[Paper Review] From a toy model to the double square root voting system

Wojciech Słomczyński, Karol Życzkowski|arXiv (Cornell University)|Jan 30, 2007

Game Theory and Voting SystemsEconomics, Econometrics and Finance37 references22 citations

TL;DR

This paper proposes the double square root voting system, where voting weights and qualified majority quotas are set such that voting power is proportional to the square root of population. Using a toy model and normal approximation, it shows that setting the quota at the inflection point of the cumulative distribution (q = 0.5 + σ) minimizes power-weight discrepancies, yielding near-perfect compliance with the Penrose square root law and equal citizen voting power across states.

ABSTRACT

We investigate systems of indirect voting based on the law of Penrose, in which each representative in the voting body receives the number of votes (voting weight) proportional to the square root of the population he or she represents. For a generic population distribution the quota required for the qualified majority can be set in such a way that the voting power of any state is proportional to its weight. For a specific distribution of population the optimal quota has to be computed numerically. We analyse a toy voting model for which the optimal quota can be estimated analytically as a function of the number of members of the voting body. This result, combined with the normal approximation technique, allows us to design a simple, efficient, and flexible voting system which can be easily adopted for varying weights and number of players.

Motivation & Objective

• To design a fair two-tier voting system where each citizen's voting power is approximately equal, regardless of state size.
• To solve the inverse problem: how to assign weights and set a quota so that voting power matches desired proportional representation.
• To identify an optimal quota that minimizes the discrepancy between voting power and voting weight in square root weighted systems.
• To demonstrate analytically and numerically that the inflection point of the normal approximation yields near-optimal power distribution.

Proposed method

• Uses a toy model with M members to analytically estimate the optimal quota q* as a function of M and population distribution.
• Applies the normal approximation to the sum of voting weights, modeling coalition weights as normally distributed.
• Derives the Banzhaf index ψ_j as the difference of two normal cumulative distribution functions, enabling analytical estimation of voting power.
• Identifies the inflection point q_n = 0.5 + σ as the optimal quota, where σ² = (1/4)∑w_i², due to minimal power-weight deviation.
• Shows that at q_n, the normalized Penrose–Banzhaf index β_j/w_j ≈ 1, with error term O(v_j⁴), indicating high accuracy.
• Uses the normal approximation to justify that q_n minimizes the mean discrepancy Δ between power and weight, ensuring transparency and fairness.

Experimental results

Research questions

• RQ1 What is the optimal quota q* that minimizes the discrepancy between voting power and voting weight in a square root weighted voting system?
• RQ2 How can the inverse problem—assigning weights and setting a quota to achieve desired power distribution—be solved efficiently?
• RQ3 Can the optimal quota be estimated analytically in a simplified model, and how does it compare to numerical solutions?
• RQ4 What role does the normal approximation play in estimating voting power and identifying the optimal threshold?
• RQ5 How does setting the quota at the inflection point q_n = 0.5 + σ affect the proportionality between voting power and weight?

Key findings

• The optimal quota q* is closely approximated by the inflection point q_n = 0.5 + σ, where σ² = (1/4)∑w_i², minimizing the mean discrepancy Δ between voting power and weight.
• At q_n, the normalized Banzhaf index β_j/w_j ≈ 1 with error O(v_j⁴), where v_j = w_j / √(∑w_i²), indicating near-perfect proportionality.
• The approximation error at q_n is significantly smaller than at q = 0.5, where second-order terms in v_j are present, leading to larger deviations.
• Setting the quota at q_n reduces absolute voting power by a factor of 1/√e ≈ 0.607 compared to q = 0.5, but improves power-weight proportionality.
• The double square root system with q_n as quota ensures that each citizen’s voting power is approximately equal, fulfilling the Penrose square root law in practice.
• The method is simple, efficient, and flexible, enabling direct application to varying numbers of players and population distributions.