Protecting billiard balls from collisions

We present a game inspired by research on the possible number of billiard ball collisions in the whole Euclidean space. One player tries to place $n$ static "balls" with zero radius (i.e., points) in a way that will minimize the total number of possible collisions caused by the cue ball. The other player tries to find initial conditions for the cue ball to maximize the number of collisions. The value of the game is $\sqrt{n}$ (up to constants). The lower bound is based on the Erd\H{o}s-Szekeres Theorem. The upper bound may be considered a generalization of the Erd\H{o}s-Szekeres Theorem.