  A coloring of the ground set of a matroid is proper if elements of the same
color form an independent set. For a loopless matroid $M$, its chromatic number
$\chi (M)$ is the minimum number of colors in a proper coloring. In this note
we study a game-theoretic variant of this parameter. Suppose that Alice and Bob
alternately properly color the ground set of a matroid $M$ using a fixed set of
colors. The game ends when the whole matroid has been colored, or if they
arrive to a partial coloring that cannot be further properly extended. Alice
wins in the first case, while Bob in the second. The game chromatic number of
$M$, denoted by $\chi_{g}(M)$, is the minimum size of the set of colors for
which Alice has a winning strategy. Clearly, $\chi_{g}(M)\geq\chi (M)$. We
prove an upper bound $\chi_{g}(M)\leq 2\chi (M)$ for every matroid $M$. This
improves and extends a result of Bartnicki, Grytczuk and Kierstead, who showed
that $\chi_{g}(M)\leq 3\chi (M)$ holds for graphic matroids. Our bound is
almost tight, as we construct a family of matroids $M_k$ (for $k\geq 3$)
satisfying $\chi (M_k)=k$ and $\chi_{g}(M_k)=2k-1$.
