I Who would win a perfect game of chess?

[lpetrich]

Endgame tablebase - Wikipedia -- solution complete up to 7 pieces. Some positions previously thought to be drawn have turned out to be winnable, though sometimes with a lot more than 50 moves. The 50-move rule states that after 50 moves with no captures or pawn moves, the game is officially drawn. That is like having a timeout in an Internet connection -- if something takes too long, then one's software will quit.

IMO, it is telling that some victories require more than 50 moves, because that suggests that a complete solution for chess will include much longer games.

 

[BWV]

Quote attributed to chess master Efim Bogolyubov: When I am White I win because I am White. When I am Black I win because I am Bogoljubow. -- he was overstating it by far, but it does seem that White has a greater chance of winning than Black, all other things being equal: First-move advantage in chess - Wikipedia states that White wins instead of Black 52% - 56% of the time.

Interesting that is a far smaller margin than the 100 games between Alphazero and Stockfish

 

[mfb]

First-move advantage in chess - Wikipedia states that White wins instead of Black 52% - 56% of the time.

52% to 56% is the score for white. Using the chessgames.com database we have 37.5% win for white, 34.9% draws, and 27.6% win for black. If someone wins, it is white 37.5/(37.5+27.6) = 58% of the time. If we use the CEGT chess engines results we get 59%. If we use the 100 games of AlphaZero against Stockfish it is 25/(25+3) = 89%.

It is worth noting that AlphaZero did lose some games against Stockfish. It won 155 times and lost 6 times in a more recent match where both ran with the same hardware conditions for 1000 games. The publication splits it up by side in figure 2: Two losses were with white, four with black. 145 wins were with white, only 10 with black. 84% of the games were draws.

AlphaZero beats Stockfish even when playing exclusively black. It also beats Stockfish with 1/10 of the time and 50% white (but not with 1/30).

 

[lpetrich]

This may deserve its own thread, but something related, at least in overall subject matter, is the Eight queens puzzle - Wikipedia -- how many ways to place 8 chess queens on a board so that they do not attack each other. The solutions can be found with an in-place tree search that uses backtracking.

This problem can also be solved for different board sizes with corresponding numbers of queens. The number is observed to increase approximately factorially, but a general formula is not known and neither is its asymptotic behavior.

There are variations, like different chess pieces: rook, bishop, knight, king. Also boards with periodic boundary conditions and higher-dimension boards, like a n*n*n cubical board. On a n*n board, one can place at most n queens, n rooks, 2n-2 bishops, n*floor((n+1)/2) knights, and floor((n+1)/2)^2 kings.

 

[lpetrich]

I have decided to create another thread for this problem.

 

[MathematicalPhysicist]

While chess hasn't been solved yet, other games have. For example, I know that in in some games, like connect four, if both players play perfectly, the player who goes first will always win. On the other hand, some games, like tic tac toe, a perfect game will result in a draw; in fact, I recently found out that this is true for checkers as well. What I'm wondering though, is if it's possible to predict which scenario a perfect game of chess would lead to even without having fully solved it yet, and if it is possible, what the answer is.

It will end in a draw.

Edit: that's what I believe.

 

[MathematicalPhysicist]

BTW that's also true in English Football, unless there's a knockout round in which case it will never end...

 

[phyzguy]

It will end in a draw.

How do you know this? I think the right answer, as mfb said in Post #3, is that, "we don't know." Do you have some knowledge the rest of us don't?

 

[MathematicalPhysicist]

How do you know this? I think the right answer, as mfb said in Post #3, is that, "we don't know." Do you have some knowledge the rest of us don't?

It's only a belief.

My belief is that if a board game is structured symmetrically, i.e the tools on the board of the black are a mirrored image of the white then in a perfect game no one has an advantage over the other and it will always end in a draw.

I don't know how to mathematically convince you, besides brute force of all the possible moves which is not humanely conceivable.
Checkers is a similar structured board game which always ends in a draw.

 

[PeroK]

It's only a belief.

My belief is that if a board game is structured symmetrically, i.e the tools on the board of the black are a mirrored image of the white then in a perfect game no one has an advantage over the other and it will always end in a draw.

I don't know how to mathematically convince you, besides brute force of all the possible moves which is not humanely conceivable.
Checkers is a similar structured board game which always ends in a draw.

There is no difficulty in setting up a simplified version of chess where White has a clear win. Despite the position being symmetrical.

And, in fact, if your hypothesis were correct you could always draw as Black against the world champion simply by maintaining the symmetry. Just copy his moves.

 

[MathematicalPhysicist]

There is no difficulty in setting up a simplified version of chess where White has a clear win. Despite the position being symmetrical.

And, in fact, if your hypothesis were correct you could always draw as Black against the world champion simply by maintaining the symmetry. Just copy his moves.

I meant the initial position of the tools is the same not necessarily to copy the moves of the white.

What sort of simplified version you had in mind?

 

[PeroK]

I meant the initial position of the tools is the same not necessarily to copy the moves of the white.

What sort of simplified version you had in mind?

For example:

Just two pieces each. Kings on e1 and e8. Rooks on a1 and a8.

White has a forced win, starting with Rxa8+

 

[PeroK]

I meant the initial position of the tools is the same not necessarily to copy the moves of white.

But, by your own logic, if black does maintain symmetry, then each subsequent position must also be a draw. Otherwise, you have to admit that the players reach a symmetrical position which is not a draw.

And, if black breaks the symmetry, how does your analysis determine which asymmetric position is the perfect play?

I remember arguing this with a boy at school, when I was about 14.

 

[mfb]

It's only a belief.

Then you shouldn't post it like a fact.

There are many cases where the first player has a winning strategy with a symmetric setup.

 

[PeroK]

As an aside, there are some humorous examples where in a match consisting of an even number of simultaneous games, one player or team tried to come out even by keeping two separate games identical.

E.g. if you are white in one game and black in another, you wait until your opponent moves as white, then you copy him in the game where you are white. Then, you wait to see what he does as Black, then copy him.

I think there was a university match where one team tried this by board 2 copying board 1 and board 4 copying board 3, with the hope of drawing the match at 2-2.

 

[mfb]

You lose some time with every move, if the other player plays slow after seeing the strategy you run out of time.

 

[MathematicalPhysicist]

For example:

Just two pieces each. Kings on e1 and e8. Rooks on a1 and a8.

White has a forced win, starting with Rxa8+

OK I should have clarified what I meant.

If in the initial position no player has an advantage on the other player by starting the game then it will finish in a draw.
In your setting clearly the white has an advantage by starting, in the 8x8 complete chess game there's no advantage for the one who starts.

 

[jbriggs444]

in the 8x8 complete chess game there's no advantage for the one who starts.

That seems an unjustified assertion. We do not know whether the starting position is a forced win for white.

 

[phyzguy]

OK I should have clarified what I meant.

If in the initial position no player has an advantage on the other player by starting the game then it will finish in a draw.
In your setting clearly the white has an advantage by starting, in the 8x8 complete chess game there's no advantage for the one who starts.

You're missing the whole point. Simply moving first is an advantage. If you don't believe it, just look at the statistics of high level chess games. Most wins are by white. The question is whether the advantage of moving first is enough to win a game where both players follow the optimum strategy. We don't know.

 

[MathematicalPhysicist]

That seems an unjustified assertion. We do not know whether the starting position is a forced win for white.

To justify it you need to know all the possible moves.

 

[jbriggs444]

To justify it you need to know all the possible moves.

To justify it, you need to have analyzed the game tree for all possible continuations of all the possible moves.

 

[MathematicalPhysicist]

You're missing the whole point. Simply moving first is an advantage. If you don't believe it, just look at the statistics of high level chess games. Most wins are by white. The question is whether the advantage of moving first is enough to win a game where both players follow the optimum strategy. We don't know.

I saw the statistics of mfb and it seems there are more draws than wins for the white.
The question is when if ever will we exhaust all the number of moves possible in chess?
How could one prove what are the optimal strategies without covering all the possible moves?

OK, I'll edit my previous post, though I am quite sure of a draw, mark my words! :-D
And another question how can one prove optimality of a game without knowing all the possible outcomes?

 

[MathematicalPhysicist]

72 of the 100 games between Alpha Zero and Stockfish were a draw and interestingly, the win/draw ratio was 25/25 when Alpha Zero played white but only 3/47 when it played black (Alpha Zero won every game). I have not seen any statistics released on the 44 million training games played regarding an advantage for white vs. black.https://arxiv.org/pdf/1712.01815.pdf

Did Alpha zero played its equal?
I.e itself?

 

[TeethWhitener]

And another question how can one prove optimality of a game without knowing all the possible outcomes?

If, for example, you can show that it's always possible for white (or black) to reach a position with a forced mate, you only have to evaluate the possible moves leading to that position.

 

[phyzguy]

And another question how can one prove optimality of a game without knowing all the possible outcomes?

This is like asking how we can know that there are an infinite number of prime numbers without having examined all possible integers. There are mathematical methods which allow you to prove things about very large (or even infinte) sets without having to examine every case. Game theory has used mathematical methods to prove that every game of perfect information has at least one optimal strategy, meaning a strategy which cannot be improved upon.

 

[cosmik debris]

Did Alpha zero played its equal?
I.e itself?

Yes, roughly 20 million I believe. It trained for four hours.

Cheers

 

[MathematicalPhysicist]

This is like asking how we can know that there are an infinite number of prime numbers without having examined all possible integers. There are mathematical methods which allow you to prove things about very large (or even infinte) sets without having to examine every case. Game theory has used mathematical methods to prove that every game of perfect information has at least one optimal strategy, meaning a strategy which cannot be improved upon.

OK, then how would you try finding this optimal strategy?
I know how to prove that there are infinite prime numbers.
I mean without some trial and error I don't see how can you find an optimal strategy?

 

[PeroK]

To justify it you need to know all the possible moves.

You are the only one who claims to have a solution. Everyone else is saying that we do not know. That chess is too complicated to decide between a white win and a draw, with perfect play on both sides.

The evidence from chess theory is that white has a small advantage by moving first, but that it less than the minimum needed to guarantee a win. A proof of this cannot be furnished simply by considerations of symmetry.

 

[MathematicalPhysicist]

You are the only one who claims to have a solution. Everyone else is saying that we do not know. That chess is too complicated to decide between a white win and a draw, with perfect play on both sides.

The evidence from chess theory is that white has a small advantage by moving first, but that it less than the minimum needed to guarantee a win. A proof of this cannot be furnished simply by considerations of symmetry.

I didn't claim to have a solution, it's just my strong belief which I see is unjustified.
So why in checkers for example the advantage in starting isn't sufficient to enforce a win?
What mathematical argument was given for this?

 

[TeethWhitener]

So why in checkers for example the advantage in starting isn't sufficient to enforce a win?
What mathematical argument was given for this?

http://science.sciencemag.org/content/317/5844/1518