A Mathematical Model for Optimal Decisions in a Representative Democracy

TL;DR

This paper develops a mathematical model to optimize decision-making in representative democracies, demonstrating conditions under which they outperform direct voting, and identifying optimal group sizes for representatives based on costs and benefits.

Contribution

It introduces a formal mathematical framework for analyzing and optimizing the structure of representative democracies, focusing on decision accuracy and group size parameters.

Findings

Representative democracy can effectively decide on multiple noisy issues.

Optimal group size for representatives is linear in the number of voters under fixed costs.

When costs and benefits are polynomial, the optimal group size remains close to linear.

Abstract

Direct democracy is a special case of an ensemble of classifiers, where every person (classifier) votes on every issue. This fails when the average voter competence (classifier accuracy) falls below 50%, which can happen in noisy settings where voters have only limited information, or when there are multiple topics and the average voter competence may not be high enough for some topics. Representative democracy, where voters choose representatives to vote, can be an elixir in both these situations. Representative democracy is a specific way to improve the ensemble of classifiers. We introduce a mathematical model for studying representative democracy, in particular understanding the parameters of a representative democracy that gives maximum decision making capability. Our main result states that under general and natural conditions, 1. Representative democracy can make the correct decisions simultaneously for multiple noisy issues. 2. When the cost of voting is fixed, the optimal representative democracy requires that representatives are elected from constant sized groups: the number of representatives should be linear in the number of voters. 3. When the cost and benefit of voting are both polynomial, the optimal group size is close to linear in the number of voters. This work sets the mathematical foundation for studying the quality-quantity tradeoff in a representative democracy-type ensemble (fewer highly qualified representatives versus more less qualified representatives).