A closed two-player Elo system depends on constants $k$ and $c$ in the following way. The two players begin with Elo rating of 0, and each match is zero-sum (w.r.t. Elo). If they enter with ratings of $x$ and $-x$, then the first player is assigned probability $p=p(x)=\frac{1}{1+e^{-2cx}}$ and the second $q=q(x)=\frac{1}{1+e^{2cx}}$; the first gains $k(X-p)$ points while the second gains $k((1-X)-q)$, where $X$ is the outcome of the match (1 for a win and 0 for a loss, and 0.5 for a tie in games allowing ties). (e.g. FIDE uses $c=\frac{\log10}{400}$ and $k$ on the order of dozens.)
Say $k=1$ (I suspect it will not matter too much for the theorem I am wondering about) and fix $c$ to be a constant. Suppose two players play $N$ matches of a win/loss game, so that $X$ in the $i$th match is the $i$th entry of some bit sequence in $\{0,1\}^N$. Suppose further we know that there are exactly $n$ 1's in the bit sequence. In terms of $N$ and maybe $n$, what is the possible range of resulting ratings (for the first player)?
Intuitively, this should "converge" in some sense to $x=x_{N,n}$ such that the resulting $p(x)$ equals $\frac{n}{N}$. Explicitly, this is $\frac{1}{1+e^{-2cx}}=\frac{n}{N}$, so $x=-\frac{1}{2c}\log\left(\frac{N}{n}-1\right)$. So, it would make sense to say that any $x$ resulting from a length-$N$ bit sequence with $n$ 1's will lie in an interval of the form $\left(x-\Delta_N,x+\Delta_N\right)$ where $\Delta_N\in o(N)$ (or even $O(1)$). Is this known to be true?
The only result I have in this ballpark is that if the first player always wins, then their Elo grows without bound, but the proof does not use anything special about the Elo formula. (I can reproduce the proof but it does not seem to be particularly relevant to the finite-limit case.)