One kind of infinite game discussed by logicians is determined by the choice of a subset X of the half-open unit interval [0, 1). Then each player in turn selects a bit from {0,1}, and if the countable bitstring thus created lies in X then the first player (W) wins; if not, the second player (B) wins. The Axiom of Determinacy states that every such game has a strategy for one player or the other. Meanwhile, the Axiom of Choice can be used to show there exists a subset X of [0,1) whose corresponding game has no strategy for either player. I'm hoping for a totally explicit game perhaps played by placing counters on unoccupied cells of a board, as in Hex (or Conway's Angels & Devils game). —Dan ----- On Saturday, March 3, 2018, 4:36:13 AM PST, Fred Lunnon <fred.lunnon@gmail.com> wrote: If your game has length \omega , how do you know who wins? WFL On 3/3/18, Dan Asimov <dasimov@earthlink.net> wrote:
I am curious: Do people know of interesting infinite two-player games, preferably without chance playing a role?
I'm mainly interested in discrete 2-player games with one player going first and with play alternating thereafter. Such that in some sense, except for that first player / second player thing, the challenge facing each player is as symmetrical as possible.
Though I will not rule out continuous games.
Here's a fun game which works with any finite number k of players: Players take turns* to name positive integers that haven't already been named. When you name an integer, you subsequently 'own' it. Play continues (possibly beyond time omega) until every integer is owned. Clearly, it must stop before time omega_1. * For a positive integer n, Turn(n) is the sum (reduced modulo k) of the base-k digits of n. For an ordinal alpha, Turn(alpha) is the sum (again reduced modulo k) of the Turns of the coefficients of alpha in Cantor normal form. This is really fair. An integer n is described as 'dominated' by a player if there exists an infinite set S containing n such that all finite sums of elements from S are owned by that player. The winner is the person who dominates the smallest integer. (A winner exists by Hindman's theorem, and is unique by the fact that you can only dominate an integer you own.) Best wishes, Adam P. Goucher
Sent: Saturday, March 03, 2018 at 7:38 PM From: "Dan Asimov" <dasimov@earthlink.net> To: "Eugene Salamin" <gene_salamin@yahoo.com>, math-fun <math-fun@mailman.xmission.com> Subject: Re: [math-fun] Infinite games
One kind of infinite game discussed by logicians is determined by the choice of a subset X of the half-open unit interval [0, 1).
Then each player in turn selects a bit from {0,1}, and if the countable bitstring thus created lies in X then the first player (W) wins; if not, the second player (B) wins.
The Axiom of Determinacy states that every such game has a strategy for one player or the other. Meanwhile, the Axiom of Choice can be used to show there exists a subset X of [0,1) whose corresponding game has no strategy for either player.
I'm hoping for a totally explicit game perhaps played by placing counters on unoccupied cells of a board, as in Hex (or Conway's Angels & Devils game).
—Dan
----- On Saturday, March 3, 2018, 4:36:13 AM PST, Fred Lunnon <fred.lunnon@gmail.com> wrote: If your game has length \omega , how do you know who wins? WFL
On 3/3/18, Dan Asimov <dasimov@earthlink.net> wrote:
I am curious: Do people know of interesting infinite two-player games, preferably without chance playing a role?
I'm mainly interested in discrete 2-player games with one player going first and with play alternating thereafter. Such that in some sense, except for that first player / second player thing, the challenge facing each player is as symmetrical as possible.
Though I will not rule out continuous games.
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participants (2)
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Adam P. Goucher -
Dan Asimov