A note on simple games with two equivalence classes of players

TL;DR

This paper provides structural characterizations and enumeration formulas for bipartite simple games, which model voting systems with two types of voters such as houses and senates.

Contribution

It introduces new structural insights and explicit formulas for bipartite simple games, enhancing understanding of voting systems with two voter types.

Findings

Structural characterizations of bipartite simple games

Explicit enumeration formulas for these games

Application to real-world voting systems

Abstract

Many real-world voting systems consist of voters that occur in just two different types. Indeed, each voting system with a {\lq\lq}House{\rq\rq} and a {\lq\lq}Senat{\rq\rq} is of that type. Here we present structural characterizations and explicit enumeration formulas for these so-called bipartite simple games.