# Does the Elo system work with different expected value functions?

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?