Sprague-Grundy Function of Matroids and Related Hypergraphs
Endre Boros, Vladimir Gurvich, Nhan Bao Ho, Kazuhisa Makino, Peter, Mursic

TL;DR
This paper extends the classical game of NIM to hypergraph NIM, providing explicit formulas for the Sprague-Grundy function for certain hypergraph classes, including all 2-uniform hypergraphs and matroids, especially self-dual ones.
Contribution
It introduces explicit formulas for the Sprague-Grundy function in hypergraph NIM, characterizing all applicable 2-uniform hypergraphs and matroids, notably including self-dual matroids.
Findings
Explicit formulas for hypergraph NIM's Sprague-Grundy function.
Characterization of 2-uniform hypergraphs and matroids where formulas apply.
Inclusion of all self-dual matroids in the characterized class.
Abstract
We consider a generalization of the classical game of called hypergraph . Given a hypergraph on the ground set of piles of stones, two players alternate in choosing a hyperedge and strictly decreasing all piles . 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 for several classes of hypergraphs. In particular we characterize all -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.
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