  Consider the following game on a graph $G$: Alice and Bob take turns coloring
the vertices of $G$ properly from a fixed set of colors; Alice wins when the
entire graph has been colored, while Bob wins when some uncolored vertices have
been left. The game chromatic number of $G$ is the minimum number of colors
that allows Alice to win the game. The game Grundy number of $G$ is defined
similarly except that the players color the vertices according to the first-fit
rule and they only decide on the order in which it is applied. The $(a,b)$-game
chromatic and Grundy numbers are defined likewise except that Alice colors $a$
vertices and Bob colors $b$ vertices in each round. We study the behavior of
these parameters for incomparability graphs of posets with bounded width. We
conjecture a complete characterization of the pairs $(a,b)$ for which the
$(a,b)$-game chromatic and Grundy numbers are bounded in terms of the width of
the poset; we prove that it gives a necessary condition and provide some
evidence for its sufficiency. We also show that the game chromatic number is
not bounded in terms of the Grundy number, which answers a question of Havet
and Zhu.
