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[H.G.Muller]

Indeed, those are the correct expressions for the best estimates.

What makes the Elo determination problem so tough is that the winning probabilities are in general not independent. (Unless the pairing network is a linear chain.) With N players there are N*(N-1)/2 win, draw and loss probabilities. (But of course W+D+L=1 for each player.) And they follow from only N parameters (their ratings), of which one can actually be eliminated because only rating differences enter the probability equation. (So an extra condition is needed to uniquely specify the ratings, e.g. that their average is zero.)

The rating model thus implies a dependency between the probabilities of the individual pairings. Even if you forget about draws, the allowed combinations of ratings form an (N-1)-dimensional manifold in the N*(N-1)/2-dimensional space of the probabilities. As for most rating models this manifold is not a (hyper-)plane, but curved, it is quite difficult to decide what you would consider an 'unbiased' prior (i.e. every point of the manifold assumed equally likely). I did not even manage solving this for a linearized rating model.