Growth of Dimension in Complete Simple Games

TL;DR

This paper proves that the dimension of complete simple games can grow exponentially with the number of players, resolving an open problem about the worst-case growth rate.

Contribution

It introduces a novel technique to establish that the worst-case dimension growth in complete simple games is exponential.

Findings

Dimension can grow exponentially with players

Complete simple games can have arbitrarily high dimension

New technique demonstrates exponential growth

Abstract

The concept of dimension in simple games was introduced as a measure of the remoteness of a given game from a weighted game. Taylor and Zwicker (1993) demonstrated that the dimension of a simple game can grow exponentially in the number of players. However, the problem of worst-case growth of the dimension in complete games was left open. Freixas and Puente (2008) showed that complete games of arbitrary dimension exist and, in particular, their examples demonstrate that the worst-case growth of dimension in complete games is at least linear. In this paper, using a novel technique of Kurz and Napel (2016), we demonstrate that the worst-case growth of dimension in complete simple games is exponential in the number of players.