# Name for a certain “product game”

Let $G,H$ be two (combinatorial impartial) games. Consider the following new game $P$: The positions are the pairs of positions of $G$ and $H$. A move in $P$ is a move in $G$, or a move in $H$, or a move in both. If we end in a terminal position of $G$, we go on and play in $H$. Similarly with $H$ and $G$.

What is the name of this "product game"? Has it been studied somewhere? In particular, what are the $\mathcal{P}$-positions of $P$ in terms of the ones for $G$ and $H$?

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In Winning Ways, this is called the "selective compound" of $G$ and $H$.