Sprague–Grundy function of symmetric hypergraphs

Abstract

We consider a generalization of the classical game of Nim called hypergraph Nim. Given a hypergraph $\mathcal{H}$ on the ground set $V=\{1,\dots ,n\}$ of n piles of stones, two players alternate in choosing a hyperedge $H\in \mathcal{H}$ and strictly decreasing all piles $i\in H$. The player who makes the last move is the winner. In 1980 Jenkyns and Mayberry obtained an explicit formula for the Sprague–Grundy function of the hypergraph Nim whose hypergraph contains as the hyperedges all proper subsets of V (that is, all except ∅ and V itself). Somewhat surprisingly, the same formula works for a very wide family of hypergraphs. In this paper we characterize symmetric hypergraphs in this family.