On Dedekind's problem for complete simple games

TL;DR

This paper presents a new enumeration formula for complete simple games with two shift-minimal winning coalitions, using a combination of the parametric Barvinok algorithm and a generation approach, simplifying previous proofs.

Contribution

It provides a shorter proof and an explicit enumeration formula for a specific subclass of complete simple games, extending recent results.

Findings

New enumeration formula for complete simple games with two shift-minimal winning coalitions

Simplified proof of existing enumeration results

Application of the parametric Barvinok algorithm to game enumeration

Abstract

We combine the parametric Barvinok algorithm with a generation algorithm for a finite list of suitably chosen discrete sub-cases on the enumeration of complete simple games, i.e. a special subclass of monotone Boolean functions. Recently, Freixas et al. have proven an enumeration formula for complete simple games with two types of voters. We will provide a shorter proof and an enumeration formula for complete simple games with two shift-minimal winning coalitions.