The rating of a perfect player

Is your perfect player allowed to sneeze, in the middle of a move, and leave his queeen en prise?

Perfect people are so boring though. Fischer was an exciting player and some people's idea of a perfect player when he was at the top of his game (remember the 6 to zero thrashing of the top players in the world?) But beyond the chess board he was.... wel,l strange.

I think a perfect player refers to a theoretical entity who can plays the objectively best moves in every position.

There's was an interesting discussion in the Rybka forums about this, appropriately titled "God's rating."

A perfect chess player is further beyond Kasparov than Kasparov is beyond me or you. No player would manage to even draw against a perfect player. The only way a perfect player would draw is vs a player who is also playing perfecty. I would estimate their rating to be OVER 9000!!!!!!!

Fortunately white is saved by its illegality.

The most successful opening moves for white, starting with the highest scoring are:

• Nf3
• d4
• g3
• c4
• e4

e4 suffers most because of the almost level score against 1. ... c5.

On this point, to a perfect engine, it will know which moves are critical errors (changing the theoretical result) and which are merely errors to a less omniscient player. So it can choose a theoretically correct move to create difficulties for its opponent even if a lesser computer would give it a worse evaluation.

There is none. Some moves have scored more than others in practice. That people  believe that the correct result of a position is a draw is a quite separate thing. And both are independent of knowing the theoretical result of a position with perfect play.

2000.  No more no less.  Perfection must eventually erode until it infinitely approaches this rating.

Once you are 800 points above another player (1200 is the start) no pts are gained.  Since perfection should achieve a win rate of 100%, everything non-perfect will infinitely be ceilinged at 1200, while everything perfect will be  should eventually approach as low as 2000 without ever reaching that low.

One of the most illogical posts I saw here.

That would make sence!

philidor.  As many approach perfection, the meaning of the numbers would change.  If a perfect computer (100% win) played Magnus Carlsen 1,000,000 times, Magnus's rating would fall to below 1200, as would everyone else the perfect player played.  Thus, over an infinite number of games, the average rating of the opponent would approach 1200, since that is what people start at.  If you can get no higher than 800 pts. better than your best opponent, then eventually all imperfect would approach 1200, and all perfect would approach 2000.  This assumes that the perfect player/computer is always able to play the best possible opponent.  If that is the case, the best possible could never be higher maintain a rating higher than 1200. 

This doesn't exacly work because ratings already exist that are higher than 1200, so by the time everyone dropped far down, the perfect player would already be well past that.  It makes more sense if everyone started at 1200 today.. or.. if a perfect player came along, beat everyone down to 1200 or below, and then retired, then another perfect player came along, the highest he should be able to make is around 2000 if he is always playing against the best.

That's ridiculous because you're assuming that perfect play would win 50% of the games and draw the other 50% against the best human player of all time, which is so erroneous as to be complete garbage - effectively, you're saying that when Kasparov is playing as White, he has a 50% chance of producing a perfect game! Even the computers that are currently not considered top competitive software would very rarely draw a game playing as Black against Kasparov - they would win the vast majority of games, and probably end up scoring about 19/20 as Black against full points as White. You want a score out of 100? Make that 97.5. And that's for a computer not considered to be part of serious competitive play - computers these days completely outclass humans. Let's call these computers Tier 3s; the Tier 2 computers, near the bottom of the World Top 100, can score approximately 95/100 against these computers, and the Tier 1 computers (e.g. Rybka and Houdini, the very best) can score about 93/100 against the Tier 2 computers. This is a gap of about 1,300-1,500 points between the top computers and the top humans.

In my own opinion, no computer so far has come anywhere remotely close to perfect play. Our search depths range up to about 18; I would say that true perfect play would require a search of about 60 moves deep (in most situations - to obtain a computer which could produce the perfect move in any situation would require a depth of many hundreds of moves, but most positions would not require that). So I would say that perfect play could be produced in most situations by a computer with a rating (60/18)*1,600=5,333 points ahead of the highest humans, i.e. approximately 8,230.

@verticle5, with all due respect, your understanding of how the rating system works is flawed in any close to realistic scenario. When a single player enters the pool of players, he/she/it finds its level without significantly altering the ratings of other players.

 That is true, but we aren't looking at realistic scenarios...  except .  If a 'perfect' player wins 100% of his games, and assuming he can play an infinite number of times ( over the course of time this would happen) there would be a gradual shift downwards of EVERYONE else's rating.  The only reason there is currently so much variance is precisely because of imperfection. 

Let's just assume that there are 50 perfect chessplayers in the world, and everyone else is equally horrible, and there is an endless supply of them.  The 50 perfect players would always draw each other, and would always beat everyone else.  The ratings of the endless supply of horrible people would always be gravitating towards 1200, and thus the perfect players would never be able go much further than 2000. 

Basically, my whole hypothetical scenario here relies on the idea that perfection is sooooo much better than 'excellent'.. that anything short of perfect is actually equally bad. Take the example of tic-tac-toe.  Is someone who always loses on the 8th move better than someone who always loses on the 6th move????  Not really, because they are both very bad. In the same way a perfect tic-tac-toe player would hardly distinguish between two people he can easily beat... a 'perfect' chessplayer would be so good that a 2500 and an 1000 rated player are not even distinguishable.  They are both equally bad in that they *should* lose 100% of the time to a "perfect" player.

But...... Bringing tic-tac-toe into the example, (assuming a rating system), we really don't really know how a perfect player would rate.  Am I perfect at tic-tac-toe because I can always make the best move? I guess... but that means I should be drawing every time. 

The question here is flawed because we are not really seeking 'perfection' of the winner as much as trying to understand the likelihood of mistakes of their opponents.  I don't care how great a computer is.. it can not beat me at tic-tac-toe.  Because the mind is limited, no one can say the same about chess.  The question of a forced win in chess doesn't even matter, because if it is possible, then a 'perfect' player could still lose as black. just as a 'perfect' tic-tac-toe player would lose to someone who got to take the first two moves ( a forced win).

So... in conclusion... if chess perfection is attainable, then it is meaningless :)

I don't think there is any evidence that a perfect player would win all his games.

He certainly won't lose any, but how do we know Carlsen or Anand won't get the odd draw?

Because they make too many mistakes.

I think they might occasionally play games where they don't make any move that loses by force. I understand that's up to their opponents as well, but still. I'm pretty sure the perfect player would drop huge points by the occasional draw. If not against humans, against houdini or something.

The prediction of the Elo statistical model is that if a player much stronger than the best human existed, he/she would still get an occasional draw against it. Just not very many. And vastly fewer fluke wins.

How many draws do you think a top GM would get against Houdini/Rybka?

It's been 3-4 years since those odds matches where strong GMs could only score a few draws here and there.  The programs are a lot stronger today.

And we're talking about something that plays much stronger than that... a so called perfect player.  There is no doubt that top GMs like Anand or Carlsen would not get an odd draw even after hundreds of games.

With current evidence, processor power increases of a factor of 2 provide about  50-60 points rating increase. This is associated with a (very rough) requirement for an increase in speed of 6 times to get 1 more ply of search depth. Obviously, if this relationship could be extrapolated indefinitely, the perfect rating would be extremely high, since search depths up to well over 100 would be useful in all games (requiring almost unimaginable processing power). But there is an intuitively understandable reason to expect increasing speed to provide dimimishing returns, and it may be possible to get a better estimate of the perfect rating from this.

Firstly imagine a comparison of two computers, identical except one is able to calculate one move deeper (N+1 versus N, say). On average this advantage would allow the first computer to gain a slight edge (call it E) over its opponent over each N moves. But because of the overlap between the trees the computers are using at each stage, it might be reasonable to believe that, on average, this edge could only be multiplied by M/N by the endgame (roughly speaking), where M is the length of the game to this point. We might also think that this edge would be enough to win if it is above a certain threshold.

The point is that if M/N is 10, a computer can achieve an edge of 10 x E by the endgame on average, but if M/N is only 5, the computer can only achieve an edge of 5 x E by the endgame. This rough logic suggests that it is not the number of extra ply that leads to an increase in rating, it is the ratio of the search depth. i.e. 11 ply versus 10 ply is as big a rating difference as 22 ply versus 20 ply. The first of these requires a 6 times increase in speed, but the second requires a 36 times increase.

Of course all of these things have large random components - the length of games varies a lot, the search depth varies with the nature of the position, the edge that is achieved by searching deeper has a random component - the shallower computer will very often pick the same move as the deeper computer and very occasionally will choose a better move by accident. [all comparisons are with respect to deeper evaluation functions].

The above rationale gives a model for predicting increase in rating from increased search depth:

dR/dD = k/D, where R is rating, D is depth and k is some constant, then

Integrating this:

R = k ln(D)

[EDIT: UNJUSTIFIABLE ASSUMPTION THAT INTEGRATION CONSTANT IS ZERO INVALIDATES REST OF THIS POST]

Using the figure of 60 points for a ply and a rating of about 3000 as the base point for a computer thinking to about 20 ply, we estimate k = 733.

If we assume a search depth of 200 ply would be close to perfection, this would give a rating of 3883 for the perfect computer, lower than I expected. The theoretical model, the calculation methodology and several of the numbers used are very rough, so there is a substantial amount of uncertainty.

Maybe because I've been up 24 hours but I just don't see the math part.  Why are you using integration and how did you get k = 733?

Add more little steps please