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I know how to calculate elo rating (in chess) and etc but why does it uses sigmoid? which makes chance of win/lose grow slower and slower if difference in elo is bigger (for example 0 difference gives 0.5, 100 difference 0.64, 700 difference 0.98 and 800 difference 0.99). What is the point of making chances grow up slower by increasing difference?

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    $\begingroup$ If the chance of win didn't grow slower, for a large difference in rating you'd have more than 1.00 chance of win, which doesn't make any sense. $\endgroup$ – Rahul Jul 8 '18 at 18:46
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The win probability has to increase more slowly as the rating difference increases, because it can never exceed $1$. In your example, if a difference of $100$ in rating means a win probability of $0.64$, surely a difference of $400$ in rating cannot mean a win probability of $1.06$! But that's what would happen if the rate of increase remained constant.

The exact distribution chosen is more an empirical question than a theoretical one. If the win probability for a difference of $100$ in rating is $0.64$, then the win probability for a difference of $200$ in rating answers the question, "If Alice beats Bob 64% of the time and Bob beats Charlie 64% of the time, how frequently does Alice beat Charlie?" But there's no theoretical reason why that must be any number in particular (except that it's probably more than 64% and certainly no more than 100%) unless you have a really good model of how chess skill works, and I don't think we do.

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Suppose we were modeling a sumo robot competition where the probability of failure of robot $A$ between times $t$ and $t+dt$, given that it functions up to time $t$ is independent of time: $$P\left(\bar A(t+dt)|A(t)\right)=\lambda_Adt$$ Similarly for robot $B$ $$P\left(\bar B(t+dt)|B(t)\right)=\lambda_Bdt$$ If one robot fails, the other pushes it out of the ring and wins. Of course, real sumo robot competition can't be so simply modeled :) But given such a model the ratio of points scored by $A$ to those scored by $B$ between $t$ and $t+dt$ is $$r_{A/B}=\frac{\lambda_Bdt}{\lambda_Adt}=\frac{\lambda_B}{\lambda_A}$$ Since this ratio is valid for all time intervals it will be the ratio for the entire struggle. We can more easily grasp differences than ratios, so we set up a logarithmic scale $$\ln r_{A/B}=\ln\lambda_B-\ln\lambda_A$$ $$\ln\lambda_A=m\left(R_0-R_A\right)$$ Now to find the mathematical expectation of $A$'s score in a bout with $B$, we get $$p_A=\frac{\text{points}_A}{\text{points}_A+\text{points}_B}=\frac1{1+\frac{\text{points}_B}{\text{points}_A}}=\frac1{1+\frac1{r_{A/B}}}=\frac1{1+\frac{\lambda_A}{\lambda_B}}=\frac1{1+e^{m\left(R_B-R_A\right)}}$$ The scale factor $m$ for ratings is arbitrary; Elo chose $$m=\frac{\ln10}{400}$$ With the reference rating $R_0$ set at some value such that hopefully nobody gets negative ratings. This is sort of the way IQ scores are curved so that the mean is $100$ and the standard deviation is $15$ points, or more exactly like loudness in decibels.

It might seem odd that we got to Elo ratings by focusing on your failure rate rather than how brilliant you were, but I remember looking at the code for Crafty back in the day when it was getting good enough to beat grandmasters at speed chess and it really didn't have any more strategic insight built into it than one might get out of reading My System by Aron Nimzovich and the programmer wasn't any stronger than an $A$ player, but it could calculate enough positions that it made few obvious tactical mistakes. Put another way, it vindicated Richard Teichmann's estimate that chess is $90\text{%}$ tactics as perhaps even an underestimate.

Of course a cynic might remark that Elo was a physicist and so set $A$'s mean score to be Fermi-Dirac distributed with energies $\epsilon_B=R_B$, Fermi energy $\epsilon_F=R_A$, and $kT=400/\ln10$.

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