Timeline for Infinite games: are they well defined?
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| Jul 17, 2010 at 13:28 | comment | added | Wadim Zudilin | Carl, I just wish to draw a distinction (for myself!) between the discreteness and continuity in this particular construction; the additional property you mention can be put as a constraint for a discrete infinite game, can't it? Yes, the theory you describe, with plays being discrete sequences, everything works perfect. What's wrong when plays of a game are functions on $[0,1]$, say? I believe that this should have been studied as well, because of well-studying of the discrete situation. My question is just in the case if you know something on this. Just my curiosity. Thanks! | |
| Jul 17, 2010 at 12:56 | comment | added | Carl Mummert | No, I don't view continuous games as part of this construction. For example, the continuous games considered by Bollobás, Leader, and Walters ([2] above) have the property that one cannot "play strategies against each other", and it is possible for both players in those continuous games to have winning strategies. I don't see how to reconcile that with the family of games I defined. | |
| Jul 17, 2010 at 9:23 | comment | added | Wadim Zudilin | @Carl, thanks a lot for your very detailed formalism and the references. It's an excellent review of the subject. Just a silly question (in view of Andres' example): aren't continuous games a part of this general construction? | |
| Jul 17, 2010 at 2:12 | history | edited | Carl Mummert | CC BY-SA 2.5 |
obvious typo: player II, not player i. Also make a cosmetic change
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| Jul 17, 2010 at 2:04 | comment | added | Joel David Hamkins | Daniel, you may be interested in Itay Neeman's work on the determinacy of similar long transfinite games. See his book at math.ucla.edu/~ineeman/long-games.htm. He an others have also considered games going beyond the countable ordinals. | |
| Jul 16, 2010 at 17:58 | comment | added | Daniel Asimov | Oops, that should have read, "There must be a first countable ordinal beta at which there are no unclaimed members of J; the player whose turn it is at beta is defined as the winner. | |
| Jul 16, 2010 at 17:54 | comment | added | Daniel Asimov | (Cont'd.) Assume "whose turn it is" has been defined for all countable limit ordinals, and hence by alternation for all countable ordinals. Then it's not even necessary to predefine the length of the game. A simple example is this: let J be any countably infinite set. Each player on their turn claims some member of J. There must be a first countable ordinal at which the last member of J is claimed: the player who did that is defined as the winner. (I hope to expand on this soon in a short article.) | |
| Jul 16, 2010 at 17:47 | comment | added | Daniel Asimov | Incidentally, there's no reason that the plays of an infinite ordinal game must be indexed by omega. Any infinite ordinal lambda will do, though let's just be concerned about countable ordinals here. Special care must be taken to know whose turn it is at each play (whose index is some ordinal < lambda). This is easily accomplished in a "fair" way as long as whose turn it is is defined for limit ordinals; this is not difficult. (Successor ordinals just follow the alternating turn rule, as for the case lambda = omega.) | |
| Jul 16, 2010 at 15:50 | history | answered | Carl Mummert | CC BY-SA 2.5 |