Set, Game, Match

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1ryan_b

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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 GamesThe objects which comprise a game are

players,moves, andpayoffs. A game is asetwhich 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

believethey 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 Doeliminatingassumptions, but in any applied context all that happens is that assumptions slide intothe axioms. I am therefore wary of axioms tricking me into being wrong about the assumptions at work, when that information really should be obvious at all times.