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2012-02-19T06:59:20  <antimatroid> damn i hate looking for counterexamples
2012-02-19T07:01:13  <mcstar> the easiest of disproofs
2012-02-19T07:01:42  <mcstar> just set up a program to search the problem space
2012-02-19T07:02:01  <mcstar> or..
2012-02-19T07:02:07  <mcstar> use your intuition
2012-02-19T07:11:20  <antimatroid> i have a program to make the games
2012-02-19T07:11:47  <antimatroid> i haven't made one to run through all combinations, there's lots and it means writing algorithms to check symmetry conditions which is hard
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2012-02-19T07:12:48  <antimatroid> i'm looking for a game which has a player n-transitive automorphism group, a player transitive (but not a player n-transitive) and strategy trivial subgroup of the automorphisms and no pure strategy nash equilibria :P
2012-02-19T07:13:36  <mcstar> thats a lot of requirements
2012-02-19T07:13:54  <mcstar> i guess you know about GAP
2012-02-19T07:14:08  <antimatroid> GAP?
2012-02-19T07:14:14  <antimatroid> it is a lot of requirements
2012-02-19T07:14:23  <antimatroid> i'm basically trying to show a theorem can't be generalised
2012-02-19T07:14:46  <antimatroid> there's a very small class of games there which i haven't been able to show they don't extend to (yet) by example
2012-02-19T07:14:58  <mcstar> gap is an advanced opensource program dealing with finite groups
2012-02-19T07:15:01  <mcstar> afaik
2012-02-19T07:15:20  <mcstar> it is part of sage
2012-02-19T07:15:30  <mcstar> or you can use just by itself
2012-02-19T07:15:46  <mcstar> anyway, i dont want to feed you disinformation, so see for yourself, it could help
2012-02-19T07:15:50  <antimatroid> i doubt it'd be much help
2012-02-19T07:15:56  <antimatroid> my game theory library is good for this stuff
2012-02-19T07:16:01  <antimatroid> hence why i wrote it
2012-02-19T07:16:03  <mcstar> haha
2012-02-19T07:16:16  <mcstar> i dont doubt that its good
2012-02-19T07:16:25  <mcstar> but i think gap is waay better
2012-02-19T07:16:29  <mcstar> :)
2012-02-19T07:16:32  <antimatroid> not for games
2012-02-19T07:16:45  <mcstar> ok, i just wanted to let you know
2012-02-19T07:16:46  <antimatroid> i give it some game bijections in a text file and it generates a symmetric game for me
2012-02-19T07:16:51  <antimatroid> cheers :)
2012-02-19T07:20:16  <UncleVasya> Do not let this guy pick a game for the next contest! :D
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2012-02-19T07:22:56  <mcstar> XD
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2012-02-19T07:28:34  <antimatroid> i know of only one example meeting the starting qualifications
2012-02-19T07:28:53  <antimatroid> and it has a pure strategy nash equilibria no matter how i assign the ordinal preferences to the pure strategy profiles :(
2012-02-19T07:29:41  <antimatroid> UncleVasya: don't trust me to make it as abstract as possible? :P
2012-02-19T07:31:49  <UncleVasya> I think your game will be very interesting... for computer to play. But not for humans to watch.
2012-02-19T07:32:09  <antimatroid> this wouldn't be that interesting for bots to play
2012-02-19T07:32:18  <antimatroid> they're fairly simple finite matrix games
2012-02-19T07:32:19  <UncleVasya> mathematicians aren't humans :p
2012-02-19T07:32:38  <antimatroid> i'm more interested in the structure of symmetric games in general
2012-02-19T07:32:54  <antimatroid> ie. how fair is a given symmetric game
2012-02-19T07:33:41  <antimatroid> it took me months to even find that example i mentioned above :|
2012-02-19T07:33:49  <antimatroid> having to possibly find yet another is a daunting task
2012-02-19T07:33:55  <UncleVasya> I remember McLeopold posted here a link to an article about program that make games.
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2012-02-19T07:34:37  <antimatroid> http://pastebin.com/X9zGf3aV
2012-02-19T07:34:37  <antimatroid> that
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2012-02-19T07:38:17  <mcstar> antimatroid: dont you have some tensor to multiply this game with, that preserves the requirements? like some similarity transform, but for your game?
2012-02-19T07:38:28  <mcstar> so it would generate other games?
2012-02-19T07:38:33  <mcstar> (maybe not all of them)
2012-02-19T07:38:37  <antimatroid> i don't understand what you mean?
2012-02-19T07:38:53  <antimatroid> you mean swap the numbers?
2012-02-19T07:39:07  <mcstar> no
2012-02-19T07:39:24  <mcstar> im not sure what the structure of the tensor is
2012-02-19T07:39:48  <antimatroid> let me define a game for you so it makes more sense
2012-02-19T07:40:36  <antimatroid> you have a set of at least two players N, for each player i you have a set of strategies A_i and a utility function u_i:A->reals where A = cartesian product of A_i's = strategy profiles of the game
2012-02-19T07:40:59  <antimatroid> then the elements of the "tensor" or table correspond to strategy profiles
2012-02-19T07:41:10  <antimatroid> and each player has a utility value for each strategy profile
2012-02-19T07:41:46  <antimatroid> so first cell is (a, c, e) and players all get payoff 1
2012-02-19T07:42:37  <antimatroid> i actually hoped that game didn't exist
2012-02-19T07:43:14  <antimatroid> as i wanted "if the automorphism group is player n-transitive and there exists a player transitive and strategy trivial symmetry group then there exists a player n-transitive and strategy trivial symmetry group" but that game breaks it
2012-02-19T07:44:08  <antimatroid> then there's a theorem saying every game where all players have 2 strategies and there's a player n-transitive and strategy trivial symmetry group then there exists a pure strategy nash equilibrium
2012-02-19T07:44:41  <mcstar> so that tensor represents the utilities of a 3 player, 2 strategy/player game
2012-02-19T07:44:48  <antimatroid> yep
2012-02-19T07:45:01  <antimatroid> row player is 1, col player is 2 and "matrix/page" player is 3
2012-02-19T07:45:18  <antimatroid> imo it should be row = n-1, col = n etc. but whatever
2012-02-19T07:45:22  <mcstar> yep, the a,b,c,d... i s alittle confusing
2012-02-19T07:45:42  <antimatroid> sorry, A_1 = {a, b}, A_2 = {c, d}, A_3 = {e, f}
2012-02-19T07:46:03  <antimatroid> it's easier when working with game bijections if the players strategy sets are distinct
2012-02-19T07:46:15  <antimatroid> as in it's easier to follow what's mapping where
2012-02-19T07:46:30  <mcstar> i see
2012-02-19T07:47:16  <antimatroid> ie. when you go to find (a, c, e), you look for 'a' first which should be the matrix, then index the row of the matrix with c and then the column with e
2012-02-19T07:47:36  <antimatroid> also, when writing out the cartesian product you usually increment the last possible element in the tuple first
2012-02-19T07:47:57  <antimatroid> so when storing in a 1d vector you would go through the matrices in logical order
2012-02-19T07:48:01  <antimatroid> rather than jumping all over the place
2012-02-19T07:48:17  <antimatroid> so in my pedantic opinion the entire game theory community writes n>2 player games out wrong :P
2012-02-19T07:49:12  <mcstar> so transitivityimplies that in every 2 indices, the tensor is simmetruc right?
2012-02-19T07:49:34  <antimatroid> defining all those terms is not easy :P
2012-02-19T07:49:50  <antimatroid> you need the notion of a game bijection
2012-02-19T07:50:52  <antimatroid> a subgroup H of the game bijections is transitive if for all players i,j there exists a bijection in H st it maps player i to j
2012-02-19T07:51:14  <antimatroid> it's player n-transitive if for all player permutations there exists a bijection in H which maps players according to it
2012-02-19T07:51:37  <antimatroid> game bijections also give a way of mapping between mapped players strategy sets, and that induces a way of mapping between strategy profiles
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2012-02-19T07:52:30  <mcstar> what is the most general game bijection?
2012-02-19T07:52:36  <mcstar> some permutation group?
2012-02-19T07:53:27  <antimatroid> consider the identity game bijection, you have p:N->N st p(i) = i for all i, and t_i:A_i->A_i st t_i(a) = a for all a \in A_i
2012-02-19T07:54:01  <antimatroid> any game bijection is just a bijection p:N->N and for all i \in N a bijection t_i:A_i->A_{p(i)}
2012-02-19T07:54:12  <antimatroid> defining notation to map strategies is incredibly gross
2012-02-19T07:54:56  <antimatroid> you have like g.s_i = t_{p^{-1}(i)}(s_{p^{-1}(i)})
2012-02-19T07:55:17  <antimatroid> don't even get me started on composition and inverses of bijections :P
2012-02-19T07:55:32  <antimatroid> but games and game bijections form a groupoid as one would hope
2012-02-19T07:56:36  <mcstar> well, a bijection can be represented by a tensor product, right?
2012-02-19T07:56:54  <antimatroid> i have the barest understanding of what tensor products are
2012-02-19T07:57:15  <antimatroid> i dunno if i have a module though
2012-02-19T07:57:19  <antimatroid> which would be needed yeah?
2012-02-19T07:57:23  <antimatroid> possibly even a vector space
2012-02-19T07:58:30  <antimatroid> (i know i should know about tensor products, it's very far up on the to learn properly soon list)
2012-02-19T07:58:47  <mcstar> i dont think you need that for permutation matrices
2012-02-19T07:59:00  <mcstar> (or tensors)
2012-02-19T07:59:26  <antimatroid> hmm okay
2012-02-19T07:59:33  <antimatroid> i'll learn that stuff in the next few months
2012-02-19T07:59:39  <antimatroid> i'm going to need to for phd stuff anyway
2012-02-19T08:00:01  <antimatroid> trying to also iron out this game theory stuff to publish a paper
2012-02-19T08:00:04  <mcstar> do you have you thesis somewhere online?
2012-02-19T08:00:12  <mcstar> r
2012-02-19T08:00:20  <antimatroid> i almost find that embarrasing now but yeah i think hang on
2012-02-19T08:01:02  <antimatroid> https://docs.google.com/viewer?a=v&pid=explorer&chrome=true&srcid=0B5joEFt1EpwZZTg1OGM3MDYtMThmNC00NWNhLTgwM2QtYTU3MmEzOTBjNmZh
2012-02-19T08:01:41  <antimatroid> i didn't do the cardinal utility stuff properly
2012-02-19T08:01:45  <antimatroid> but it has the right idea
2012-02-19T08:02:38  <antimatroid> i've thrown the "fully" symmetric classification from that out too
2012-02-19T08:03:00  <mcstar> i cant see that
2012-02-19T08:03:56  <mcstar> ok
2012-02-19T08:04:05  <mcstar> but why did i need to log in?
2012-02-19T08:04:48  <antimatroid> dunno?
2012-02-19T08:04:51  <antimatroid> blame google? :P
2012-02-19T08:06:34  <mcstar> so you have some own results in there?
2012-02-19T08:07:07  <antimatroid> only fairly trivial ones to do with symmetric games
2012-02-19T08:07:19  <antimatroid> the paper will just be on those
2012-02-19T08:07:34  <antimatroid> and it's more giving a nice overall coverage
2012-02-19T08:07:46  <antimatroid> i also point out an incorrect classification which i see people use a lot (including wikipedia :P)
2012-02-19T08:07:52  <mcstar> are you going to share credit with your advisor?
2012-02-19T08:08:05  <antimatroid> well, technically hte games are symmetric that meet the definition, but they obviously got it wrong
2012-02-19T08:08:13  <antimatroid> i don't think so, but i will ask his opinion on that
2012-02-19T08:08:19  <antimatroid> i realise that's a bit weird, but i don'
2012-02-19T08:08:24  <antimatroid> t think he actually has any authorship
2012-02-19T08:08:29  <antimatroid> and i suspect he'll agree
2012-02-19T08:08:54  <antimatroid> if he thinks he should i honestly don't care
2012-02-19T08:09:33  <mcstar> ill read some chapter headings, to actually have a feeling of what you are doing, but not now, i have others thing sto do
2012-02-19T08:09:56  <antimatroid> i haven't seen matchings as i define them used before too
2012-02-19T08:10:20  <antimatroid> which i show as an equivalent (and perhaps more useful) way of defining standard symmetric games than how this other guy did
2012-02-19T08:10:58  <antimatroid> i formalised some of my "conversation" about them with what i'm writing up for the paper
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2012-02-19T08:58:03  <antimatroidl> i want to rename by game bijections but what to?
2012-02-19T08:58:41  <antimatroidl> it's really a bijection g:NxA->NxA where N is the set of players and A is the set of strategy profiles
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