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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?

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Let me answer your second question, why they switched to the logistic function. The Elo rating system comes from the Chess world. The initial Elo's premise was a normal distribution, but since more chess statistics became available, FIDE (The World Chess Federation) realized that it was better to consider the logistic function. Have a look here for a description of Elo's rating system.

Regarding your first question, if your formula will work, well, c is the logistic parameter used to fine-tune your rating system. The value c = 400 is for the chess world. But in reality, you can pick any which fits your rating requirements. The same happens with the Distribution you pick. First, maybe you will have to draw your players' initial ratings and see which distribution they follow.

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  • $\begingroup$ My intent is to ask why they had to change the assumption for the underlying distribution from a normal distribution to a logistic distribution. If my formula would still work, then shouldn't the underlying distribution be irrelevant to the Elo ranking system? If my formula does not work, then the explanation of why it doesn't work would explain the importance of updating the assumption from normal distribution to logistic distribution. $\endgroup$ – Victor Oct 28 '20 at 2:51

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