  We introduce and study a Maker-Breaker type game in which the issue is to
create or avoid two disjoint dominating sets in graphs without isolated
vertices. We prove that the maker has a winning strategy on all connected
graphs if the game is started by the breaker. This implies the same in the
$(2:1)$ biased game also in the maker-start game. It remains open to
characterize the maker-win graphs in the maker-start non-biased game, and to
analyze the $(a:b)$ biased game for $(a:b)\neq (2:1)$. For a more restricted
variant of the non-biased game we prove that the maker can win on every graph
without isolated vertices.
