Answers.Body: <p>I'm arguing below that it is logically problematic to even define such a thing.</p>
  <p>The ELO of the strongest computers is really defined only relative to a ladder of weaker computers, starting from the ELO of best human players. A program A winning a certain percent of points against a 2775 ELO grandmaster will be assigned ELO 2875. A program B winning the same percentage of points against A will be assigned ELO 2975. And so on, up to maybe 3700 ELO of AlphaZero. The differences between ladders are better to be reasonable, as, e.g., pitching AlphaZero against a human would lead to 100% of wins in practice, and no numeric value of ELO can be assigned on that basis.</p>
  <p>Now, how do we assign ELO to God? Inevitably, by measuring it against a weaker program, say W, that plays imperfectly. But since God is omniscient, not only it knows the best move in each position, it also can exploit ideally any imperfection in W's play. So, either W itself plays perfectly, in which case every game will be a draw, or God will win 100% of games. In the latter case, it is impossible to define the ELO of God relative to W.</p>
  <p>So the only way to define the ELO of God is to say that it is equal to the ELO of any program W that itself plays perfectly, but is not necessarily omniscient. This ELO, again, can only be assigned relative to some weaker program V. But that ELO difference is defined not so much by the fact that W plays a perfect (i.e., evaluation-preserving) move at every position, but by how well it exploits the fact that V plays imperfectly - i.e., two programs W,W' that both play perfectly can have a different relative ELO against V. But on the other hand, they must have the same ELO, as they draw all games against each other (and God).</p>
  <p>All these problems are really exacerbated by the fact that we cannot even meaningfully assign an ELO difference to two <em>deterministic</em> programs by pitching them repeatedly against each other, as all the outcomes will be the same... So we need something like randomizing over starting positions (in which case the ELO difference will be severely dependent on the ensemble of these positions), or use non-deperministic programs... That may solve the problem of the highest rung of the ladder, but still it is clear that the &quot;number of rungs&quot; in a ladder reaching perfect play from human level is not well defined - it depends on the programs in the ladder and how adapted they are against each other's weaknesses.</p>
  
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Answers.Text: "either W itself plays perfectly, in which case every game will be a draw, or God will win 100% of games": No. W might play perfect games some of the time, by sheer luck (and it wouldn't necessarily require a great deal of luck if W's game was _almost_ perfect).
Answers.UserDisplayName: TonyK
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Answers.Text: @TonyK, the notion of "sheer luck" assumes that W is nondeterministic - i.e., might behave in different ways under the same conditions (position+clock). In that case, what you are saying is correct, assuming that God has no control of the randomness generator used by W :)
Answers.UserDisplayName: Kostya_I
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Answers.UserDisplayName: Kostya_I
Answers.Body: <p>It would need at least couple of centuries even for quantum computers to solve or probably it wouldn't be solved at all. Chess is highly complex with over 10^120 positions. So, assuming that chess would be solved is super highly near to impossible.</p>
  
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Answers.UserDisplayName: Sparsh Sharma
Answers.Body: <p><a href="https://en.chessbase.com/post/how-god-plays-chess" rel="nofollow noreferrer">Source</a></p>
  <blockquote>
  <p>So the question arises: what happens if you let the system ponder for eight hours, or for eighty, or run on a far larger number of processors? My prediction: it will not cross Elo 4000 — nothing ever will. The draw in chess will prevent that from happening: a 3900 program will always be able to hold the game, however strong the opponent.</p>
  </blockquote>
  <p>By the way current computers aren't close to perfect play in general. They might be really hard to beat from the starting position, but give them a more dubious position and they can still lose (and fail to win the reverse).</p>
  
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Answers.Text: Accepted for Friedel (and Thompson) being reknowned experts on the matter and, last but not least, the story having me in stitches. Only time can tell where the Elo of computers will go (quantum computers, anyone?).
Answers.UserDisplayName: Hauke Reddmann
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Answers.Text: What are we going to achieve with these talks. Machines are not going to play football.. Or sprints
Answers.UserDisplayName: ShadYantra
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Answers.Text: An ad article to sell their 2018 tablebase product wherein author makes an assumption on leela and fails to provide any data is good enough to close this divine question?
Answers.UserDisplayName: Peter Fischer
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Answers.Text: This is nice answer, but for perspective it is good to remember that ~20 years ago many experts thought that a 2700/2800 chess playing entity would always be able to play for a draw with White and get it, however strong the opposition. The recent boom of super-strong computers has shown humans how little they understood about the game and moved the supposed frontier by 1000 rating points (!), but maybe in 2030 we will be confidently saying: "we can see that today's 3900 softwares are far from perfect, but surely a 4900 elo software would be unbeatable!", and so on...
Answers.UserDisplayName: Evargalo
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Answers.Text: @Evargalo do you have a source for how 2800 players can play for a draw and always get it? My justification for how today's 3900 software are unbeatable from the starting position is the ICCF results - virtually every game at top level is drawn.
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Body: <p>Assume a 32-piece tablebase would exist. It would be a Chess God. Surely there have been estimations of its ELO?</p>
  <p>According to the very related <a href="https://chess.stackexchange.com/questions/8630/is-this-really-limitation-of-absolute-elo/8637#8637">question</a>, two problems arise:</p>
  <p>a) At any point the practical MaxELO&lt;=400+second best rated player.</p>
  <p>b) ELO drift over time.</p>
  <p>Thus, a) I'd like to know the <em>theoretical</em> maximum (not the practical as given in the linked answer), b) at some arbitrary but fixed time (maybe the one at where it was answered somewhere in the internet).</p>
  <p>c) <a href="https://en.wikipedia.org/wiki/Swedish_Chess_Computer_Association" rel="nofollow noreferrer">Wiki data</a> for chess computers for convenience. No S-curve to see yet (maybe due to b)...)</p>
  
Comments.ContentLicense: CC BY-SA 4.0
Comments.CreationDate: 2022-07-19T08:28:12.450Z
Comments.Text: And what about other variants.. ELO is the level of difficulty I feel
Comments.UserDisplayName: ShadYantra
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Comments.Text: Ad a: Looks to be a function of (among others) "fiat", ELO given to entities new to the system. Ad c: Yes, ELO is distributed over population in a sigmoid, but what makes you assume the course of peaks over time would?
Comments.UserDisplayName: Peter Fischer
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Comments.Text: @PeterFischer: Easy - current chess computers should be near perfect play by now, so "law of diminishing return" should set in. Of course I might underestimate chess big time and maybe I'm totally wrong...
Comments.UserDisplayName: Hauke Reddmann
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Comments.CreationDate: 2022-07-19T19:13:00.283Z
Comments.Text: I figured the course of peaks over time was roughly linear (exp. search depth divided by exp. hardware progress). Anyway, a lower bound for gods rating was suggested once at ~5230, based on an asymptotic draw rate of 1, meaning there would be a challenger to god near his level.
Comments.UserDisplayName: Peter Fischer
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Comments.CreationDate: 2022-07-19T19:28:16.497Z
Comments.Text: Recalled correctly and found it! Jeremy Rutman's work, posted here: https://blogs.harvard.edu/doctorsplendiferous/2021/02/08/gods-chess-rating/ The underlying math looks a bit suspicious though, and it's original is older than 2021 (18 or 19 I think). If that helps...
Comments.UserDisplayName: Peter Fischer
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Comments.CreationDate: 2022-07-20T01:30:38.560Z
Comments.Text: The Elo system (not ELO, it's a man's name, not an acronym) seems to be based on the dubious idea that, if we know how A scores against B and how B scores against C, then we can deduce how A scores against C. A 32-man tablebase would of course play God to a draw, but God would do much better against slightly weaker players, because He knows how to play to their weaknesses. A 32-man tablebase choosing randomly among all game-theoretically equal moves might let even a weak player off with a draw, e.g., it might consider 1. h3 and 2. Rh2 just as good as the Ruy Lopez or Queen's Gambit.
Comments.UserDisplayName: bof
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Comments.Text: A 32 piece tablebase would never lose, but it could be trivial to draw against. There's no "trying to win a drawn position" built into tablebases, any move that keeps the position drawn is considered equal to any other. The same happens when you want to train defending an endgame against the tablebase -- it's often way too easy because it doesn't try to win.
Comments.UserDisplayName: RemcoGerlich
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Comments.CreationDate: 2022-07-20T07:18:42.883Z
Comments.Text: @bof: Good point, but then, traditionally, God is benevolent :-) (Couldn't have used Old Nick metaphorically, since that dude often out-gambits himself :-)
Comments.UserDisplayName: Hauke Reddmann
Tags: <theory><rating><tablebases>
Title: Estimation of "God's Elo"
ThreadText: # Estimation of "God's Elo"
  
  (Asked by: username_0 on 2022-07-19)
  
  Assume a 32-piece tablebase would exist. It would be a Chess God. Surely there have been estimations of its ELO?
  
  According to the very related [question](https://chess.stackexchange.com/questions/8630/is-this-really-limitation-of-absolute-elo/8637#8637), two problems arise:
  
  a) At any point the practical MaxELO<=400+second best rated player.
  
  b) ELO drift over time.
  
  Thus, a) I'd like to know the *theoretical* maximum (not the practical as given in the linked answer), b) at some arbitrary but fixed time (maybe the one at where it was answered somewhere in the internet).
  
  c) [Wiki data](https://en.wikipedia.org/wiki/Swedish_Chess_Computer_Association) for chess computers for convenience. No S-curve to see yet (maybe due to b)...)
  
  **Tags:** theory, rating, tablebases
  
  > **Comments:**
  > **username_1**: And what about other variants.. ELO is the level of difficulty I feel  
  > **username_2**: Ad a: Looks to be a function of (among others) "fiat", ELO given to entities new to the system. Ad c: Yes, ELO is distributed over population in a sigmoid, but what makes you assume the course of peaks over time would?  
  > **username_0**: @PeterFischer: Easy - current chess computers should be near perfect play by now, so "law of diminishing return" should set in. Of course I might underestimate chess big time and maybe I'm totally wrong...  
  > **username_2**: I figured the course of peaks over time was roughly linear (exp. search depth divided by exp. hardware progress). Anyway, a lower bound for gods rating was suggested once at ~5230, based on an asymptotic draw rate of 1, meaning there would be a challenger to god near his level.  
  > **username_2**: Recalled correctly and found it! Jeremy Rutman's work, posted here: https://blogs.harvard.edu/doctorsplendiferous/2021/02/08/gods-chess-rating/ The underlying math looks a bit suspicious though, and it's original is older than 2021 (18 or 19 I think). If that helps...  
  > **username_3**: The Elo system (not ELO, it's a man's name, not an acronym) seems to be based on the dubious idea that, if we know how A scores against B and how B scores against C, then we can deduce how A scores against C. A 32-man tablebase would of course play God to a draw, but God would do much better against slightly weaker players, because He knows how to play to their weaknesses. A 32-man tablebase choosing randomly among all game-theoretically equal moves might let even a weak player off with a draw, e.g., it might consider 1. h3 and 2. Rh2 just as good as the Ruy Lopez or Queen's Gambit.  
  > **username_4**: A 32 piece tablebase would never lose, but it could be trivial to draw against. There's no "trying to win a drawn position" built into tablebases, any move that keeps the position drawn is considered equal to any other. The same happens when you want to train defending an endgame against the tablebase -- it's often way too easy because it doesn't try to win.  
  > **username_0**: @username_3: Good point, but then, traditionally, God is benevolent :-) (Couldn't have used Old Nick metaphorically, since that dude often out-gambits himself :-)  
  
  ---
  
  ## 3 answers
  
  ### Answer by username_8 on 2022-07-20. Score: 4 | Accepted answer
  
  [Source](https://en.chessbase.com/post/how-god-plays-chess)
  
  > So the question arises: what happens if you let the system ponder for eight hours, or for eighty, or run on a far larger number of processors? My prediction: it will not cross Elo 4000 — nothing ever will. The draw in chess will prevent that from happening: a 3900 program will always be able to hold the game, however strong the opponent.
  
  By the way current computers aren't close to perfect play in general. They might be really hard to beat from the starting position, but give them a more dubious position and they can still lose (and fail to win the reverse).
  
  > **Comments:**
  > **username_0**: Accepted for Friedel (and Thompson) being reknowned experts on the matter and, last but not least, the story having me in stitches. Only time can tell where the Elo of computers will go (quantum computers, anyone?).  
  > **username_1**: What are we going to achieve with these talks. Machines are not going to play football.. Or sprints  
  > **username_2**: An ad article to sell their 2018 tablebase product wherein author makes an assumption on leela and fails to provide any data is good enough to close this divine question?  
  > **username_9**: This is nice answer, but for perspective it is good to remember that ~20 years ago many experts thought that a 2700/2800 chess playing entity would always be able to play for a draw with White and get it, however strong the opposition. The recent boom of super-strong computers has shown humans how little they understood about the game and moved the supposed frontier by 1000 rating points (!), but maybe in 2030 we will be confidently saying: "we can see that today's 3900 softwares are far from perfect, but surely a 4900 elo software would be unbeatable!", and so on...  
  > **username_8**: @username_9 do you have a source for how 2800 players can play for a draw and always get it? My justification for how today's 3900 software are unbeatable from the starting position is the ICCF results - virtually every game at top level is drawn.  
  
  ---
  
  ### Answer by username_5 on 2022-07-29. Score: 11
  
  I'm arguing below that it is logically problematic to even define such a thing.
  
  The ELO of the strongest computers is really defined only relative to a ladder of weaker computers, starting from the ELO of best human players. A program A winning a certain percent of points against a 2775 ELO grandmaster will be assigned ELO 2875. A program B winning the same percentage of points against A will be assigned ELO 2975. And so on, up to maybe 3700 ELO of AlphaZero. The differences between ladders are better to be reasonable, as, e.g., pitching AlphaZero against a human would lead to 100% of wins in practice, and no numeric value of ELO can be assigned on that basis.
  
  Now, how do we assign ELO to God? Inevitably, by measuring it against a weaker program, say W, that plays imperfectly. But since God is omniscient, not only it knows the best move in each position, it also can exploit ideally any imperfection in W's play. So, either W itself plays perfectly, in which case every game will be a draw, or God will win 100% of games. In the latter case, it is impossible to define the ELO of God relative to W.
  
  So the only way to define the ELO of God is to say that it is equal to the ELO of any program W that itself plays perfectly, but is not necessarily omniscient. This ELO, again, can only be assigned relative to some weaker program V. But that ELO difference is defined not so much by the fact that W plays a perfect (i.e., evaluation-preserving) move at every position, but by how well it exploits the fact that V plays imperfectly - i.e., two programs W,W' that both play perfectly can have a different relative ELO against V. But on the other hand, they must have the same ELO, as they draw all games against each other (and God).
  
  All these problems are really exacerbated by the fact that we cannot even meaningfully assign an ELO difference to two *deterministic* programs by pitching them repeatedly against each other, as all the outcomes will be the same... So we need something like randomizing over starting positions (in which case the ELO difference will be severely dependent on the ensemble of these positions), or use non-deperministic programs... That may solve the problem of the highest rung of the ladder, but still it is clear that the "number of rungs" in a ladder reaching perfect play from human level is not well defined - it depends on the programs in the ladder and how adapted they are against each other's weaknesses.
  
  > **Comments:**
  > **username_6**: "either W itself plays perfectly, in which case every game will be a draw, or God will win 100% of games": No. W might play perfect games some of the time, by sheer luck (and it wouldn't necessarily require a great deal of luck if W's game was _almost_ perfect).  
  > **username_5**: @username_6, the notion of "sheer luck" assumes that W is nondeterministic - i.e., might behave in different ways under the same conditions (position+clock). In that case, what you are saying is correct, assuming that God has no control of the randomness generator used by W :)  
  
  ---
  
  ### Answer by username_7 on 2022-07-26. Score: -1
  
  It would need at least couple of centuries even for quantum computers to solve or probably it wouldn't be solved at all. Chess is highly complex with over 10^120 positions. So, assuming that chess would be solved is super highly near to impossible.
  
  
  ---
  
  