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I'm trying to implement a variant of the ELO system, for a game I'm working on. Giving two players $A$ and $B$ with ratings $R_A$ and $R_B$ respectively, the expectation of $A; E_A$ is given by the formula: $$E_A = \frac{1}{1+10^{\frac{R_B-R_A}{Y}}}\tag{*}$$

The Expectation of $B$ is similarly given by: $$E_B = \frac{1}{1+10^{\frac{R_A-R_B}{Y}}}$$

The different possible game outcomes are given scores: A win is $1.0$, a loss is: $0.0$, and a draw is $0.5$. The actual score of $A$ is $S_A$.

After a match between $A$ and $B$, the new ranking of $A; R'_A$ is given by: $$R'_A = R_A + K(S_A - E_A)$$

Question: For any value of $Y$, what value of $K$ should I choose, such that that value is a constant and makes the player's rating as reliable as possible.

I'm currently using $K = \sqrt{Y}$

$(*)$ Most Chess ELO algorithms use a value of $Y = 400$

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  • $\begingroup$ The $*$ was meant to provide information. -_- $\endgroup$ – Tobi Alafin Dec 17 '16 at 17:30
  • $\begingroup$ Look at the bottom of the question. $\endgroup$ – Tobi Alafin Dec 17 '16 at 17:34
  • $\begingroup$ Suggest an edit. $\endgroup$ – Tobi Alafin Dec 17 '16 at 17:38
  • $\begingroup$ As for the question: Wiki has a discussion on the $K$ value problem: en.wikipedia.org/wiki/Elo_rating_system#Mathematical_issues (though only for $Y=400$) $\endgroup$ – Winther Dec 17 '16 at 17:41
  • $\begingroup$ I read it. $K$ values were seemingly chosen arbitrarily without a mathematical explanation. Also most organisations used different $K$ values for different ranges. Furthermore, due to the pecularities of my game system, I don't use $400$. $\endgroup$ – Tobi Alafin Dec 17 '16 at 17:44

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