The way Conway likes to present it -- of course this only applies to Conway-style combinatorial games (perfect information, outcome is just win/lose, etc.) -- is as follows. First of all, identify a game with its initial position, so we don't need separate notions of "game" and "position". Now a game is defined by the moves available to the two players, and a move is identified as the game (i.e., position) reached by making that move. So a game is a pair of sets of games.
Conway suggests (not in WW but in ONAG) that rather than embedding such a thing in ZFC set theory or whatever, we should think of it as a sort of deviant set theory with two different kinds of membership. I don't think this viewpoint is very widely shared.
Kinda related and possibly of interest to you: the axiom of determinacy.
The axiom of determinacy is very interesting.
I am reading a book called Probability and Finance: It's Only a Game, by Glenn Shafer and Vladimir Vovk, which has two insights which have caused my head to explode.
The first insight is that the environment is a player in the game. The basic game is between unequal players, Skeptic and World. The second insight is players can be decomposed into other players. They decompose World into a variety of other players, mostly by the kinds of moves they make.
It seems reasonable that this relationship must work two ways - something like all games roll up into Agent v. Reality games. If that is true, then I would accept the axiom of determinacy, because I have a real hard time imagining that Reality would not have the winning strategy over a long enough game.
I am making an investigation of game theory, and wanted to get my intuitions about this down; this is a better location than most for the job.
Sets of Games
The objects which comprise a game are players, moves, and payoffs. A game is a set which contains the specified objects. Combinatorial games are a set of sets, which may include further payoffs. Game space is every possible combination of these objects; there can at least in theory be infinitely many players, infinitely many moves, and an infinite variety of payoffs. Therefore, gamespace is infinite.
What about information?
Information is special - conventionally it is specified as a part of the game. With the game defined as a set in this way, we can say that the information for a game is specified by the set of which the players believe they are a member. In a game with perfect information, the players each believe the correct set of objects. With incomplete information, the players each believe a subset of the correct set. Differential information means the players each believe a different game set. Errors are represented by believing a set with elements not in the real set.
To Do