International Journal of Game Theory

, Volume 42, Issue 2, pp 411–437

# On Dedekind’s problem for complete simple games

Article

## Abstract

We state an integer linear programming formulation for the unique characterization of complete simple games, i.e. a special subclass of monotone Boolean functions. In order to apply the parametric Barvinok algorithm to obtain enumeration formulas for these discrete objects we provide a tailored decomposition of the integer programming formulation into a finite list of suitably chosen sub-cases. As for the original enumeration problem of Dedekind on Boolean functions we have to introduce some parameters to be able to derive exact formulas for small parameters. 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 a new enumeration formula for complete simple games with two minimal winning vectors.

## Keywords

Boolean functions Dedekind’s problem Voting theory Complete simple games Application of the parametric Barvinok algorithm

## Mathematics Subject Classification

05A15* 91B12 94C10 52B20

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