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Quantifying the Balinski-Young Theorem: Structure and Probability of Quota Violations in Divisor Methods for Three States

Joseph Cutrone, Tyler C. Wunder

TL;DR

The paper tackles the problem of quota violations in divisor methods for three states under the Balinski–Young framework. It develops exact structural classifications and analytic tests for both lower and upper quota violations, deriving probability formulas based on standard quotas $q_i$ and joint distributions, and validates them with empirical simulations. For $n=3$ and divisor functions with $d(s) \in [s, s+\tfrac{1}{2}]$, upper quota violations do not occur in Adams, Dean, or Huntington–Hill, while lower quota violations occur only on the largest state, with concrete feasible regions and area-based probabilities. The work integrates geometric and probabilistic techniques to quantify how often quota violations arise, offering a rigorous toolset for understanding fairness in apportionment rules and a foundation for extending to more states and other divisor families.

Abstract

The apportionment problem asks how to assign representation to states based on their populations. That is, given census data and a fixed number of seats, how many seats should each state be assigned? Various algorithms exist to solve the apportionment problem, but by the Balinski-Young Theorem, every such algorithm will be flawed in some way. This paper focuses on divisor methods of apportionment, where the possible flaws are known as quota violations. This paper presents a detailed analysis of quota violations that can arise under divisor methods for three states. The study focuses on quota violations in Adams's, Jefferson's, Dean's, and the Huntington-Hill methods when allocating $M$ seats. Theoretical results are proved about the behavior of these methods, particularly focusing on the types of quota violations that may occur, their frequency, and their structure. The paper then introduces tests to detect quota violations, which are then employed to construct a probability function which calculates the likelihood of such violations occurring given an initial three state population vector whose components follow varying distributions.

Quantifying the Balinski-Young Theorem: Structure and Probability of Quota Violations in Divisor Methods for Three States

TL;DR

The paper tackles the problem of quota violations in divisor methods for three states under the Balinski–Young framework. It develops exact structural classifications and analytic tests for both lower and upper quota violations, deriving probability formulas based on standard quotas and joint distributions, and validates them with empirical simulations. For and divisor functions with , upper quota violations do not occur in Adams, Dean, or Huntington–Hill, while lower quota violations occur only on the largest state, with concrete feasible regions and area-based probabilities. The work integrates geometric and probabilistic techniques to quantify how often quota violations arise, offering a rigorous toolset for understanding fairness in apportionment rules and a foundation for extending to more states and other divisor families.

Abstract

The apportionment problem asks how to assign representation to states based on their populations. That is, given census data and a fixed number of seats, how many seats should each state be assigned? Various algorithms exist to solve the apportionment problem, but by the Balinski-Young Theorem, every such algorithm will be flawed in some way. This paper focuses on divisor methods of apportionment, where the possible flaws are known as quota violations. This paper presents a detailed analysis of quota violations that can arise under divisor methods for three states. The study focuses on quota violations in Adams's, Jefferson's, Dean's, and the Huntington-Hill methods when allocating seats. Theoretical results are proved about the behavior of these methods, particularly focusing on the types of quota violations that may occur, their frequency, and their structure. The paper then introduces tests to detect quota violations, which are then employed to construct a probability function which calculates the likelihood of such violations occurring given an initial three state population vector whose components follow varying distributions.
Paper Structure (26 sections, 21 theorems, 30 equations)

This paper contains 26 sections, 21 theorems, 30 equations.

Table of Contents

  1. Introduction
  2. Preliminary Notation and Definitions
  3. Results
  4. Background Information
  5. Assumptions
  6. Introductory Lemmas and Definitions
  7. Structure and Probability of Quota Violations for Three States in Adams's, Dean's, and Huntington-Hill's Method
  8. Characterization of Quota Violations in Three States
  9. Lower Quota Violation Criteria Test for 3 States
  10. Probability Function for Lower Quota Violations in Three States
  11. Motivating Example
  12. Calculation of the Feasible Region
  13. Probability Formula for Lower Quota Violations
  14. Empirical Results
  15. Structure and Probability of Upper Quota Violations in Jefferson's Method for Three States
  16. ...and 11 more sections

Key Result

Lemma 4.1

(Symmetry) A divisor method is preserved under reordering. That is, if $A(p_1, \dots , p_n)=(a_1, \dots, a_n)$ then for any re-indexing, $(1, \dots, n) \mapsto (\sigma(1), \dots, \sigma(n))$, A($p_{\sigma(1)}, \dots , p_{\sigma(n)}) = (a_{\sigma(1)}, \dots, a_{\sigma(n)})$.BY01

Theorems & Definitions (34)

  • Definition 2.1
  • Lemma 4.1
  • Lemma 4.2
  • Theorem 4.4
  • Theorem 4.5
  • Theorem 5.1
  • proof
  • Lemma 5.2
  • proof
  • Lemma 5.3
  • ...and 24 more