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.
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.
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.