version of tic-tac-toe). This assumption could be wrong, but for the moment lets assume it is true. Now if two perfect players were playing each other the result would always be a draw. The draw percentage would be 100%. Suppose we make a graph of chess rating (x-axis) vs percentage of draws (y-axis). The games used for this graph should be games between two players that have a fairly close rating (maybe 50 points of one another). We know that the percentage of draws among beginners is about 3% and among the top players it is above 50%. I wonder what this line or curve looks like. Maybe we can extrapolate the line to 100% draws and see what rating it corresponds to. Would this be the appoximate rating of a perfect chess player. Has anyone seen a graph like this?
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I have not seen a graph like this, but I doubt the data would cooperate to the extent of producing a straight line. How will you extrapolate if it is a curve? What if it appears that the percentage of draws approaches 100% only asymptotically as the ratings incease? That would imply that the perfect player has an infinite rating.
Or what if (like winning percentage as a function of rating difference) the graph is concave up until 50% and then concave down thereafter? The part of the graph we have (i.e. up to 50%) will look radically different from the remainder, making extrapolation very dicey.
That said, your idea is probably more likely to produce a reasonable result than mine is.  I'm curious to see such a graph, regardless of what we think it means about perfect chess. It might well be concave up everywhere, which is to say that the percentage of draws actually approaches 100% more and more quickly as the ratings increase. I won't even venture a guess on this score.
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omar
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Re: Rating of a perfect chess player
« Reply #8 on: Oct 22nd, 2004, 5:50pm » |
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I wasn't expecting this to get into a discussion of what is a perfect player. But since it hasn't ever been discussed to death (at least on this forum) lets do it.
The model of a perfect player which Toby has described is what I generally think of as a perfect player. It randomly samples only the perfect moves from any given position. Maybe we can call it the 'random perfect player' (RPP) There can also be perfect players which use some heuristics to select their move, but always select from the set of perfect moves for any given position. Lets call these the 'heuristic perfect players' (HPP).
Now in order to have ratings for our perfect players the field of players must contain some 'non-perfect players' (NPP); otherwise all games would have the same result and all our perfect players would always have the initial rating we assigned them.
If our field of players includes non-perfect players it is possible for some of the perfect players to have a higher rating than other perfect players. The moves selected by some of the HPP may lead more often to positions where the NPP are more likely to blunder (i.e. selecting a move from the losing set of moves) and thus resulting in more wins for the HPP. It is also possible that the moves selected by some of the HPP may lead more often to positions where the NPP are less likely to blunder and thus result in more draws for the HPP. So there will be a range of ratings for the perfect players. Now keep in mind that even the lowest rated perfect player will kill you if you make even one mistake throughout the game; after all it is a perfect player So against most of the NPP all of the perfect players will win. Only against some of the NPP which are on the verge of playing almost perfectly will some perfect players win more often while others draw more often. So I would not expect the range of ratings for the perfect players to be very wide when compared to the difference between the average player in the field and the lowest rated perfect player.
I would suspect that the rating of the RPP will be somewhere in the middle of this range.
So the bottom line is: it is possible to define at least conceptually what a perfect player is; there can be a range of ratings among the perfect players; and that (at least rating wise) the RPP is probably a good example of an ideal perfect player.
Now as I said in my first post, I am only interested in an appoximate rating of a perfect chess player. The word approximate is very important here. So in this case there is no need for us to really even define what a perfect player is (but it is fun to think about it ).
All Im proposing is making a graph of ratings vs 'draw percentage', extrapolating the line and reading off the rating where the line crosses 100% draws. The number might not be accurate, but I figure it is a lot easier to try doing this than to solve chess . It can't hurt to see what comes out; who cares if it is not very accurate.
So has anyone ever seen such a graph?
Omar
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99of9
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Re: Rating of a perfect chess player
« Reply #9 on: Oct 22nd, 2004, 6:35pm » |
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on Oct 22nd, 2004, 4:06pm, Fritzlein wrote:| That is a reasonable definition of a perfect player. I would predict, however, that such a player, active on the FIDE tour, would have a lower rating than Kasparov. |
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I agree. Basically one's ability to hold a 0 net score against this beast relies on the player being able to not blunder for an entire game. It appears that humans may be able to do this most of the time now in chess, therefore they have surpassed Random-Perfect. I don't think we have surpassed Random-Perfect in arimaa yet (if arimaa is a draw, and we certainly haven't if arimaa is a win).
By the way: I participated in a computer Rock-Scissors-Paper (Roshambo) competition. Random-Perfect does badly in such competitions, because although it never loses and always draws (by playing randomly), it never exploits weak players. So that is another game where I think we have surpassed Random-Perfect.
http://www.cs.unimaas.nl/~donkers/games/roshambo03/
I agree however that the "answer" omar's method will provide is the answer to Ron's player's strength. Because humans are always trying to trap one another, and actively force their opponent toward a loss, they are probably on the way toward approximating Ron's player in some human-context kind of way.
Ron, here's one way you could have a non-context-dependent way of defining "good" choices of moves:
Explore the full game subtree corresponding to each satisfactory move. Count the number of leaves that are wins, subtract the number of leaves that are losses, and the highest total score is the move you should play.
This is still not the best player a human could ever face, but it would be pretty scary! I think this player would win any non-learning human chess tournament today. (the only problem is that since this player is deterministic, once someone drew, humans could learn the correct path and it would forever more be a draw - that's why I specify non-learning.)
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| « Last Edit: Oct 22nd, 2004, 6:40pm by 99of9 » |
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omar
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Re: Rating of a perfect chess player
« Reply #10 on: Oct 25th, 2004, 10:22am » |
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on Oct 22nd, 2004, 4:23pm, Fritzlein wrote:
I have not seen a graph like this, but I doubt the data would cooperate to the extent of producing a straight line. How will you extrapolate if it is a curve? What if it appears that the percentage of draws approaches 100% only asymptotically as the ratings incease? That would imply that the perfect player has an infinite rating.
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Actually I was also wondering what the graph might be like. I don't think the graph will be concave down or asymptotically approach 100%. Because at some finite rating it has to actually hit 100%. Think of the set of end nodes being sampled at a certian rating when the draw percentage is say 90%. As the rating (of both players) increases we are sampling a smaller subset of these end nodes. A subset which most likely contains fewer non-drawing nodes. Convex down would mean that we are throwing out drawing nodes faster than non-drawing nodes.
My guess is that it will probably be a curve that is convex up the whole way from 0 to maxRating.
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omar
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Re: Rating of a perfect chess player
« Reply #11 on: Oct 27th, 2004, 5:10pm » |
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on Oct 22nd, 2004, 6:35pm, 99of9 wrote:
I agree. Basically one's ability to hold a 0 net score against this beast relies on the player being able to not blunder for an entire game. It appears that humans may be able to do this most of the time now in chess, therefore they have surpassed Random-Perfect.
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If humans are able to surpass Random-Perfect then we may never see a day when humans are consistently defeated no matter how good the computers get. Kasparov once said in an interview that he worries about the day when humans will never be able to win a game against the best computer. But this may not happen if we can reach Random-Perfect. Chess may well be a complex game in which humans can reach a level of perfect play. This may have already happend in Checkers. See:
http://www.wylliedraughts.com/Tinsley.htm
But I don't think that humans have surpassed random perfect (most of the time) in chess yet. If that was the case then most of the games among the top rated humans would be natural draws. Currently over 60% of such games are draws, but many of them are due to mutal draws. If mutual draws were not allowed I think it would be less than 50% draws.
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99of9
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Re: Rating of a perfect chess player
« Reply #12 on: Oct 27th, 2004, 7:17pm » |
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on Oct 27th, 2004, 5:10pm, omar wrote:If humans are able to surpass Random-Perfect then we may never see a day when humans are consistently defeated no matter how good the computers get.
But I don't think that humans have surpassed random perfect (most of the time) in chess yet. If that was the case then most of the games among the top rated humans would be natural draws.
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No, I disagree.
Humans do not need to *become* perfect to hold their own ELO-wise against random-perfect. They just have to be able to:
1) hold their own (draw) most of the time against R-P
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2) be able to beat the rest of the field more often than R-P does.
I believe that even I might have the chess ability to achieve number 1. This may sound crazy, but think about it. You have to remember the massive difference between RANDOM-perfect and INTELLIGENT-perfect. R-P is just trying to stay in the drawing regime, it is not trying to achieve anything but that. Every time I threaten a piece swap, it will most likely move some other random piece, and allow me to make the swap. The more pieces I swap, the easier it becomes for even me to draw an endgame against a player playing without any purpose at all.
Thinking about the fact that R-P does not even try to lay traps, I'm pretty sure all grandmasters would be able to get draws against it, as long as they didn't try for a win. So it would perform exactly mid-field in any high level tournament (or slightly above that if someone blundered), as long as everyone knows who they're playing. In form Kasparov/Kramnik/Anand would still beat lower GMs, so condition #2 would also be achieved.
Intelligent-Perfect is an entirely different kettle of fish. I agree with Kasparov that there will come a point where humans can hardly ever get a draw against a beast of this nature (or even a 100-ply searcher). I think this is what you're really going to get an estimate of when you talk about draw percentages converging to 100%.
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| « Last Edit: Oct 27th, 2004, 7:19pm by 99of9 » |
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MrBrain
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Re: Rating of a perfect chess player
« Reply #13 on: Oct 28th, 2004, 9:53am » |
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Random perfect would not be as strong as what most would consider perfect. This reminds me of a story I read (I believe in the excellent book on computer checkers "One Jump Ahead") where a parameter was reversed in a chess computer in a computer tournament. (Going from memory here, so the story may not be exact, but this is the gist of it.)
When the computer found winning positions in the endgame database, instead of picking the move to win in the least number of moves, it instead picked the move to win in the most moves. As a result, the computer played very bizarre looking moves that made everyone think that the computer blundered. But eventually, the 50-move draw rule comes into effect forcing the computer to eventually mate.
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Fritzlein
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Re: Rating of a perfect chess player
« Reply #14 on: Oct 29th, 2004, 10:39am » |
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on Oct 22nd, 2004, 3:59pm, RonWeasley wrote:I wasn't part of the previous discussions about perfect play, but I would like to propose a variation on 99's definition.
1) when in a theoretically winning position, chooses the variation forcing the win in the fewest number of moves. If there is more than one with equal number, chooses the variation where the opponent is most likely to make a mistake resulting in a win in fewer than the original number of moves.
2) when in a theoretically drawn position, chooses the variation maintaining the draw where the opponent is most likely to make a mistake resulting in a winning position.
3) when in a losing position, chooses the variation where the opponent is most likely to make a mistake resulting in a non-losing position.
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This is a reasonable definition of perfect play, and I think it corresponds well with most people's intuitions about perfect, but this definition has the possible flaw that an "imperfect" player can get a higher rating. According to the Elo system, the more points you score the higher your rating is. So suppose you are in a theoretically drawn position where the trickiest (against this particular human opponent) sound move is still fairly boring and will win only 20% of the time and draw 80% of the time, whereas making the best dangerous, unsound move will win 80% of the time and lose 20% of the time. Your perfect player will choose the former because it is only choosing among moves which preserve the draw, but a more aggressive player who will risk losing scores more points on average (0.8 vs. 0.6) and thus earns a higher Elo rating.
This sort of conundrum complicates the question "What is the Elo rating of a perfect player", because the notion of having as high a rating as possible may actually contradict the notion of perfect play, unless you want to actually define perfect play in terms of Elo rating, in which case you have to specify the opposition.
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