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?

Metagame is not a well-defined game. The set of "all two person finite games," which player 1 needs to pick an element from, is not mathematically defined, much in the same way that the "set of all sets" or even the "set of all finite sets" are not well defined sets.

Therefore, we cannot classify the Metagame as a finite or infinite game, because Metagame is not a game to begin with. However, if we come up with a new theory of metagames and define finite and infinite in much the same way as we do in normal game theory, then we can see that the Metagame is finite, because we no longer run into the ambiguous situation of allowing a player to choose the Metagame as their first move.

Posted by Avin on 2005-04-13 13:37:34