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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 ₌₁ (G) ('₌₁ (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 '₌₁ (P₂ Pₙ) =n for n 1, ₌₁ (P₂ Pₙ) equals n, n-1, n-2, for 1 n 4, 5 n 12, and n 13, respectively. For the disjoint union of P₂ Pₙs, we show that ₌₁' (₈=₁ᵏ (P₂ Pₙ) ᵢ) =k n (n 1), and that ₌₁ (₈=₁ᵏ (P₂ Pₙ) ᵢ) equals k n, k n-1, k n-2 for 1 n 4, 5 n 12, and n 13, respectively.
Forcan et al. (Fri,) studied this question.