Dimension of the Lisbon voting rules in the EU Council: a challenge and new world record

TL;DR

This paper investigates the complexity of the EU Council's voting rules, establishing a new lower bound for its dimension and providing a large-scale representation, highlighting the system's intricate structure.

Contribution

It demonstrates that the EU Council's voting system has a dimension of at least 7, surpassing previous records, and offers a heuristic method to approximate its representation.

Findings

The voting system's dimension is at least 7.

A heuristic approach yields a representation with 13,368 weighted games.

The Boolean dimension of the system is exactly 3.

Abstract

The new voting system of the Council of the European Union cannot be represented as the intersection of six or fewer weighted games, i.e., its dimension is at least 7. This sets a new record for real-world voting bodies. A heuristic combination of different discrete optimization methods yields a representation as the intersection of 13368 weighted games. Determination of the exact dimension is posed as a challenge to the community. The system's Boolean dimension is proven to be 3.