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 the solution, it has been assumed that the second player's
"move" may be the same as the first player's, in that the second
player makes his "move" by choosing Metagame as the game to play. This ability is not defined in the rules provided in the
definition of the game, and contradicts the fact that, if selecting
the game to play was a move, then the second player could not make the first move as it was already done by the first player. 

The definition of an infinite game is [1] a game that
does not have a knowable beginning or ending, and [2] the rules of the game ensure the game is infinite. Metagame does indeed fit the definition of the first part, in that its ending is not knowable, that is, until the first player chooses a finite game to play, yet it fails in the second, in that its own rules do not ensure the game is infinite. (E.g., the first player chooses tic-tac-toe, the game ends after a sparring of five to nine moves either in a win or draw. Thus Metagame is not infinite.

But is it finite? The definiton of a finite game is [a] it has a definite beginning and ending and [b] rules exist to ensure the game is finite.
Metagame fails in [a] in that the ending is not well defined. Though Metagame defines the winner as the player who wins the game, it does not define how the game is won, or even what constitutes a draw. Therefore, Metagame is, by definition, not finite.
Though there exist the labels "finite games" and "infinite games", not all games are one or the other. Just as is Metagame, some are neither. And, by its own rule, Metagame can not be a game picked within Metagame as a game to play as it is not defined as finite.

Posted by Dej Mar on 2013-03-11 12:58:10