The Instigator
Grape
Pro (for)
Winning
18 Points
The Contender
imabench
Con (against)
Losing
0 Points

There is an Ideal Opening Move in Chess

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Post Voting Period
The voting period for this debate has ended.
after 5 votes the winner is...
Grape
Voting Style: Open Point System: 7 Point
Started: 4/16/2012 Category: Miscellaneous
Updated: 4 years ago Status: Post Voting Period
Viewed: 7,243 times Debate No: 22880
Debate Rounds (4)
Comments (31)
Votes (5)

 

Grape

Pro




Resolved:
There is an ideal opening move in chess.

Chess is this game: http://en.wikipedia.org...

An opening move is the first move made by either Black or White.

A move is considered "ideal" if it enables the player to guarantee a tie or a win. Obvious examples of "ideal" moves are those that lead to forced mates or those that stalemate the game.

The burden of proof is on Pro is to show that an ideal opening move exists.

First round is acceptance.
imabench

Con

I accept this debate and will argue that there isnt an ideal opening move in Chess since the result of any Chess game is based extremely on chance and mistakes made until much later in the game.

On the other hand My little brother has whipped my a** at chess and I have tried every chess move in the book and still cannot win so I will be arguing that there is no move that gurantees a tie or a win either.

Other than that,

Debate Round No. 1
Grape

Pro

Introduction:

I will build up my case through a series of theorems ending with a proof of the resolution. I will do my best to make it clear and easy to follow.


Theorem 1: Chess is a finite game.



Definition 1: Board State

A board state is an ordered 64-tuple set giving the occupier, if any, of every piece on the board. Unoccupied spaces are marked by blanks. It starts at A1, then goes to A2, etc... until it reaches H8. Some convention could be adopted for naming pieces (W for White, B for Black, R for Rook, B for Bishop, # for empty, etc.). Thus the starting position is {WR, WK, ...BR}. An integer can be assigned to each board state (1 for the starting position, 2 for some first move, 3 for some other first move, etc up to about 10^45 if we only worry about legal positions).

Lemma 1: There are finitely many positions in chess.

There are 32 pieces on a chess board (only 4 of which are unique) and 64 positions. There are only a limited number of ways to assign finite number of pieces to a finite number of spaces (though this number is large). A large number of these board states are not accessible from the starting position, are identical because they involve switching non-unique pieces, or involve less than 32 pieces (the minimum number of pieces on the board at the beginning of a turn is 3).

Lemma 2: There are only finitely many possible turns in chess.

A chess game ends if fifty moves have passed without a piece being captured. If a king is captured or only kings remain, the game ends. This means that no chess game can be longer than 1500 moves (the maximum time to exchange the other 30 pieces).

Definition 2: Game State

A game state is an ordered pair giving the board state and turn number of a chess game. For example, {92385983475928740239756183610947261, 56} might be a game state. Not every game state is necessarily legal (example: {1, 2}). A game state is called an end state if it results in the game's ending (there is a checkmate or draw).

Conclusion:

The game states exhaust the possibilities for chess. A game with the game state given is fully described. The number of game states is finite (equal to the product of the number of positions and the number of turns). Therefore, the game of chess is finite.



Theorem 2: If a position that ends the game is accessible from the position at the beginning of a player's turn, that player can guarantee that the game will end.

Lemma 3: There is no chance in chess.

There is chance in a game if a player does not know the outcome of a move before deciding to make it. When a player attacks in Risk, he or she does not know how many armies will remain at what territories afterward. In chess, a piece always moves to the desired position.

Lemma 4: Chess is a game of perfect information.

A game of perfect information is a game in which every player has access to all the information about the game. A game that hinges on imperfect information is Stratego. In Stratego, the value of each player's pieces is concealed from the other player. In chess, both players can see everything relevant to the game on the board before their eyes; there are no hidden variables.

Lemma 5: Every position has finitely many positions that are accessible from it.

This is entailed by Claim 1. No matter what moves were allowed, a move could only change the current game state to a new game state (and where the current turn number was n, the new turn number would have to be n + 1. Though time travel could make chess very interesting!).

Conclusion:

If a position is a game ending position, a player can determine that it is because there is perfect information. There is finite set of positions accessible from a given position in chess, so the player can always determine if one of them is a game ending position by analyzing each one. The player can also guarantee that a move will lead to the desired position because there is no chance.

Final Proof: There is an ideal opening move in chess.

Lemma 6: With each move, the number of accessible game states is reduced.

This follows trivially from Theorem 1. Regardless of what moves are allowed, the turn number is increased by 1, and so the number of game states degrees by the cardinality of the set of board states. Because it is not legal to simply rearrange the pieces however one chooses each turn, the possibilities are in fact reduced far more rapidly.

Lemma 7: An ending state is accessible from every game state that is not itself an ending state.

This simply follows from the fact that the game does indeed end.

Lemma 8: A player can only move into an end state that results in a tie or a win.

This is a logical consequence of the rule that one may not put oneself in checkmate.

Definition 3: Chess Tree

From every game state A, there is a finite set of game states reachable from it (the legal max is 218 and the minimum is of course 0 for end states). Call the members of this set B1 ... BN. From each of B1...BN, there is a new set of game states accessible for that position. This process is recursive until the set of game states accessible from a final game state is empty. These can be arranged in a topological structure called a chess tree, which will be used to visualize the game. Such a tree would look like this, but it would be of much greater complexity:

http://i.investopedia.com...


Assumption: There is a path down the tree that will allow White to win or tie the game no matter what Black does.

White can try tracing every path down the tree. If any path leads to a game state in which it is Black's turn and Black has an ideal move, that path is abandoned and a new path is attempted. If there is a path leading to a White tie or win regardless of what Black does, then every move in that tree, including the first move, is an ideal move for White (because, per the definition, it enables White to guarantee a victory). If such a tree exists, the resolution is true.

Suppose that no such path exists for White. That means that if Black plays correctly, there is no way that White can win the game or intiate a tie. But the game must end, so that means that Black can guarantee a win or a tie from the starting position. Any correct move that Black makes from the starting position would be considered an ideal move.


Conclusion:

The result was first proved (more rigorously, but for different chess rules) by Ernest Zermelo in 1913. It should be stressed that no one knows what the ideal opening move in chess is or whether it belongs to White or Black (though it seems more plausible tha it is White's... if it is Black's then the starting positon is a Zugzwang), but we can be sure that such a move does indeed exist. If we are persistent and do not die in some terrible disaster, humans will eventually solve the game of chess just as we have solved the game of checkers. Any game that is basically finite can eventually be solved by looking at all the possibilities. This is obvious in the case of Tic-Tac-Toe, the solution to which can easily be visalized:

http://www.csc.csudh.edu...

Chess is a mathematically trivial game in the same way, it's just a great deal more computationally complex.
imabench

Con

There are a finitely limited number of positions in Chess = Agreed

There are finitely many turns in Chess = Agreed

There are only finitely many possible turns in chess. = Which is 1500 but many games end before then. Agreed

The game of chess is finite = Turn wise, Agreed

============================================================================

There is no chance in Chess = I disagree, Chess is a game of perfection meaning that the one who loses the game is the one who makes the most critical mistakes as the game progresses. A person may THINK they know all of the consequences of a move before they make it but they may be mistaken.

Chess is a game of perfect information. = I agree but read below

" In chess, both players can see everything relevant to the game on the board before their eyes; there are no hidden variables."

There is a difference though in variables being hidden and variables not being recognized.... And in Chess the latter is the key to winning the game. Chess is a game of open information, but whether or not the player can recognize, process, and take into account of that information though makes Chess a game of imperfection.

"The player can also guarantee that a move will lead to the desired position because there is no chance."

But they cant, and thats the key. In every Chess game there are dozens of move with up to a hundred possibilities in ever ROUND of Chess, and people playing Chess can slip up or simply not notice something due to the sheer number of moves that are available.

============================================================================

With each move, the number of accessible game states is reduced. = I disagree

With Tic Tac Toe, you have 9 possible moves, then 8, then 7, then 6, so on so on. But with Chess one move opens up other moves for the next turn that you have.

Heres an example, at the beginning of the game you only have 20 possible moves, 16 of those are by pawns, and the remaining 4 are to the 4 places you can move your 2 knights. If you move a pawn that allows you to move your bishop or your queen though, then on your next turn you have MORE moves to choose from.

Chess is a game where the number of moves you can make increases from the beginning, then decreases once pieces start to get eliminated. So Turn NUMBER may decrease by 1 but the turns you can choose from in any turn can increase or decrease dramatically

Lemma 6: With each move, the number of accessible game states is reduced.
There is more than one way to access many game states though. If the game state is to have both pawns in front of rooks moved forward be two spaces ahead then there are 2 ways to do so, you can start with the left or the right. Whichever you choose though can lead to the same state.

"White can try tracing every path down the tree. If any path leads to a game state in which it is Black's turn and Black has an ideal move, that path is abandoned and a new path is attempted.... If such a tree exists, the resolution is true."

But such a tree couldnt exist. Chess is a game that is won by facing an opponent who makes more mistakes then you do, not by making a series of moves that guarantees success because any move can be countered or overlooked by the opponent. Chess is a game of chance because even though all the pieces are in open view, the number of pieces and the number of moves allows for players to overlook, not notice, or underestimate the significance of both their own moves or the opponents.

Point is, Chess cannot have such a tree because the number of moves, the number of counter moves, and the randomness of the game is so large that there is no single move that guarantees winning or tying a game of chess.

" Any game that is basically finite can eventually be solved by looking at all the possibilities."

The Pro's whole argument comes down to this, because there it a limited number of turns one has, like in tic-tac-toe, then there must exist an ideal opening move in chess and that the reason we havent found it yet is because its more complex. But its deeper than that because Chess is limited in the number of turns you have, like tic-tac-toe, but the number of MOVES you can make increases and then decreases through the game, whereas tic-tac-toe only decreases.

============================================================================

Lets look at the opening move in Tic-Tac-Toe that guarantees success in the picture below....
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You may have noticed that there isnt a picture..... Thats because Tic-Tac-Toe, like Chess, is a game of making less mistakes than your opponent in order to win. I could play the Pro in a game of tic-tac-toe and put an X in the most ideal spot, and the Con would put his O in the next best spot. But if Im an idiot I may place my next move elsewhere, and thus screwup the ideal sequence of moves needed to win if it allows my opponant, the Con, to counter the ideal move.

The same thing works in Chess because I could play the ideal move but screw up later on since Chess is a game built off of people making mistakes and the opponent exploiting it.

"A move is considered "ideal" if it enables the player to guarantee a tie or a win." - Pro
"There is an Ideal Opening Move In chess" - Resolution of Debate by the Pro

Now there could very well be a series of moves that greatly increases a players chance of winning a game of Chess, I will concede that. But there isnt an Ideal move in the OPENING of Chess because

1) It does not guarantee a win or a tie, it only increases the odds of it
2) The player could make a mistake and then that cold later cause him to lose the game even though he made the "correct" first move
3) Any move by one side in Chess can be countered, negated, or even overlooked by the other side which could nullify the ideal set of moves that would increase the odds of winning
4) Chess is a game that is won by the imperfection of the opponent and the consequences of their own mistakes,
5) Chess is a game that in the beginning the number of possible moves INCREASES and then starts to decrease.
6) Whether or not a person wins a game of chess doesnt lie in the color of the pieces or the moves they make, it is the person who is playing the game and whether or not he will make more mistakes than their opponent.
7) An ideal move in Chess would only increase the possibility of winning, it would not gurantee a win or a tie.

And most importantly

7) Any opening move could not guarantee a win or a tie because whether you win or lose it depends entirely on moves made after the opening move. Meaning that even though an ideal opening move may exist, it does not guarantee a tie or a win because of the randomness and chance of the game.

There is no ideal opening move in Chess, there only might be an opening sequence of multiple moves that could increase the chances of winning or tying, but there isnt a single, ideal, opening move that guarantees a win or a tie.
Debate Round No. 2
Grape

Pro

Introduction:

Con has conceded my first theorem. I will take this to be conceded for the remainder of the debate, since were Con to challenge it later, his arguments would contradict one another.

Con's argument mainly concerns theorem two. I will show that Con misunderstands theorem two and its associated lemmas and takes them to mean something very strong, but only a weak interpretation is required for my proof.

Analysis of the Resolution:

There are two critical words in the definitions whose meaning I think was very clear, but which I did not discuss previously:

"A move is considered "ideal" if it enables the player to guarantee a tie or a win."

The move must enable the player to guarantee a tie or a win. If the player, due to human error, does not capitalize on this opportunity, that does not matter. Presumably an ideal opening move has been made and yielded a loss countless times in the history of the game of chess.

"The burden of proof is on Pro is to show that an ideal opening move exists."

The move must exist in the game. It is not necessary that anyone be able to determine what it is.

1. There is no chance in chess.

Con confuses chance in human behavior with chance in the game. While a human being can play a blunder, the blunder was always the purely deterministic result of an intentional move that turned out to be wrong. There is only chance in the game itself if the randomness built into the rules of the game. When a player rolls in Monopoly, he or she has no idea what the result will be. When the same person moves a rook to B2 in chess, they is certainty that the rook will end up at B2. Thus there is no chance in chess.

If it is any easier, imagine that we are at a computer chess championship and the rate of hardware errors for the computers is negligible. But I am really talking about chess in an abstract mathematical sense, since chess as a game is itself an abstract concept.

2. In chess, both players can see everything relevant to the game on the board before their eyes; there are no hidden variables.

Con makes the same mistake as above. Humans might fail to notice something on the board, but I am not arguing that humans recognize everything on the board. I'm just arguing that it's there. That's all that's necessary for an ideal move to exist. In Stratego, there can't be a perfect move because the rules of the game forbid the player from having the information necessary to determine if the move is perfect in principle. A perfect move would be the result of luck and could not be guaranteed.

3. The player can also guarantee that a move will lead to the desired position because there is no chance.

Same error as above, though this one is a bit of a stretch. A normal human being will not fail to move a piece to the correct position when making a move.

4. With each move, the number of accessible game states is reduced.

I didn't say immediately accessible game states. I mean all possible future positions. This number necessarily must go down at least by the cardinality of the set of board states, because at turn n the maximum number of accessible states if all moves are legal is (1500 - n)(|{Board States}|) and at turn n + 1 it is (1500 - (n + 1))(|{Board States}|). The number of states immediately legally accessible can range from 0 to 218 and goes up and down throughout the game.

Con also said that the same state can be accessible from different paths. The proof gets more difficult to explain if we take this into account (because divergent branches can meet back up) but I don't see how it's relevant to the result. We might as well just allow branches to have redundant game states.

This lemma might not be essential anyway.

5. White can try tracing every path down the tree. If any path leads to a game state in which it is Black's turn and Black has an ideal move, that path is abandoned and a new path is attempted.... If such a tree exists, the resolution is true.

I am totally lost on how this tree doesn't exist. It demonstrably does. The tree I describe is nothing more than a way of arranging the game states described in Theorem 1 based on which ones are legally accessible from one another by what series of moves. Just because a human would fail to comprehend the whole thing doesn't mean it can't exist. This is like saying pi isn't a real number because no human can fully comprehend it.

Con is effectively arguing based on the size of the tree, but that is of no relevance in principle. He also says preposterous things like, "[T]he randomness of the game is so large that there is no single move that guarantees winning or tying a game of chess," which is just silly because it is obviously possible to have a checkmate-in-one scenario.

6. Any game that is basically finite can eventually be solved by looking at all the possibilities.

Con's chief objection to this is that, "Chess is limited in the number of turns you have, like tic-tac-toe, but the number of MOVES you can make increases and then decreases through the game, whereas tic-tac-toe only decreases." I fail to see the relevance of this. Imagine a tree for tic-tac-toe: it starts out bushy at the bottom and gets sparser as you reach the highest branches. Now imagine a tree for chess: it gets get thicker or sparser in a seemingly random pattern as it goes to toward the top. The significant similarity is that both trees end. They are finite. You could always start at the trunk and trace your way to the tip of each twig. If two people were playing a game, taking turns determining which way to go up the branch, and the winner was the one who moved last, there would always be a perfect strategy for both trees. The structure of the trees would not matter (much of my argument was intended to show that the topological structure of the game does not matter).

Con goes on to seemingly deny that ideal play in tic-tac-toe is possible. There is a perfect algorithm for playing tic-tac-toe:

http://en.wikipedia.org...

The fact that a human may fail to play correctly, is, again, irrelevant to whether or not an ideal move exists. It does, and in the case of tic-tac-toe it can be found by consulting the list.

Con's 8 Points:

1)

There are no odds in chess. A player does not have a certain probability of winning from a certain position as he might have in Yatzhee. Con is confusing our ability to guess at who will win with probabilistic features of the game.

2)

The move only has to enable him to guarantee victory, not assure that he will. See above.

3)

This is plainly false. Chess has forced mates. No matter what the defending player does in a forced mate scenario, he loses.

4)

Con's whole argument rests on this ill-defined notion of "imperfection" and "mistakes." This is, honestly, an unsophisticated way of looking at the game. "Players" and "opponents" are really just a heuristic for thinking about what is essentially a purely mathematical structure.

5)

This is irrelevant, see above.

6)

See 4)

7)

See 1)

8) (sic)

This is an obvious misunderstanding. If you are in a "mate-in-two" scenario, the first move is just as essential as the mating move. This is true for forced mates of any sequence. If you fail to make the first move properly, you've blown the whole sequence. I'm arguing that the whole game is like that.

Conclusion:

Con seems not to believe that any game can be solved. In his world, there can be no perfect play for player two in the game "Who can name the highest number?" because if I am player one and say "62" he might say "55." That doesn't change the fact that "if your opponent said n, say n+1" is a perfect strategy for player two. Chess likewise has a perfect strategy independent of whether we can determine it. Con's other crucial error is to understand games in the unsophisticated sense of real events. The mature, proper understanding of a game is independent of actual instances of its being played. A real game of chess might not be deterministic, but chess is. This is a crucial distinction that has been lost on Con.
imabench

Con

"Con has conceded my first theorem. I will take this to be conceded for the remainder of the debate"
I AGREED with the theorem, there is a difference

============================================================================

Wording of Resolution:

"The move must enable the player to guarantee a tie or a win. If the player, due to human error, does not capitalize on this opportunity, that does not matter"

If a move only enables a player to win or tie, then it still does not determine once and for all what the final outcome will be. It increases your odds of winning, but it doesnt guarantee the desired result because what determines a win, a tie, or a loss are the moves that come AFTER an initial move. For a move to be an "ideal" move it must guarantee victory or a tie, it can't just enable it because then that leaves the move open to compromise, being countered, and not yield the desired result.

"an ideal opening move has been made

and yielded a loss countless times in the

history of the game of chess."


Then the "ideal" move does not guarantee victory or a tie. It holds little significance at all since the "ideal" move can be canceled out both by human error from you and from a counter measure from the opponent.

"The move must exist in the game. It is not necessary that anyone be able to determine what it is."

Im not asking you to determine what it is. Im only asking how an ideal move can exist when so much of the game is based on the mistakes made by you and your opponent by not recognizing certain factors over the course of the game. After that then I ask how there could be an ideal OPENING move since what determines if someone wins or loses is determined by all the subsequent moves by you along with the counter measures by the opponent.

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Chance in Chess

"Con confuses chance in human behavior with chance in the game. While a human being can play a blunder, the blunder was always the purely deterministic result of an intentional move that turned out to be wrong"

The game is controlled by people though, if someone makes an ill-advised move it is ill advised not because the human did it randomly, but because he did not take into account all of the information available to him when he made his move because the player overlooked a variable. Something that you said was impossible

" In chess, both players can see everything relevant to the game on the board before their eyes; there are no hidden variables"

Two seconds later...

"Humans might fail to notice something on the board, but I am not arguing that humans recognize everything on the board. I'm just arguing that it's there. That's all that's necessary for an ideal move to exist."

There exists the possibility for variables to not be taken into account. This could then cause the player make an ill-advised move which would then compromise any move they made before which allegedly guarantees them victory. This means that the move meant to guarantee them victory was lost. Thus there cant be an ideal move since there exists the potential to overlook variables which would forfeit their "guarantee" of winning

"A perfect move would be the result of luck and could not be guaranteed."

A perfect move doesnt exist in Chess either because the results of all the moves made after the initial move through the course of the game isnt guaranteed either since Chess is a game of luck and making less mistakes than your opponent.

============================================================================

"A normal human being will not fail to move a piece to the correct position when making a move."

Oh thats what you meant by desired position, I mistook that to mean something else. I apologize for my goof

============================================================================

Game States

"I didn't say immediately accessible game states. I mean all possible future positions"

Well you should have specified that before -_____-

"The number of states immediately legally accessible can range from 0 to 218 and goes up and down throughout the game"

Yes I agree to this, however the way you worded your arguments before it sounded like you were saying that the number of possible game states can only go down.... Thank you for clarifying

============================================================================

A Chess tree

I mistook this argument as the Pro claiming that there exists a path from the beginning of the game to the end that always guarantees a win or a tie. I apologize for my second goof.

============================================================================

Finite-ness of the game of Chess

"Con's chief objection to this is that, "Chess is limited in the number of turns you have... But the number of MOVES you can make increases and then decreases through the game, whereas tic-tac-toe only decreases." I fail to see the relevance of this."

In Tic-Tac-Toe you only eliminate available spaces with your pieces to try to get three in a row and you get as many pieces as you need. Pieces cannot be moved, removed, captured, etc.

In Chess though the object of the game is much more complex. You start with many pieces then lose them over time, spaces are only temporarily occupied and can be taken, and the amount of moves you have to make at any given turn increases as the game progresses than decreases as pieces are eliminated. Why is this relevant?

In Tic Tac Toe an ideal move does exist, and once you make that move your opponent cannot terminate that piece, or

The Pro compares Chess to Tic-Tac-Toe by saying that since both trees eventually come to an end, if someone goes through the tree and wins then that implies there is an ideal first move

" If two people were playing a game..... and the winner was the one who moved last, there would always be a perfect strategy for both trees."

Just because one wins a game of chess it doesnt imply that a perfect strategy exists because that particular route to victory that person just took could have been countered in numerous ways, along with any other path to victory that other people have taken. There are thousands and thousands of ways for someone to win a game of chess which means in the game tree there would be thousands and thousands of ways to go through the whole tree and win. This in no way suggests there is an ideal move that guarantees victory.

"the fact that a human may fail to play correctly, is, again, irrelevant to whether or not an ideal move exists"

There is a difference between a move that increases your odds of winning and a move that guarantees victory. If a move were truly ideal then it would still be able to guarantee victory despite human error, and there is no such OPENING move in either Chess or in Tic-Tac-Toe since both games are entirely decided by later moves in the game.

Running low on characters, ill jump to the Conclusion.

There could very well be an opening move in Chess that increases your odds of winning. But chess is a game of so many different variables that there isnt a single opening move that guarantees victory. Furthermore a move that is only one of many, many other moves needed to enable victory is a far cry from being an ideal opening move that guarantees a victory and a tie.

Back to you Pro :)

Debate Round No. 3
Grape

Pro

Introduction:

This debate has become somewhat disorganized, so I'll pick out the key points so I can respond concisely. I will try to do so as charitably as possible. Con still seems to have numerous misunderstandings. His case rests on trying to differentiate chess from other solvable games based on irrelevant differences.

Theorem 1:

Con his agreed to this. I ask that it not be contested next round since I will not be able to reply.

The Resolution:

Con's argument here rests on his overlooking of the word "enable." You can always voluntarily throw a game of chess (unless you are in an odd configuration where a loss is literally impossible). A guaranteed win is enabled if the player cannot fail to win if he makes the right moves. The fact that players do in fact make the wrong moves is irrelevant.

Con's interpretation of an ideal move is silly. As I said in Round 3, this means that no game can have a perfect strategy, even "Who can name the highest number?" because a player can fail to employ that strategy. Surely whether or not something is often done correctly has no bearing on whether or not there is indeed a correct way.

Chance in Chess:

Con is still confusing the difference between uncertainty in the game and human error. It is perfectly consistent to say that a person will fail to take information into account that is available to him. The rules of the game allow Player A to see that a rook threatens his bishop. If he doesn't notice and makes the wrong move as a result, that is a result of his error, not uncertainty within the game about whether the rook was threatening this bishop. Against, contrast with Risk and Stratego, games of genuine uncertainty.

Con does agree that players will always move successfully. But maybe some people are so stupid they can't even do that, or play chess at all? Why does Con arbitrarily choose the average human as his standard? It is surely plausible that chess grandmasters (or even skilled amateurs) never fail to perceive events on the board. What his argument really hinges on is our inability to comprehend every possible sequence of events. But this has nothing to do with chess either; it's a coincidence of biology (or really physics). A computer program has been designed that is unbeatable at checkers, and the only reason that chess has not been similarly solved is that it is more complex and requires more computing power. He is taking a purely human-centric approach to a game that exists in the abstract.

If it is easier, pretend that there is a race of supermen who are smarter than the smartest computer and never make mistakes. If they play chess, it will still be the same game, but none of these factors will apply. It is silly to define solubility in terms of human brain or computational power. Otherwise abstract mathematical problems (what is the 9347109374th prime?) could switch from solvable to unsolvable based on human technology. But solubility is itself an abstract mathematical concept and should not change when a computer is invented.

Game States & Chess Trees:

Con seems to agree to my prior claims. However, there is a discrepancy:

Con: "I mistook this argument as the Pro claiming that there exists a path from the beginning of the game to the end that always guarantees a win or a tie."

I did say that! You examine the tree for White and see if there is a path that leads to a win or tie no matter what Black does. If there is, White has a perfect strategy. If not, then no matter what White does, Black can counter. But that means Black has a perfect strategy, because Black can always counter White! It's a really elegant proof when you understand it.

Finiteness of Chess:

The real debate seems to lie under this heading, though it is a misnomer because Con agrees that chess is finite and thinks there are complicating factors.

Con seems to agree that Tic-Tac-Toe is solvable (though he seems to contradict himself on this), and argues that chess is meaningfully different. I interpret his argument like this:

"Tic-Tac-Toe is different from chess because in Tac-Tac-Toe, nothing is ever undone. In chess, pieces can be removed, moved, and switched, but in Tic-Tac-Toe they can only be added."

But my argument is indifferent to this entirely. It only concerns game states. What the differences between those game states are determined by is irrelevant to the solution to the game. An equivalent game to chess (which would be much harder to understand) would consist of nothing more than a series of numbers representing game states and functions represented moves. The board, the pieces, and the legal moves are all irrelevant to the fact that the game states can be arranged in a topological structure according to which the correct combinations are legally accessible from one another. I emphasize the word topological because the final design of this tree is the only important thing. Con has nothing more than an unproven intuition that more complex rules meaningfully alter the inherent nature of the game. The only reason I need to appeal to the rules at all is to justify Theorem 1, which is very obviously correct.

Con says: "Just because one wins a game of chess it doesnt imply that a perfect strategy exists because that particular route to victory that person just took could have been countered in numerous ways, along with any other path to victory that other people have taken."


This is to miss the point entirely. I am sure that the huge majority of paths can be countered. The question is, can all of them? As we see above, either answer implies an ideal move! If there is a sure path to victory, then each move that leads down it is an ideal move. If there is no path, then the opponent has a series of ideal moves to victory: counter every time!


Con also says, " If a move were truly ideal then it would still be able to guarantee victory despite human error..."

I think this is a preposterous standard because a player can always throw a game.

Con must agree that there a situations in which a player can force checkmate later in the game. This is the premise on which chess puzzles rest. I defy Con to tell us that there is no ideal move for Black in this position: http://0.tqn.com... Yet surely he can fail to make it. Con must give up this argument to stay in the bounds of a reasonable discussion.

Conclusion:

My initial proof and subsequent explanation meet the "weak MIT standard" of proof: while not fully deductive from axioms, it is laid out in common language rigorously enough so as to be immune to attack from a well caffeinated mathematician.

The Con position is based on mistaken intuitions about the game. He brings in a series of folk game theoretic notions like luck, mistakes, and odds. Luck and odds are rigorously defined in terms of probability and they do not exist in chess. Someone who claims to have lost chess because of luck or who says the game is broken because of a mistake is surely a sore loser. In my opinion, a great deal of appreciation for the mathematical beauty of the world is lost by restricting oneself to simplistic conceptual schemes such as this. Accordingly, Con has offered no proof whatsoever that his objections are actually relevant. I have thoroughly examined each of them an explained why that are not.

Voting Points:

These are my suggestions on who you should vote for. I'll explain why I think this is good to do in the comments if anyone cares. I urge Con to include this also.

Conduct: Tie

I think that conduct was good on both sides.

Spelling and Grammar: Pro

Con makes numerous spelling and grammatical errors, and his case is badly organized. I have occasional typos, but I am generally far more articulate.

Arguments: Pro

See my conclusion :D

Sources: Tie

Sources were not important to this debate, so I urge a tie vote.
imabench

Con

Resolution:

"A guaranteed win is enabled if the player cannot fail to win if he makes the right moves. The fact that players do in fact make the wrong moves is irrelevant."

Whether or not a player makes a wrong move determines whether or not a move is "ideal". If a player makes all the right moves then any move could be considered an ideal move since it "enables" the player to win. An ideal move though guarantees a player will win or tie a game of chess regardless of any future errors. Chess is a game controlled by humans, so human error will always be a factor which eliminates the possibility of there being an ideal opening move.

"As I said in Round 3, this means that no game can have a perfect strategy"

There is a difference between having a perfect strategy, having a strategy that increases your odds of winning, and a strategy that guarantees victory or a tie no matter what...... It is perfectly possible that a move exists in Chess that boosts your odds of winning, but that is a far cry from being able to claim that there is an opening move that guarantees a win.

Chance in Chess:

" If he doesn't notice and makes the wrong move as a result, that is a result of his error, not uncertainty within the game..."

I agree. But that error still leads to chance because you never know when your opponent is going to screw up. Stratego creates chance through deception, Chess creates chance through failing to take into account all of the variables.

"Con does agree that players will always move successfully"

I certainly do not!

" It is surely plausible that chess grandmasters (or even skilled amateurs) never fail to perceive events on the board."

Its plausible but it isnt reality. There are computers that can beat chess grandmasters which means that even chess grandmasters fail to account for variables. The ironic part is that occasionally the same chess grandmasters can beat the computers who beat them, meaning that even the computers fail to properly take into account all of the variables in a game of chess.

http://www.nytimes.com...
http://gambit.blogs.nytimes.com...
http://www.nytimes.com...
http://www.engadget.com...
http://www-03.ibm.com...

"If it is easier, pretend that there is a race of supermen who are smarter than the smartest computer and never make mistakes. If they play chess, it will still be the same game, but none of these factors will apply"

If such a day comes, there is still the fact that there are many different ways to win at chess, and each one of those paths to victory consist of moves that can be countered in many different ways as the game progressed.

" solubility is itself an abstract mathematical concept and should not change when a computer is invented."

But what happens when solubility is defined as increasing the odds to win rather than guarantee it? There are moves in every game ever invented that increases or decreases your odds of winning a game, and the math behind solubility of a game doesnt come as a clear cut path to victory. It comes in the form of making moves that over time stack the odds in your favor as time progresses. Solubility doesnt revolve around revealing an unbeatable series of moves to win, its about using a series of moves that you can make which boost your chances of winning over time.

Game States + Chess Trees

" You examine the tree for White and see if there is a path that leads to a win or tie no matter what Black does."

But any move that white makes can be countered by black, in the particular branch though black failed to take into account all the variables and made mistakes that other players could counter successfully. There are thousands and thousands of paths to victory for white, but none of those are the ideal strategy because all of them can be countered in numerous ways at numerous stages.

Finite-ness of Chess

" Con has nothing more than an unproven intuition that more complex rules meaningfully alter the inherent nature of the game. "

The number of game states in chess does lead to other game states, much like tic-tac-toe. However the nature of how tic-tac-toe is played compared to chess does change how the transition from one change of game state to another is affected. What makes chess fundamentally different from tic-tac-toe isnt that chess is more complex or has more pieces, its because Chess is won differently than tic-tac-toe.

In tic-tac-toe, three in a row and that is how you win. In chess though you win by cornering or capturing the opponents King piece. What does this mean though in terms of transition of game states between both games?

The reason that there is a way to win at tic-tac-toe is because in tic-tac-toe game states rely a great deal on prior game states. Chess on the other hand does not rely extensively on prior game states to determine who wins or loses. In chess the game states in the past do not significantly determine the outcome of the game because game states in chess are made to allow access to other game states to increase the odds of winning.

Tic-tac-toe goes from state to state to try to win, chess on the other hand goes from state to state just to stack your odds in the long term. That is why there isnt an ideal opening move in Chess, because the opening move does so little to affect the outcome of the game.

" I am sure that the huge majority of paths can be countered. The question is, can all of them?"

Yes, all of them can be countered. At any point in any game a key piece could be eliminated or a different piece could be moved instead of another to protect against a move. There is no invincible way to win a game of Chess.

" If there is no path, then the opponent has a series of ideal moves to victory: counter every time!"

Chess isnt a game of defense though, its a game of both offense and defense by both sides. Black can counter against any move or any piece, whether they choose to or not determines the outcome of the game. White wins if black makes more errors in assessing the variables, black wins if white makes more errors. Just because Black can counter any move by White does not mean that Black has a series of moves to victory. White and Black both have to play offense and defense.

"I defy Con to tell us that there is no ideal move for Black in this position"

There is an ideal move for black in that position, however there are many aspects of this scenario that shows how this isnt an ideal opening move.

Number 1) White set himself up for this more than black did anything
Number 2) Whites last move was the fatal error, literally any other move would nullify the "ideal" move by Black
Number 3) There are over 100 other ways that White could have made his first two moves, of those 100+ moves this is the ONLY game state which allows for this easy checkmate by Black.

And most importantly

Number 4) Blacks initial opening move does not win the game, it is the move made after it.

Conclusion:

Chess is played by making moves that increase your odds of winning over time, an ideal opening move does not exist in chess because all moves in chess can be countered and negated. Pro himself said that an ideal move has resulted in a loss thousands of times in the past, an ideal move guarantees victory regardless of human error or moves made by the opponent. In chess though both of these exist and both of these are evidence for why there isnt an ideal opening move in chess.

I thank the Pro for a wonderful debate and I thank all the voters for reading :D
Debate Round No. 4
31 comments have been posted on this debate. Showing 1 through 10 records.
Posted by lgreeff 2 years ago
lgreeff
A simpler way to look at this debate would be from the other end as it were. Imagine we had a database of all possible legal games. We then sort the database for wins for white, wins for black and ties. To start our game, we simply choose a game from either the wins for white or wins for black database depending on which color we are.

Each successive move will eliminate games from the database as they are no longer possible. Since we are playing only games that result in a win it doesn't matter what moves the opponent makes. The opponents moves simply serve to reduce the possible outcomes. The very first move already reduces the database to 1 20th since there are only 20 possible opening moves for white.

Whether or not there is only one opening move that always leads to a win is unknown, it may turn out that there are a few options, or all opening moves could force a win with advantage going to who starts first or second. Or it may turn out that only a draw can always be forced.

A comparative analysis of the three databases would reveal if there were overlapping opening moves. If there are opening moves in only the win databases that are unique to either black or white then an ideal opening move to force a win does exist.

The only way to determine this would be to have a database of all possible legal games. Sorting this to sensible games would make a huge difference to the total number of games. Or in other words games that force a win.

This approach reduces the debate to a series of eliminations leading to a win. Much easier to conceive.

(Based on an idea of mine for a chess computer I had in 1980)
Posted by hereiam2005 3 years ago
hereiam2005
The debate is very good, but there was one flaw if con exploited could lead to his win.

The resolution is "there is an opening move" => indicate a single move at the beginning of the match by White, since Black's is called defense move instead. ( note the singular form of move)

Now white *assume* that there is a path that guarantee a win or a draw for White. If such path does not exist, no matter what White does always leads to a loss to Black if Black play ideally => every Black's opening defense move are considered ideal. This consequence is not negated by Pro.
Because a loophole in the resolution, Pro did not fully prove that the ideal opening move by White exist => resolution fail.
Posted by johnlubba 4 years ago
johnlubba
Hi Grape. :)

So I'm wondering now, If the ideal move seems possible based on a calculation of finite moves, but is yet un-known. Then why on earth have you won all the points in the debate, you concede yourself that it is un-known with only a hope of it being discovered in the future.

I am confused here.
Posted by Grape 4 years ago
Grape
"This suggests to me, but does not come close to proving, that every opening move in chess is ideal. It's pure speculation, but I wonder if it could be proved or disproved." - RoyLatham

I have a similar intuition, but I doubt that there is any good justification for it based on what we currently know.

"I am just wondering if this was a concession by Pro in the first round.

If we are persistent and do not die in some terrible disaster, humans will eventually solve the game of chess just as we have solved the game of checkers.

Hence meaning the ultimate move has yet to be discovered." - johnlubba

My argument was that there is an ideal move. The ideal move does not have to be known to us. It may never be known, and finding it would require immense computational resources.
Posted by johnlubba 4 years ago
johnlubba
I have read the first round by both contestants, What a interesting debate, It's much to complex for me to follow it completely.

I am just wondering if this was a concession by Pro in the first round.

If we are persistent and do not die in some terrible disaster, humans will eventually solve the game of chess just as we have solved the game of checkers.

Hence meaning the ultimate move has yet to be discovered.
Posted by RoyLatham 4 years ago
RoyLatham
Interesting debate. Pro would always win, I think, because the resolution only asserts that the move exists, not that it can be discovered.

The definition of "opening move" is odd. White always moves first, so the opening move by Black is actually a two-move sequence with Black responding to White. There are 20 opening moves by White and 20 responses by Black. So if Black has any ideal opening move, then any opening move by White can be led by Black to a path that at least yields a draw. Thus if Black has an ideal opening move, White cannot have an opening move that guarantees a win. White may still have an opening move that leads to a draw.

This suggests to me, but does not come close to proving, that every opening move in chess is ideal. It's pure speculation, but I wonder if it could be proved or disproved.

Incidentally, tic tac toe is game in which optimal play by both sides guarantees a draw. http://ostermiller.org... Both sides have ideal opening moves.
Posted by One_Winged_Rook 4 years ago
One_Winged_Rook
I'm a bit behind on this one, but I wanted to comment to imabench. I was very impressed by his debate on prostitution against Danielle, and I thought he should have used that tatic here... he should have just concede totally that it is theorectically possible that there is an Ideal opening move... it does exist... he had to argue that although that does theorectically exist, the question should have just been raised if it's within the realm of possibility that human's will EVER be able to know it, or, assuming they can't... how long it will take before a computer will be able to solve it, if ever.... good job Grape and to imabench, try debating more like you did against Danielle
Posted by THEBOMB 4 years ago
THEBOMB
@strikingfury

There are so many contradictions in that statement I do not know where to begin...
Posted by strikingfury 4 years ago
strikingfury
definitely no ideal move. i play chess and im rated in about the 1800s and i always play something different . it doesnt matter what u play. as long as both people dont make a mistake than they can draw
Posted by Grape 4 years ago
Grape
"[I]t is hard to lose a debate where what you are debating is a proven fact..." - blazeratman

You'd be surprised. Someone lost a four color theorem debate a while ago.
5 votes have been placed for this debate. Showing 1 through 5 records.
Vote Placed by InVinoVeritas 4 years ago
InVinoVeritas
GrapeimabenchTied
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Reasons for voting decision: "He brings in a series of folk game theoretic notions like luck, mistakes, and odds. Luck and odds are rigorously defined in terms of probability and they do not exist in chess." This sums it up well. And, indeed, Grape was more articulate in his arguments, with fewer mistakes.
Vote Placed by blazeratman 4 years ago
blazeratman
GrapeimabenchTied
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Reasons for voting decision: Blowout for Pro, but then it is hard to lose a debate where what you are debating is a proven fact, I was well aware of the mathematical proofs, seemed to me like Pro knew there was an Ideal Opening Move in this debate too :)
Vote Placed by miketheman1200 4 years ago
miketheman1200
GrapeimabenchTied
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Reasons for voting decision: This was a total win for pro. Con failed to make any convincing arguments, and made alot of grammatical errors. I felt that pro had better conduct as well.
Vote Placed by Thaddeus 4 years ago
Thaddeus
GrapeimabenchTied
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Reasons for voting decision: Clear win for Pro. Most of Cons arguments were based off misunderstandings of Pro's case. Pro clearly demonstrated that in a finite game, with no hidden variables, there will be an ideal move.
Vote Placed by FourTrouble 4 years ago
FourTrouble
GrapeimabenchTied
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Reasons for voting decision: Con's arguments made no sense to me. First, Con's argument that an ideal move cannot "guarantee" a tie/win seems like a blatant misreading of the word "enable" in the resolution. Second, I don't see how human error is relevant to the discussion (Chess is a formal system). Third: How does the existence of responses to moves change the fact that an ideal move still exists for every position? Some more thoughts in comments...