Simple games with minimum
Sascha Kurz, Dani Samaniego
TL;DR
This work addresses counting non-isomorphic simple games with a single minimal winning vector within the broader context of Dedekind's problem for monotone Boolean functions. The authors develop an algebraic parameterization using a size vector $n ext{_bar}$ and a minimal-winning matrix $M$, and apply generating functions together with Polya's enumeration to factor out isomorphic copies. They derive explicit formulas and recurrences for $SG^{neg v, neg n}(n,t,1)$ and its extensions to $SG(n,t,1)$, and determine the dimension of these games in terms of the number of equivalence classes $t$ and the presence of null or veto players, supported by concrete examples (e.g., $n=9$, $t=3$ yielding 14 cases without null/veto). The results provide a precise enumeration framework for simple games with minimum and offer a pathway to extensions to higher minimal-winning-vector counts and related voting structures.
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.