  We consider an online vector balancing game where vectors $v_t$, chosen
uniformly at random in $\{-1,+1\}^n$, arrive over time and a sign $x_t \in
\{-1,+1\}$ must be picked immediately upon the arrival of $v_t$. The goal is to
minimize the $L^\infty$ norm of the signed sum $\sum_t x_t v_t$. We give an
online strategy for picking the signs $x_t$ that has value $O(n^{1/2})$ with
high probability. Up to constants, this is the best possible even when the
vectors are given in advance.
