  We prove that the difference between the paint number and the choice number
of a complete bipartite graph $K_{N,N}$ is $\Theta(\log \log N )$. That answers
the question of Zhu (2009) whether this difference, for all graphs, can be
bounded by a common constant. By a classical correspondence, our result
translates to the framework of on-line coloring of uniform hypergraphs. This
way we obtain that for every on-line two coloring algorithm there exists a
k-uniform hypergraph with $\Theta(2^k )$ edges on which the strategy fails. The
results are derived through an analysis of a natural family of chip games.
