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
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Axioms? Update: I have achieved skepticism about the value of axioms for what I want to accomplish. I am separately reading a book about game-theoretic probability which argues that game theory should supplant measure theory as the base for probability, and one of the merits is that when people make mathematical decisions about probability, they often cite measure theory as justification—wrongly. The thing is that people treat axioms as eliminating assumptions, but in any applied context all that happens is that assumptions slide into the 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.
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How do operations work in this context? Unknown.
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I fully expect that this has been covered elsewhere—Winning Ways for Your Mathematical Plays is a good candidate.
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.
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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.