Problem Setup: Consider the intervals $I_R = [a_R, b_R]$ and $I_B = [a_B,b_B]$. Let $F_R$ and $F_B$ denote the CDF of distributions with support on the intervals $I_R$ and $I_B$. I draw $k$ red and blue points from $F_R$ and $F_B$ respectively. Let $\mathscr{R}$ and $\mathscr{B}$ denote the set of these $k$ red and blue points respectively. I consider the following matching procedure: I randomly pick a blue point in the set $\mathscr{B}$ and match it to the closest red point in set $\mathscr{R}$ where the distance is measured using the function $c(r,b) = |r-b|^\alpha$ for $\alpha > 1$. After I match, I remove the blue point and its matched red point from set $\mathscr{B}$ and $\mathscr{R}$ respectively. I continue with this procedure till I have emptied the sets $\mathscr{R}$ and $\mathscr{B}$.

Let us denote the average matching cost (averaged across $k$ points) be to $\mathsf{RND}(k)$. In the case where $k \to \infty$, the optimal matching cost is the optimal transport value between $F_R$ and $F_B$ with the cost function $c(r,b)$. Let us denote that optimal transport value as $\mathsf{OPT}$.

If it helps, one may assume for simplicity that both $F_R$ and $F_B$ are continuous distributions on the interval $I_R$ and $I_B$ respectively with densities bounded above and below.

Questions: I want to know under what conditions does $\lim_{k \to \infty} |\mathsf{RND}(k) - \mathsf{OPT}| = 0$ and $\lim_{k \to \infty} |\mathsf{RND}(k) - \mathsf{OPT}| \neq 0$? More specifically, I am looking for conditions on $F_R$ and $F_B$.

1. I believe that if $F_R = F_B$, then it might be the case that $\lim_{k \to \infty} |\mathsf{RND}(k) - \mathsf{OPT}| = 0$? While I intuitively think that this is the case, I am unable to find a proof for this.
2. If the supports $I_R$ and $I_B$ are disjoint, then $\lim_{k \to \infty} |\mathsf{RND}(k) - \mathsf{OPT}| \neq 0$? I am unable to prove this for general $F_R$ and $F_B$, but my intuition comes from the case when $F_R = \text{Unif}(I_R)$ and $F_B = \text{Unif}(I_B)$. In this case I think it can be easily argued that $\lim_{k \to \infty} |\mathsf{RND}(k) - \mathsf{OPT}| \neq 0$?
3. Extrapolating the previous point further, can we show that $\lim_{k \to \infty} |\mathsf{RND}(k) - \mathsf{OPT}| \neq 0$ under the following condition: We have that $F_R(x) \geq F_B(x)$ for all $x \in (-\infty, \infty)$ and there exists an interval $I^\prime$ with positive measure such that $F_R(x) > F_B(x)$ for $x \in I^\prime$. I think this condition subsumes the case when the intervals $I_R$ and $I_B$ are disjoint.
4. (Perhaps Ill Defined) Is there a more general set of assumptions on $F_R$ and $F_B$ such as $F_R \neq F_B$ which are sufficient to show that $\lim_{k \to \infty} |\mathsf{RND}(k) - \mathsf{OPT}| \neq 0$?
5. If there a good reference which shows that the optimal transport map (outside of a zero measure set) does an assortative match, i.e., point $r$ gets matched to $F_B^{-1}(F_R(r))$ ?

Any help or pointers with this or related questions/papers would be greatly appreciated. Thanks :)