Let's say we have a set $A=\{a_1,a_2, ..., a_{2n}\}$ of $2n$ numbers, $n$ of those are random variables that are $N(-1, \sigma^2)$ and the other are $N(1,\sigma^2)$. We define the 2 means of $A$ as the 2 numbers that minimizes the sum of squared differences to the numbers of A such that each number is mapped only to the closest mean, i.e.: $$\min_{m_1,m_2\in\mathbb{R}}{\sum_{a_i\in A}{\min\left\{(a_i-m_1)^2,(a_i-m_2)^2\right\}}}$$
What is the expected value of those 2 means ($m_1$ and $m_2$)?
If $\sigma$ is small enough it should be very close to $-1$ and $1$, since most variables would be mapped to the correct mean, however as $\sigma$ get's larger more variables would be mapped to the second mean (i.e, points from $N(1,\sigma)$ would be closer to $-1$) so we expect $m_1,m_2$ to get closer to each other, but how can we calculate exactly how much closer using $\sigma$?