There is an Ideal Opening Move in Chess

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.