This subject is discussed in Prof. Mark E. Glickman, A Comprehensive Guide To Chess Ratings.
It appears as though there is very little distinction between the shape of the logistic distribution in Figure 2 and the normal distribution in Figure 4. Figure 5 shows both curves superimposed, with the logistic distribution drawn as a solid line. In fact, statistics professor Hal Stern in a 1992 article[11] has shown that when analyzing paired comparison data it makes virtually no difference whether one assumes the logistic distribution or the normal distribution for differences in players' strengths. So, empirically, the choice between the Bradley-Terry model and the Thurstone-Mosteller model is a moot issue. Mathematically, however, the Bradley-Terry model tends to be more tractable to work with. This is the most likely reason that most organizations administering a probabilistic rating system (e.g., FIDE, USCF) use the Bradley-Terry model, which uses the logistic distribution assumption, rather than the Thurstone-Mosteller model, which uses the normal distribution assumption.
It is easy to see from the "FIDE Rating Regulations 2022" that the original Elo tables are still in use by FIDE.
The following distributions are discussed by Elo in The Rating of Chess Players.
- 8.4 Logistic Probability As a Rating Basis
- 8.5 Rectangular Distribution as a Ratings Basis
- 8.8 Binomial Distribution and Small Examples
- 9.1 The Maxwell-Bolzmann Distribution and Chess Ratings
Since the Elo system is self-correcting using the K-factor formula, any reasonable form of the probability distribution function can be used as a starting point for a rating system (ch. 8.75).