Chess is a Draw

[lkaufman]

To my thinking, "chess is a draw" is in the same category as something like "there are no humans living permanently on the moon". This can't be proven; it is theoretically possible that when we landed on the moon, one or two astronauts secretly remained, drilled deep below the surface, brought enough supplies to last half a century, and have remained hidden there ever since. Hard to disprove this rigorously, but I think almost everyone would agree that the probability of the statement being true is at least 99.9999999%. Similarly we can use statistical arguments using engines to show that chess is a draw with over 99.9999999% probability, we just can't prove it 100%. Some may enjoy the challenge of a 100% proof, but for me I'm willing to accept 99.9999999% as good enough to make all decisions in life. I wouldn't call a claim false just because there might be one chance in a billion or so that it is.

There is no evidence that the claim is false.
What would be false is the claim that it has been deductively established as true. Most of science does not care about such a claim in the first place (and daily life even less).

Most of us just accept certain things as obvious without a formal proof; we are not lizard people, the moon is not made of green cheese, the earth isn't hollow, etc. Things like this can't be totally proven impossible, one can always construct some silly argument as to why they could be true. Similarly chess might be a mate in 385 moves with perfect play, but no strong player believes this, there would at least almost certainly be some hint by now, some line that at least forced a meaningful advantage. If there were a betting market on this, and a way to settle the bet, the odds would be astronomical assuming one had to bet serious money. I'm not opposed to the search for a formal proof, I just consider it rather pointless, we should just assume that chess is a draw unless someone at least finds a line that hints otherwise.

[towforce]

This is the first time I've heart of photonic (optical) computers. Reading the wiki article, there's certainly a possibility of a big gain, but also some serious challenges to be overcome - link. It's not likely that they'll be topping the supercomputer league table (Top500) any time soon.

One issue for new computing technologies like optical or quantum: ENIAC, an early computer built in 1945, was big, expensive, unreliable, slow, difficult to program, had a tiny amount of memory, consumed a huge amount of power, and was generally absolute rubbish in comparison with today's computers. However, at the time it was the best computing device ever, and so large amounts of resource flowed into the development of the technology. I cannot see the same thing happening for new computing methods because their performance is so much poorer than even cheap computers that only need milliamps of power (e.g. smart watches / fitness trackers worn on the wrist) right now.

[syzygy]

To my thinking, "chess is a draw" is in the same category as something like "there are no humans living permanently on the moon". This can't be proven; it is theoretically possible that when we landed on the moon, one or two astronauts secretly remained, drilled deep below the surface, brought enough supplies to last half a century, and have remained hidden there ever since. Hard to disprove this rigorously, but I think almost everyone would agree that the probability of the statement being true is at least 99.9999999%. Similarly we can use statistical arguments using engines to show that chess is a draw with over 99.9999999% probability, we just can't prove it 100%. Some may enjoy the challenge of a 100% proof, but for me I'm willing to accept 99.9999999% as good enough to make all decisions in life. I wouldn't call a claim false just because there might be one chance in a billion or so that it is.

There is no evidence that the claim is false.
What would be false is the claim that it has been deductively established as true. Most of science does not care about such a claim in the first place (and daily life even less).

Most of us just accept certain things as obvious without a formal proof;

Sure, but my point is that you we should not accept that statement X has a formal proof only because X seems to be almost certainly true.

Most of us do not care about whether something has a formal proof. Most statements inherently are incapable of having a formal proof anyway. No law of gravity can be deductively proven. We cannot prove that the laws of nature that are valid today are still valid tomorrow. But we are pretty sure they will be and certainly take life decisions based on this assumption.

But there are areas of knowledge where statements can be deductively decided, and there are people that take an interest in these questions. These are not strange people or "lizard people" as you choose to call them that are unable to cross a street without a formal proof that it is safe. They just have interests that are not shared by everyone.

we are not lizard people, the moon is not made of green cheese, the earth isn't hollow, etc. Things like this can't be totally proven impossible, one can always construct some silly argument as to why they could be true. Similarly chess might be a mate in 385 moves with perfect play

But there is fundamental difference between the statement that the moon is made of cheese and the statement that chess is a mate in 385. The truth of the statement that chess is a mate in 385 moves can, at least in theory, be established purely by deduction and without reference to any observable real-world phenomena. No induction is necessary.

I'm not opposed to the search for a formal proof, I just consider it rather pointless

Chess is not any less pointless, and this is not an attack on anyone who loves to spend time on chess. People have intereststs. Yours are not mine, and that is good. I respect that you have your interests.

[jefk]

Well I don't believe a forced mate in 364 or 548 moves or whatever is possible in chess using the concept of game equilibrium.
Certain games exist in a balanced state or "drawing range" where no player can force a win through perfect play. Within this range, the game is permanently drawn.
Factors like board size, piece movement rules, and other properties determine the size of a game's drawing range. Chess likely has a relatively large drawing range
and also for this reason i strongly believe that chess is such a balanced 'equilibrium' game.

A balanced game's equilibrium isn't just a perfect balance of strengths, but allows for minor fluctuations that don't break the overall drawn state.
For example, just as a concept if blaanced positions remain within a value of e.g 0.4 pawn (withNnue eval) it is still a balanced position,
and only when such a value exceeds about 1.5 or so (rough estimate) we can start to think of forced wins (or checkmates).
Instead of trying to prove a forced win where one doesn't exist, it's sufficient to classify games as drawn if within
an equilibrium range. Some determination approaches could include:

- Analyzing results from large, high-quality opening books generated by minimax to identify balanced and imbalanced openings.
- Examining balanced endgames from master-level games.
-Using Monte Carlo tree search to evaluate positions from
the end of the opening tree, in many situations hitting
endgame table bases.

Based on many findings so far, games like checkers, international draughts, chess, and possibly Go (depending on 'komi) appear to have equilibrium properties and fall within a stable drawn range.
Systems like four-in-a-row do not. Opening book analysis like from the Chinese database supports chess as inherently drawn from the starting
position due to balanced lines. Of course, more exhaustive evaluation would provide stronger conclusions or convince more people (whereas i'm already convinced as i tried to convey here several times in this forum).
Anyway the concept of game equilibrium explains why forced mates beyond a certain point seem improbable.

(*) due to drawing rules (50 move, 3 pos repetition, and stalemante) and the endgame (often in endgames with opposite bishops even two pawns extra are not enough,
besides that we can think of many examples of situations where Black can setup a fortress which is impenetrable for White)

[jefk]

Well I don't believe a forced mate in 364 or 548 moves or whatever is possible in chess using the concept of game equilibrium.
In game theory (and real life) there are certain games which are in a balanced state or "drawing range" and where no player can force a win through perfect play. Within this range, the game is permanently drawn.
Factors like board size, piece movement rules, and other properties determine the size of a game's drawing range. Chess likely has a relatively large drawing range
and also for this reason i strongly believe that chess is such a balanced 'equilibrium' game.

A balanced game's equilibrium isn't just a perfect balance of strengths, but allows for minor fluctuations that don't break the overall drawn state.
For example, just as a concept if balanced positions remain within a value of e.g 0.4 pawn (withNnue eval) it is still a balanced position,
and only when such a value exceeds about 1.5 or so (rough estimate) we can start to think of forced wins (or checkmates).
Instead of trying to prove a forced win where one doesn't exist, it's sufficient to classify games as drawn if within
an equilibrium range. Some determination approaches could include:

- Analyzing results from large, high-quality opening books generated by minimax to identify balanced and imbalanced openings.
- Examining balanced endgames from master-level games.
-Using Monte Carlo tree search to evaluate positions from
the end of the opening tree, in many situations hitting
endgame table bases.

Based on many findings so far, games like checkers, international draughts, chess, and possibly Go (depending on 'komi) appear to have equilibrium properties and fall within a stable
drawn range. Systems like four-in-a-row do not because after a while there is zugzwang, whereas in the game
of thess there continuously are sufficient degrees of freedom (move choices for Black) to avoid a forced loss. . Opening book analysis like from the Chinese database supports chess as inherently drawn from the starting
position due to balanced lines. Of course, more exhaustive evaluation would provide even stronger (or even definitive) conclusions or convince more people (whereas i'm already convinced as i tried to convey here several times in this forum).
Anyway the concept of game equilibrium explains why forced mates beyond a certain point seem improbable.
And the burden of proof to demonstrate that chess is not such an equibrium situation imo lies with the
mathematicians, not with people who already know from experience that Chess is a Draw.

(*) due to drawing rules (50 move, 3 pos repetition, and stalemante) and the endgame (often in endgames with opposite bishops even two pawns extra are not enough,
besides that we can think of many examples of situations where Black can setup a fortress which is impenetrable for White)

[smatovic]

Hmm, an example for deductive (syzygy), inductive (Kaufman), abductive (Kaan) reasoning?

Towforce's paradox might help, "If it is not a draw, then where is the win?" The deductive approach "just" needs one forced win to prove the others false.

--
Srdja

[lkaufman]

To my thinking, "chess is a draw" is in the same category as something like "there are no humans living permanently on the moon". This can't be proven; it is theoretically possible that when we landed on the moon, one or two astronauts secretly remained, drilled deep below the surface, brought enough supplies to last half a century, and have remained hidden there ever since. Hard to disprove this rigorously, but I think almost everyone would agree that the probability of the statement being true is at least 99.9999999%. Similarly we can use statistical arguments using engines to show that chess is a draw with over 99.9999999% probability, we just can't prove it 100%. Some may enjoy the challenge of a 100% proof, but for me I'm willing to accept 99.9999999% as good enough to make all decisions in life. I wouldn't call a claim false just because there might be one chance in a billion or so that it is.

There is no evidence that the claim is false.
What would be false is the claim that it has been deductively established as true. Most of science does not care about such a claim in the first place (and daily life even less).

Most of us just accept certain things as obvious without a formal proof;

Sure, but my point is that you we should not accept that statement X has a formal proof only because X seems to be almost certainly true.

Most of us do not care about whether something has a formal proof. Most statements inherently are incapable of having a formal proof anyway. No law of gravity can be deductively proven. We cannot prove that the laws of nature that are valid today are still valid tomorrow. But we are pretty sure they will be and certainly take life decisions based on this assumption.

But there are areas of knowledge where statements can be deductively decided, and there are people that take an interest in these questions. These are not strange people or "lizard people" as you choose to call them that are unable to cross a street without a formal proof that it is safe. They just have interests that are not shared by everyone.

we are not lizard people, the moon is not made of green cheese, the earth isn't hollow, etc. Things like this can't be totally proven impossible, one can always construct some silly argument as to why they could be true. Similarly chess might be a mate in 385 moves with perfect play

But there is fundamental difference between the statement that the moon is made of cheese and the statement that chess is a mate in 385. The truth of the statement that chess is a mate in 385 moves can, at least in theory, be established purely by deduction and without reference to any observable real-world phenomena. No induction is necessary.

I'm not opposed to the search for a formal proof, I just consider it rather pointless

Chess is not any less pointless, and this is not an attack on anyone who loves to spend time on chess. People have intereststs. Yours are not mine, and that is good. I respect that you have your interests.

Of course it is fine to pursue such interests. I certainly was not referring to people who like to prove things as "lizard people", that was just an example of an absurd belief held by a tiny percentage of people (but in the millions per surveys!) which cannot be disproven. But I suspect that if a proof that chess is a draw ever becomes possible, it would require a massive commitment of resources, which would seem to be not worth the cost just to prove what is already obvious. But if you are a billionaire and want to devote your wealth to such a project, that would be your right.

The related goal that I do find interesting is this: What is the smallest/most acceptable change to chess rules (or start position) that would eliminate draws without obviously favoring either side? Of course any drawless chess would be a forced win for one side, but as long as even the best engines devoting days to it could not make a determination with any confidence, that should be good enough at least for now. Currently my candidate would be "White can only castle short, Black can only castle long, Black wins draws". As far as I can tell, it's too close to tell which side to prefer. Perhaps someone has an even better solution.

[towforce]

...I suspect that if a proof that chess is a draw ever becomes possible, it would require a massive commitment of resources, which would seem to be not worth the cost just to prove what is already obvious. But if you are a billionaire and want to devote your wealth to such a project, that would be your right.

I agree that one cannot crunch out the entire game tree of chess with today's technology - but nobody has demonstrated to me, to my satisfaction, that this would be necessary to prove that chess is a draw. I think that it probably can be done with today's technology.

The "proof" that I was shown in a previous thread (a peer reviewed paper from the 1960s IIRC) only really proves that, for a chess-like game, as the size of the board increases, the number of moves needed to achieve [something like a goal]* increases at [something like a polynomial or exponential rate]*. For me, while this proves that the game gets a lot more complex as the board size and number of pieces increases, it doesn't prove that the result of the game cannot be proven without crunching out the whole game tree.

I could be wrong, but I think that proving that the standard opening position in chess is drawn is probably easier than most people realise: the problem with this forum is that most members are "game tree people".

*apologies - I might not have remembered the parts in square brackets with 100% fidelity

[towforce]

I'm going to make a point for the "chess might be a win for white" side:

Problem: we've studied a massive number of openings, and no way has yet been found to force a decent advantage from any of these openings.

Question: might we have missed an advantage that's there?

Unfortunately, I have to say "yes". I might need to provide more detail, but I'm going to be brief right now:

(1) We know that hand-crafted, and even NN evaluations, don't see everything

(2) We know that deep selective search doesn't see everything

So it's possible that something has been missed. However, quite a lot is being seen, and if an advantage exists, it must eventually reach a position in which the computers can see the advantage.

[syzygy]

But I suspect that if a proof that chess is a draw ever becomes possible, it would require a massive commitment of resources, which would seem to be not worth the cost just to prove what is already obvious.

With that I fully agree. I don't believe chess will ever be solved, so a project to try to solve it would be pointless. (If someone has a brilliant idea or some unexpected new technology is developed, then I will reconsider; I would be happy to be proven wrong.) And since we can't solve chess, the next best thing is to develop heuristic engines and make them play as well as we can. So that's what we do. But that we don't know the "true" outcome of chess, in the sense of actually solving the game, does (very slightly) bug me. (But if I could choose what open math problem gets solved, chess would not be anywhere near the top of the list.)

The related goal that I do find interesting is this: What is the smallest/most acceptable change to chess rules (or start position) that would eliminate draws without obviously favoring either side? Of course any drawless chess would be a forced win for one side, but as long as even the best engines devoting days to it could not make a determination with any confidence, that should be good enough at least for now. Currently my candidate would be "White can only castle short, Black can only castle long, Black wins draws". As far as I can tell, it's too close to tell which side to prefer. Perhaps someone has an even better solution.

That is an interesting proposal. I would expect black to have an advantage (I would be surprised if white could force a win), but my intuition on this is not worth much.