Simple games with minimum

TL;DR

This paper investigates simple games with a unique minimal winning vector, providing a closed formula for their count and analyzing their dimension relative to players and symmetries.

Contribution

It introduces a closed formula for counting simple games with minimum and examines their structural properties based on players and symmetries.

Findings

Derived a closed formula for the number of simple games with minimum.

Analyzed the dimension of these games in relation to players and equivalence classes.

Connected the enumeration to Dedekind's problem for monotone Boolean functions.

Abstract

Every simple game is a monotone Boolean function. For the other direction we just have to exclude the two constant functions. The enumeration of monotone Boolean functions with distinguishable variables is also known as the Dedekind's problem. The corresponding number for nine variables was determined just recently by two disjoint research groups. Considering permutations of the variables as symmetries we can also speak about non-equivalent monotone Boolean functions (or simple games). Here we consider simple games with minimum, i.e., simple games with a unique minimal winning vector. A closed formula for the number of such games is found as well as its dimension in terms of the number of players and equivalence classes of players.