Let $w_j, v_j \geq 0$ satisfying $\sum_{j=1}^{k} w_j = \sum_{j=1}^{k} v_j = 1$. Are we able to find an upper bound for $$ \min _{\sigma \in S(k)}\left\{\sum_{j=1}^{k}\left|w_{\sigma(j)}-v_{j}\right|\right\}, $$ where $S(k)$ denotes the set of all permutations?
My guess is that an upper bound is 2 and my intuition would be as follows (not rigorous):
Since we are taking the minimum of all permutations, then if there is a maxima, we must have that $w_1 = \cdots = w_k = 1/k$. In this case $v_1 = 1, v_2=\cdots=v_k=0$ would yield the maximum of the expression, which is $2-2/k$.