On the Construction of High Dimensional Simple Games

TL;DR

This paper investigates the complexity of representing high-dimensional simple games in voting systems, revealing that certain systems inherently require an exponential number of threshold functions, with implications for their succinctness.

Contribution

It establishes a connection to coding theory to determine the minimal number of threshold functions needed for specific voting systems, showing some require exponential complexity.

Findings

Certain voting systems require exponentially many threshold functions

A novel relation to coding theory helps determine minimal representation complexity

No advantage in representation succinctness for some high-dimensional simple games

Abstract

Voting is a commonly applied method for the aggregation of the preferences of multiple agents into a joint decision. If preferences are binary, i.e., "yes" and "no", every voting system can be described by a (monotone) Boolean function $\chi\colon\{0,1\}^n\rightarrow \{0,1\}$. However, its naive encoding needs $2^n$ bits. The subclass of threshold functions, which is sufficient for homogeneous agents, allows a more succinct representation using $n$ weights and one threshold. For heterogeneous agents, one can represent $\chi$ as an intersection of $k$ threshold functions. Taylor and Zwicker have constructed a sequence of examples requiring $k\ge 2^{\frac{n}{2}-1}$ and provided a construction guaranteeing $k\le {n\choose {\lfloor n/2\rfloor}}\in 2^{n-o(n)}$. The magnitude of the worst-case situation was thought to be determined by Elkind et al.~in 2008, but the analysis unfortunately turned out to be wrong. Here we uncover a relation to coding theory that allows the determination of the minimum number $k$ for a subclass of voting systems. As an application, we give a construction for $k\ge 2^{n-o(n)}$, i.e., there is no gain from a representation complexity point of view.