Refined multi-dimensional models to deal with cycles have been proposed already:guyhaw wrote:I recommend John Beasley's book 'The Mathematics of Games'. There, he gives a critical review of the concept of rating, ELO etc.
He gives a scenario involving a circular running track (the usual kind) where, quite legitimately, A beats B, B beats C and C beats A ... a counterexample to the assumption that players can be ordered in a linear spectrum.
http://dx.doi.org/10.1016/j.jspi.2004.05.008
Abstract: The Bradley–Terry model is widely and often beneficially used to rank objects from paired comparisons. The underlying assumption that makes ranking possible is the existence of a latent linear scale of merit or equivalently of a kind of transitiveness of the preference. However, in some situations such as sensory comparisons of products, this assumption can be unrealistic. In these contexts, although the Bradley–Terry model appears to be significantly interesting, the linear ranking does not make sense. Our aim is to propose a 2-dimensional extension of the Bradley–Terry model that accounts for interactions between the compared objects. From a methodological point of view, this proposition can be seen as a multidimensional scaling approach in the context of a logistic model for binomial data. Maximum likelihood is investigated and asymptotic properties are derived in order to construct confidence ellipses on the diagram of the 2-dimensional scores. It is shown by an illustrative example based on real sensory data on how to use the 2-dimensional model to inspect the lack-of-fit of the Bradley–Terry model.
I suggested another model in that post to the computer-go mailing list:
http://computer-go.org/pipermail/comput ... 13189.html
The lack of fit of the Elo-rating model to computer-go data is particularly strong, with different kinds of programs using radically different algorithms.
Rémi
