  We study the Maker-Breaker domination game played by Dominator and Staller on
the vertex set of a given graph. Dominator wins when the vertices he has
claimed form a dominating set of the graph. Staller wins if she makes it
impossible for Dominator to win, or equivalently, she is able to claim some
vertex and all its neighbours. Maker-Breaker domination number $\gamma_{MB}(G)$
($\gamma '_{MB}(G)$) of a graph $G$ is defined to be the minimum number of
moves for Dominator to guarantee his winning when he plays first (second). We
investigate these two invariants for the Cartesian product of any two graphs.
We obtain upper bounds for the Maker-Breaker domination number of the Cartesian
product of two arbitrary graphs. Also, we give upper bounds for the
Maker-Breaker domination number of the Cartesian product of the complete graph
with two vertices and an arbitrary graph. Most importantly, we prove that
$\gamma'_{MB}(P_2\square P_n)=n$ for $n\geq 1$, $\gamma_{MB}(P_2\square P_n)$
equals $n$, $n-1$, $n-2$, for $1\leq n\leq 4$, $5\leq n\leq 12$, and $n\geq
13$, respectively. For the disjoint union of $P_2\square P_n$s, we show that
$\gamma_{MB}'(\dot\cup_{i=1}^k(P_2\square P_n)_i)=k\cdot n$ ($n\geq 1$), and
that $\gamma_{MB}(\dot\cup_{i=1}^k(P_2\square P_n)_i)$ equals $k\cdot n$,
$k\cdot n-1$, $k\cdot n-2$ for $1\leq n\leq 4$, $5\leq n\leq 12$, and $n\geq
13$, respectively.
