Speaking of strategies, consider this game (which arose from my failed attempts, including some conversations with the incredibly smart algebraist George Bergman, to define an infinite version of Hex on the hexagonally tessellated plane): Given a countable set X -- WLOG let X = Z+ = {1,2,3,...,n,...} -- players alternately remove an element of their choice, always picking from only the remaining elements. The players know each other's plays. The turns proceed, indexed by countable ordinals. The player who picks the last remaining element is the winner. By the well-ordering principle (equivalent to the Axiom of Choice), the winning player is well-defined. [Note: There is a natural way to define whose turn it is at any limit ordinal. Namely, put the turn's ordinal index into Cantor normal form (< http://en.wikipedia.org/wiki/Cantor_normal_form#Cantor_normal_form >) and add up the integer coefficients. If the sum is even, it's the First player's turn; if odd, it's the Second's.] Is there a strategy for the First or Second player? Or not? (The formal definition of a strategy for a game of perfect information and well-ordered play like this one is: For whichever player P the strategy is for: Each time it's P's turn, the strategy dictates what P's play should be -- so that if the strategy is followed at each of P's turns, then P is guaranteed to win. So it's a function from the set of all possible sequences of previous plays before any turn of P, to the set of all possible plays P has at that juncture. It's very easy to show that for any *finite* game of perfect information between two players, one of them has a strategy. (Exercise.) --Dan ___________________________________________________________________________ P.S. Then there's that proposed set-theory axiom, the Axiom of Determinacy, that says that given any subset A of [0,1] consider the game G_A for two players, who alternately choose either a 0 or a 1 -- knowing each other's plays -- to jointly produce a countable binary string -- and thus an element of [0,1]. If that element is in A, First wins; otherwise Second. It's known that this is inconsistent with the Axiom of Choice, since that can be used to construct games with no strategy. On Aug 16, 2014, at 11:00 AM, Adam P. Goucher <apgoucher@gmx.com> wrote:
In tic-tac-toe, there are at least two very different optimum strategies for the first player, depending on whether you choose to play in the centre or in the corner