Sprague-Grundy Function of Matroids and Related Hypergraphs

Abstract

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