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 $Y = 400$