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We consider a generalization of the classical game of $NIM$ called hypergraph$NIM$. Given a hypergraph $\cH$ on the ground set $V = \{1, \ldots, n\}$ of $n$piles of stones, two players alternate in choosing a hyperedge $H \in \cH$ andstrictly decreasing all piles $i\in H$. The player who makes the last move isthe winner. In this paper we give an explicit formula that describes theSprague-Grundy function of hypergraph $NIM$ for several classes of hypergraphs.In particular we characterize all $2$-uniform hypergraphs (that is graphs) andall matroids for which the formula works. We show that all self-dual matroidsare included in this class.

Sprague-Grundy Function of Matroids and Related Hypergraphs