Question: In many Elo rating systems, the expected value for player A with rating $R\_A$ against an opponent with rating $R\_B$ is

$$E\_A = \frac1{1+10^{(R\_B - R\_A)/c}}$$

The original Elo system is based on a normal distribution instead of a logistic distribution, so it used the error function instead of the logistic function in the equation for $E\_A$.

Would the Elo system still work if we chose a simpler function like this?

$$E\_A = \frac{R\_A}{R\_A + R\_B} = \frac1{1 + \frac{R\_B}{R\_A}}$$

For example, if c=400, and 1000 rating players in the logistic Elo system still have a 1000 rating in this system, then 1400 rating players in the logistic Elo system would have a rating of 10000 in this system.

If this still works, then why would they have switched to the logistic function? Is it that they wanted player ratings to follow a normal distribution, and ratings in this simpler version would follow a log-normal distribution?

Answer: 
This subject is discussed in Prof. Mark E. Glickman, [A Comprehensive Guide To Chess Ratings](https://www.glicko.net/research/acjpaper.pdf).

> 
> 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).
> 
> The original Elo system is based on a normal distribution instead of a logistic distribution
> 
> 
> ...
> 
> 
> If this still works, then why would they have switched to the logistic function?
> 
> 
> 

There hasn't been a switch. The FIDE uses Elo's original system for which a table is computed. See the [Fide handbook](https://handbook.fide.com/chapter/B022017).

The win-probability (or score,since it includes draws), is computed as

$$p(\text{A win from B})= \Phi\left( \frac{\text{Elo}\_A - \text{Elo}\_B}{200 \cdot \sqrt{2}} \right)$$

For example, a $100$ Elo difference will relate to a $0.6381632$ win probability for the higher valued player. Or $0.64$ as it is in the tables of the FIDE.

The use of the logistic function/distribution is a useful trick to make an estimate of the values in the table. Also, several other competitions may be using the logistic function. There is not really a large difference between the two and it is just a practical trick to make computations easy.

> 
> Would the Elo system still work if we chose a simpler function like this?
> 
> 
> $$E\_A = \frac{R\_A}{R\_A + R\_B} = \frac1{1 + \frac{R\_B}{R\_A}}$$
> 
> 
> 

The system is arbitrary, so yes this would work as well. However, the computations with a linear Elo scale might be easier.

* For instance, one could compute an average Elo rating for a pool of chess players and approximated the win probability of a player based on the difference with the average Elo.
* Another example is that differences in Elo rating are more easier to
evaluate. If some top player has rating 13102 and another has a
rating 4996, than it is a bit awkward to compute the difference in
the level between the players because this involves a division
instead of a subtraction.
* Also, the updating works as a zero sum game. Whenever a player increases by some value $x$, then another player decreases by a value $x$. If you would do this for an exponentially increasing score, then the people at the top will be changing their scores only very slowly (in some sense the system does have a way to reduce the speed of changes in scores for high level players, but it does not reduce the speed *that* much as what would be the case with an exponentially increasing score)