Suppose that $X \sim \mathcal{N}(r + \frac{1}{N}, s) $ and $Y \sim \mathcal{N}(r, s)$ for some $r, s \approx 1$ and $N \approx 10^6$.
What are good approximate formulas for the quantity $$\frac{ \mathbb{E}(X 1_{X > n})}{\mathbb{E}(Y1_{Y > n}) + \frac{1}{N}}$$ as a function of $r$ in the limit where $r \approx n$?
The problem arises as one potential answer here: Assessing the efficiency of a single vote in a multiparty presidential election
You can express the expectation values of normal distributions, with mean $\mu$ and standard deviation $\sigma$ truncated at 1 as
$$E[y_{\text{truncated at } 1}] = \mu + \sigma \frac{\phi\left(\frac{1-\mu}{\sigma}\right) }{1-\Phi\left(\frac{1-\mu}{\sigma}\right)} \approx \mu + \sigma \frac{\phi\left(0\right) }{1-\Phi\left(0\right)} $$
where the approximation comes from $\mu, \sigma \approx 1$.
In the case of $\mu + 1/N$
$$E[x_{\text{truncated at } 1}] \approx \mu + 1/N + \sigma \frac{\phi\left(-\frac{1}{\sigma N}\right) }{1-\Phi\left(-\frac{1}{\sigma N}\right)}$$
this we may approximated by a linear approximation of ${\phi\left(x\right) }/(1-\Phi\left(x\right))$ around $x=0$, then we get
$$E[x_{\text{truncated at } 1}] \approx \mu + \frac{1}{N} + \sigma \frac{\phi\left(0\right)}{1-\Phi\left(0\right)} - \frac{1}{N} \left(\frac{\phi\left(0\right)}{\left(1-\Phi\left(0\right)\right)}\right)^2 \approx E[y_{\text{truncated at } 1}] + \frac{1}{N} \left( 1 - \frac{2}{\pi} \right)$$