simple_model_notes.txt

A logistic function is extremely good for predicting winrate given
province-count difference (when supply is fixed).
A linear fit, on the other hand, is quite bad.

The dependence on provinces-in-supply is roughly like you'd expect:
the fewer provinces remaining, the more ahead the player who has more.
Also, at equal count, it gives an advantage to the current
player. By eye, it looks roughly like an inverse relationship.

Since the logistic model is fantastic for fixed supply, it makes sense
to see how its parameters vary as the supply varies. They seem roughly
inverse to the remaining supply, which is nice. (Maybe remaining supply + 1?)

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In fact, the plotted function is something very similar to the cumulative
distribution function of a binomial distribution, which itself is
approximated by the cumulative distribution function of a normal.
This function has a sigmoid shape, so it makes sense that a logistic
function could give a reasonably close (but not quite right) fit.
This also supports the roughly inverse relationship of the parameters with
the number of remaining provinces.