One reason for preference of the logistic over the normal distribution has to do with the relative difficulty of integrating them. Match predictions are given by the cumulative distribution conditional on the ELO difference:

For the normal distribution, no analytical expression is available and this requires numerical integration or extensive tables. For the logistic distribution, an analytical expression can be obtained (see Wikipedia).

The converse, ELO inference given the match results, requires Bayesian statistics

Code: Select all

` P(ELO|result) = P(result|ELO) x P(ELO) / P(result)`

where P(ELO) is the so-called conjugate distribution of playing strength across the population. Here, the normal distribution is self-conjugate: if playing strength is normally distributed, then match results are also normally distributed (with sqrt(2) times the standard deviation of playing strengths). For the logistic distribution, the conjugate prior is a complicated Dirichlet distribution. So which distribution is most convenient depends your application.

In practice, it doesn't matter much. You can rescale the logistic distribution in two ways to make it more similar to the normal distribution. First, you can rescale the logistic's scale parameter by Sqrt(3)/Pi (=.55) so that it has the same standard deviation as the normal distribution. You can also rescale the logistic's scale parameter by 1/2 Sqrt(Pi/2) (=.63) so that it has the same slope as the normal distribution for delta-ELO=0. In both cases, the relative differences in the cumulative distribution are always less then -1% and +3% (percent, not percentage points!).