A note on the growth of the dimension in complete simple games

TL;DR

This paper investigates the exponential growth of the dimension and Boolean dimension in complete simple games, including those with two voter types, addressing previously open questions about their complexity.

Contribution

It proves that the dimension and Boolean dimension can be exponential in complete simple games, even with restricted voter types, resolving open problems in the field.

Findings

Dimension can be exponential in complete simple games.

Boolean dimension can also be exponential in general complete simple games.

The results apply even to games with two voter types.

Abstract

The remoteness from a simple game to a weighted game can be measured by the concept of the dimension or the more general Boolean dimension. It is known that both notions can be exponential in the number of voters. For complete simple games it was only recently shown that the dimension can also be exponential. Here we show that this is also the case for complete simple games with two types of voters and for the Boolean dimension of general complete simple games, which was posed as an open problem.