Who would win a perfect game of chess?

[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

 

[phyzguy]

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?

I didn't say you can find it, only that we know it exists. The fact that there are a huge number of moves and finding the optimal strategy is so difficult is why we don't know the answer to the OP's question. You can say, "I think it will be a draw," but there is no reason for the rest of us to believe you.

 

[PeroK]

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?

Checkers is a simpler game and has been crunched by a computer.

There's no reason that the solution for chess is the same. It might turn out that white can win.

 

[jbriggs444]

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.

That sounds like the essence of alpha-beta pruning. If you are trying to decide between a forced win for white and a forced draw for black then black can prune away all moves other than to a known forced draw. White can prune away all moves other than to a known forced win.

 

[dRic2]

Spoiler

Although it might be intellectually/mathematically pleasing to think about it, what an horrible game would be a "perfect" game ?

I'm not criticizing the OP for the question (very legit), but I noticed that this kind of discussions seem to reduce chess to just "maths" (logic). A non-player (an outsider) might get the wrong idea! Chess is about art and phycology and culture and a lot of other stuff!

Now hate me for this useless comment

 

[MathematicalPhysicist]

Now when I think of it, white will always win, it's not the same as in checkers.

Try to play against the computer in level 10 with the hints with white in chess.com. I know it's not a proof, but it seems plausible that in this game the one who starts will win, if he played perfectly.
It took 109 moves, if I were to repeat these game will the moves of the black changed?

I have exams on Condensed Matter Physics and Particles theory 2 so no more time chatting here.
I stand corrected.

 

[MathematicalPhysicist]

Spoiler

Although it might be intellectually/mathematically pleasing to think about it, what an horrible game would be a "perfect" game ?

I'm not criticizing the OP for the question (very legit), but I noticed that this kind of discussions seem to reduce chess to just "maths" (logic). A non-player (an outsider) might get the wrong idea! Chess is about art and phycology and culture and a lot of other stuff!

Now hate me for this useless comment

You are in the wrong website.
:-D

 

[bahamagreen]

Last time I did programming was mainframe Fortran IV...

Given 64 squares and 32 pieces
Let all states of the board (including improper ones) be represented by a base 31 number of up to 64 places.
Assign the values 0-31 to the pieces, even white, odd black, as "digits" of this number.
Example of the initial board state at the beginning of a game might be something like:

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30

Treating these states as either numbers or strings, exclude states that represent:
- no pieces
- one piece
- two pieces where at least one of them is not a king
- more than one king of either color
- bishops on incorrect color square
- pawns on their back row
- too many of same color piece (due to pawn promotion... this would calculate how many duplicates of a piece are possible and allowed based on comparing the possible number of pawns promoted vs pawns not yet promoted and available to reach the back row)
- addition impossibilities in a game

Distinguish states that represent proper moves.
Set the initial starting game state as move number zero

For each set of all proper final move states* (see below) where the set's move number is one less than the present move number but not negative
For each piece
Test all states that re-position the piece to every square and eliminate states from the list that:
- re-position an incorrect color piece for that move sequence in the game
- re-position a piece to the square it is already occupying
- move a piece through an occupied square, inclusive (expect knights, and proper captures, castles)
- move a piece improperly per game rules of piece movement (including second same color castle)
(above piece movement tests done algebraically)
Mark or associate all proper final move states* for this move number with an indication of the move number and the moved piece number
Increment the move number
Repeat
This provides a disordered list of all proper piece position move states (including capture, castle, and promotion) for all move numbers of all proper games

Relate these proper move states into strings of all proper games
Using the proper final move states with their indication of the move number and the moved piece number

Set move number to one

For all proper states of the move number
Test all proper states where move number is one greater
If greater move number state is a proper move from the previous move state number, mark or associate the two
(include test for third of same move, move number greater than 50 if desired)
Increment move number
Repeat

Use the proper move associations to produce all possible proper games state number lists
Filter this list for however you define a "perfect game"

 

[PAllen]

The 50-move rule limits the maximum number of moves to just under 6000. There's a thread about it here:

https://www.physicsforums.com/threads/exact-number-of-possible-chess-games.730486

The 50 move rule is often removed for theoretical purposes. For example, tablebase analysis includes forced mates with hundreds of moves without capture or pawn move.

 

[PAllen]

Chess is unsolvable with traditional computers

Claude Shannon noted that a true chess solution would require storing 10^120 moves. This gets into age of the universe type computational times and impossible storage requirements with any conceivable computer technology other than maybe a huge quantum computer

No, that number would have to be analyzed but not stored. The result of analysis so far could use one of the compact tablebase representations. I once worked out this would only require a number of bits similar to atoms in the moon to play any position perfectly.

 

[PAllen]

Chess is kind of weakly solved when 7 pieces (including the two kings) remain on the board (discarding the castling moves and en passant). I don't know if they're working on 8 pieces, etc. An idea to have another indication that chess may be a draw (or a win/loss) with perfect play is to set symmetrical starting positions with the few pieces on the board and see the outcome of perfect moves.

I also guess that it leads to a draw, but it's just a pure guess.
Some chess variants are easier to deal with than chess (giveaway or suicide chess), whilst others are more complicated (crazyhouse).

Another very interesting question is whether all starting position of chess 960 (Fischer random chess) lead to the same outcome than regular chess, with perfect play.

I would say it is strongly solved for 7 or fewer pieces. The winner as well as best play for both sides, for any such position, can be generated ( even though the sequence itself is not stored).