Most two person games are finite; for example, chess has rules that don't allow an infinite game, and tic-tac-toe obviously ends after at most 9 plays.

Let's define a new two person game: the "Metagame". The first player first picks any two person finite game (e.g., chess or tic-tac-toe). Then, the second player sets up the board (or whatever is needed) and makes the first move in that game, and the Metagame winner will be whoever wins that game.

The question: is Metagame finite or infinite?

(In reply to No paradox by David Shin)

If we "Assume Metagame is infinite" shouldn't we be able to show that it is infinite?  We can't.

The first player must pick a finite game (not Metagame) so the second player makes the first move and it eventually terminates.

So if it always terminates, Metagame is finite.

This is a contradiction.

[Similarly "Assume Metagame is finite" leads to a contradiction.]

I think the way out of this is to say that some games are finite and some are not.  But to realize that a game that is not finite does not have to be infinite either.  That way Metagame cannot be picked by the first player, but it is not infinite.

Posted by Jer on 2005-04-13 17:12:08