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/
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