My basic idea, which is undeveloped is rather like this - you partition all players into 2 (or more) broad styles. One of the variables in the rating tells you how much (from 0 to 1) he plays at one extreme. The rating system itself somehow determines which style you are and it's an abstract quality that we don't necessarily have to understand. (sort of like learning algorithms that build neural networks that we don't understand but it works.)
In chess, which I use as an example because I understand it much better, there are kinds of playing styles that interact. You can have very good tactical players (who love gambit play sometimes) and you have very slow positional style. Some very good players are not particularly good at tactics. I don't know what conclusions you can draw about who is expected to win matches between various styles, but I'll bet there is a measurable non-transitive relationship somewhere there. Of course an outright learning algorithm, if given enough games, might be able to predict winning expectancy better than straight ELO. - Don On Wed, 2007-01-31 at 05:54 -0600, Nick Apperson wrote: > I feel that what we need essentially is a set of functions that tell > us expected winning percentages with certain matchups. In an extreme > example, we could imagine 3 rock, paper, scissors players. One always > plays rock, one always scissor and one always plays paper. In this > case, we would be able to define a ranking function as a relative > ranking, but absolutre ranking would not exist. And this is > consistent with our whole approach here that skill is not transitive. > A simple 2D ranking system could work like this: > > let W = chance that player 1 will beat player 2 > 1-W = chance that player 2 will beat player 1 > > our skill is expressed in R and T (for theta) > > ELO = (R1-R2)+k*sin(T1-T2) where k is some constant, R1, R2, T1, T2 > are R of player 1 and 2, theta of player 1 and 2 respectively > > > This would result in a nontransitive 2D skill map. There are many, > more compex functions that could be worked out and this has the nice > property that it easily collapses into a 1D map by merely setting > everyone's theta to the same value. Essentially R is "general skill" > and theta is a rock paper scissor type thing where one strategy is > better against certain other types of strategies. > > > On 1/31/07, Vlad Dumitrescu <[EMAIL PROTECTED]> wrote: > Hi, > > On 1/30/07, Don Dailey <[EMAIL PROTECTED]> wrote: > > It would be interesting if it would be possible to construct > a 2 > > dimensional > > model statistically. A 2 dimensional system would not be a > perfect fit > > either, > > but would simply be a better approximation. So in some > way a players > > "strength" could be expressed by 2 numbers instead of > 1, and the 2 > > numbers > > together would predict your chances of beating another (2 > dim) player > > more accurately that a 1 dimension system could. And of > course you > > could > > extend this. But I don't have a clue how one would > construct such a > > system > > or if it's even possible - but it seems like more > information should be > > better > > than less. > > Unfortunately, having more than one dimensions makes > comparisons > impossible - if an ordering relation is defined over the > domain, then > this domain is "one-dimensional" with regard to that > relation. > > In other words, one can't compare vectors, just scalars. So > the > multi-dimensional "strength vector" has to be turned into a > scalar (by > for example a weighted sum) and we're back where we > started... > > best regards, > Vlad (master of the obvious :-) > _______________________________________________ > computer-go mailing list > computer-go@computer-go.org > http://www.computer-go.org/mailman/listinfo/computer-go/ > > _______________________________________________ > computer-go mailing list > computer-go@computer-go.org > http://www.computer-go.org/mailman/listinfo/computer-go/ _______________________________________________ computer-go mailing list computer-go@computer-go.org http://www.computer-go.org/mailman/listinfo/computer-go/
