I've been stuck for weeks spending my free time on this little problem here:
Suppose we have a matrix $V$, where the entry $V_{ij} $ represents the observed win ratios in games where player i played in position 1 against player j in position 2 (and we'll set $V_{kk}=1/2$).
How do we assign a rating from a vector $X$ of n cardinal numbers (probably with the worst player having a rating of 1?) to each of the n players such that $\frac{x_i}{x_i+x_j}$ is made as consistently close to the observed win ratio $V_{ij}$ as possible? And yes, we're generating a rating based off of the Luce model.
I tried solving this using the usual least squares, but unless there's some telescoping trick I'm missing it's not going to be easy. I'm confused as to what would be the best choice of the cost function, have looked at the test statistic we see in G-tests (with the nice natural log which simplifies the calculus once you're optimizing the double sigma (and it's a double sigma because you have to worry about both i and j). I might have gotten somewhere with that approach, but I'm very concerned with its viability so I'd prefer to see someone else's take first).
Thanks in advance!