Yes, I was speaking only about prior sequences of moves that are possible according to the rules of the game -- and yes, that could occur with the lucky player following that strategy. I did not say it, but thank you, Allan, for stating it precisely. ((( Also, I know which player in the integer-choosing game (if any) has a winning strategy, and can prove it. It's kind of interesting. Anyone else up for the challenge? ))) --Dan On Aug 16, 2014, at 4:13 PM, Allan Wechsler <acwacw@gmail.com> wrote:
I am with those who have said, in different words, that Henry's plausible-seeming query eventually founders on the difficulty of rigorizing the notions of "attack" and "defense". I suspect that these notions are intuitions that come from the *embodiment* or *model* of a given game. But when the game is stripped of its model, leaving only an abstract edge-colored directed graph of positions and legal moves, "attack" and "defense" lose their intuitive force, while the nature of what constitutes a good strategy is unchanged. One can even imagine re-embodying a given abstract game with a new model, in which some originally defensive moves now engage our "attack" intuitions.
Dan's transfinite game eludes me entirely, but it gives me an occasion to quibble with his definition of a strategy. For me, at least, a strategy need not produce an answer for *all possible sequences* of moves to that point, but rather, only for *all sequences possible for a player following the strategy.* Perhaps this is what Dan meant and I am reading too strictly. That is, my strategy need not give me a well-defined response even if I am required to let five-year-old cousin choose my move every once in a while.