Walking in circles

Consider the unit circle S1S1 with distance function dd measured along the circle. We show that for every selection of 2n2n points x1,…,xn,y1,…,yn∈S1x1,…,xn,y1,…,yn∈S1 there exists i∈{1,…,n}i∈{1,…,n} such that ∑k=1nd(xi,xk)≤∑k=1nd(xi,yk). We also discuss a game theoretic interpretation of this result.