Penrose voting system and optimal quota

TL;DR

This paper analyzes the Penrose voting system, showing that an optimal quota exists where voting power aligns with weights, and that this quota decreases as the number of countries increases.

Contribution

It demonstrates the existence of an optimal quota in the Penrose system that balances voting power and weights, providing insights into the system's fairness.

Findings

Optimal quota exists for generic population distributions.

Voting power becomes proportional to weights at this quota.

The optimal quota decreases as the number of voting countries increases.

Abstract

Systems of indirect voting based on the principle of qualified majority can be analysed using the methods of game theory. In particular, this applies to the voting system in the Council of the European Union, which was recently a subject of a vivid political discussion. The a priori voting power of a voter measures his potential influence over the decisions of the voting body under a given decision rule. We investigate a system based on the law of Penrose, in which each representative in the voting body receives the number of votes (the voting weight) proportional to the square root of the population he or she represents. Here we demonstrate that for a generic distribution of the population there exists an optimal quota for which the voting power of any state is proportional to its weight. The optimal quota is shown to decrease with the number of voting countries.