# What are the Elo formulas when assuming performance to be logistically distributed?

I'm struggling to find a definite source on the Elo rating system but from what I understand:

• it originally assumes players performance to be normally distributed with $$\sigma = 1$$
• the expected score of player A is $$E_A = \frac1{1+10^{(R_B - R_A)/c}}$$
• the update rule is $$R_A = R_A + K (\text{actual score} - E_A)$$

I also read that the chess people use the logistic distribution instead of the normal distribution in the assumption of how performance is distributed. But I can't find any second set of formulas for when a logistic distribution is assumed. (When the formulas stay the same there shouldn't be a need for a different distribution because nothing would change.)

Here is one model giving rise to the Elo formula in which the difference in the random components of performance is logistically distributed.

Suppose player a's skill is $$R_a$$

Player a's performance is $$\alpha_a = R_a + \epsilon_a$$ , which we can interpret as performance being the sum of skill and other factors summarised by $$\epsilon_a$$.

If he plays against b, he'll win if he performs better, $$\alpha_a - \alpha_b > 0$$, which is equivalent to $$\epsilon_a - \epsilon_b > R_b -R_a$$

Assume that the random components are independent and distributed according to the Gumbel distribution $$\text{Gumbel}(\mu, \beta)$$ and $$\text{Gumbel}(\mu, \beta)$$. Then $$u = \epsilon_a - \epsilon_b$$ follows a mean-zero logistic distribution $$\text{Logistic}(0, \beta)$$ with scale parameter $$\beta$$ and

$$\Pr(\text{a wins}) = \Pr(u > R_b -R_a) = \Pr(u < - (R_b - R_a)) = \frac{1}{1 + \exp\left( \frac{R_b - R_a}{\beta}\right)}$$

Which is the same as the Elo function you've written down for $$\beta = \frac{c}{\log 10}$$