2023 | Book

# Delegate Apportionment in the US Presidential Primaries

## A Mathematical Analysis

Authors: Michael A. Jones, David McCune, Jennifer M. Wilson

Publisher:

Book Series : Studies in Choice and Welfare

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This book provides a comprehensive mathematical description and analysis of the delegate allocation processes in the US Democratic and Republican presidential primaries, focusing on the role of apportionment methods and the effect of thresholds—the minimum levels of support required to receive delegates. The analysis involves a variety of techniques, including theoretical arguments, simplicial geometry, Monte Carlo simulation, and examination of presidential primary data from 2004 to 2020.

The book is divided into two parts: Part I defines the classical apportionment problem and explains how the implementation and goals of delegate apportionment differ from those of apportionment for state representation in the US House of Representatives and for party representation in legislatures based on proportional representation. The authors then describe how delegates are assigned to states and congressional districts and formally define the delegate apportionment methods used in each state by the two major parties to allocate delegates to presidential candidates.

Part II analyzes and compares the apportionment methods introduced in Part I based on their level of bias and adherence to various notions of proportionality. It explores how often the methods satisfy the quota condition and quantifies their biases in favor or against the strongest and weakest candidates. Because the methods are quota-based, they are susceptible to classical paradoxes like the Alabama and population paradoxes. They also suffer from other paradoxes that are more relevant in the context of delegate apportionment such as the elimination and aggregation paradoxes. The book evaluates the extent to which each method is susceptible to each paradox. Finally, it discusses the appointment of delegates based on divisor methods and notions of regressive proportionality.

This book appeals to scholars and students interested in mathematical economics and political science, with an emphasis on apportionment and social choice theory.

#### Description of Delegate Allocation Rules

##### Chapter 1. Apportionment in the US Presidential Primaries
Abstract
This chapter summarizes the goals of the book, which are to describe and analyze mathematically the methods used to allocate delegates in the Democratic and Republican Party presidential primaries. We begin by defining the apportionment problem and describing how it occurs at several stages in the primary process. We introduce quota-based and divisor apportionment methods and discuss how delegate apportionment differs from apportionment applied both to congressional representation and to proportional representation in legislative systems. We conclude with a brief history of delegate allocation in the primaries and a summary of crucial features that make delegate apportionment unique.
Michael A. Jones, David McCune, Jennifer M. Wilson
##### Chapter 2. The Democratic Party Primary
Abstract
We detail the process by which delegates are allocated to presidential candidates in the Democratic Party presidential primaries. First, we examine how delegates are apportioned among the states prior to the state contests and then explain the process by which they are subsequently divided into statewide and district-wide delegates. Then, using data from recent primaries, we illustrate the use of Hamilton’s method with a threshold, as prescribed by the Democratic Delegate Selection Rules, to allocate delegates to candidates based on their share of the popular vote, and explain how the geometry of the simplex can be used to visualize the allocation. We conclude with a brief discussion of the use of ranked choice voting in Democratic primaries.
Michael A. Jones, David McCune, Jennifer M. Wilson
##### Chapter 3. The Iowa and Nevada Democratic Caucuses
Abstract
We describe the delegate selection process for the Democratic presidential caucuses, focusing on the 2020 Iowa and Nevada elections. These caucuses use procedures that are significantly different mathematically from the primary elections discussed in Chap. 2.
Michael A. Jones, David McCune, Jennifer M. Wilson
##### Chapter 4. The Republican Party Primary
Abstract
We provide an overview of the delegate allocation process in the Republican Party presidential primary. We include a broad outline of the process at the national level and formally define and compare the seven different apportionment methods used at the state level to assign delegates to candidates based on their share of the primary vote. These seven methods include Hamilton’s method and six methods that have not been well-studied. We illustrate these methods using data from the 2008–2016 primaries. We also discuss the use of thresholds, and the process for assigning delegates to congressional districts.
Michael A. Jones, David McCune, Jennifer M. Wilson

#### Analysis of Delegate Allocation Rules

##### Chapter 5. Properties of the Apportionment Methods Used in the Primaries
Abstract
The seven delegate apportionment methods used in the presidential primaries are analyzed and compared, according to whether they satisfy basic properties such as homogeneity and proportionality. We consider how often the delegate apportionment methods satisfy the quota condition and quantify their degree of bias for and against the strongest and weakest candidates by analyzing delegate bias, delegate thresholds, and the behavior of the methods in close elections. These ideas are then related to the sensitivity of the apportionment methods to small changes of the vote totals. We also determine which methods satisfy majority and leader criteria, and discuss the methods’ propensity to support candidate coalitions. We conclude by comparing overall vote share to seat share for the strongest candidates in recent state primaries.
Michael A. Jones, David McCune, Jennifer M. Wilson
Abstract
In this chapter, we catalog the paradoxes to which the seven delegate apportionment methods are susceptible. Because all of the methods used in the Democratic and Republican primaries are quota-based, they are susceptible to classical paradoxes like the Alabama and population paradoxes. The methods also suffer from other paradoxes that are more relevant in the context of delegate apportionment such as the elimination and aggregation paradoxes. We evaluate the extent to which each method is susceptible to each paradox. We use simulations, data from recent presidential primaries, and simplicial geometry to investigate the likelihood that each paradox occurs. For each paradox, we focus on which candidates are most likely to be affected and on the effect of thresholds.
Michael A. Jones, David McCune, Jennifer M. Wilson
##### Chapter 7. Exploring Alternative Ways to Allocate Delegates
Abstract
This chapter explores alternate models for allocating delegates to candidates. We review shift-quota and divisor methods  and different criteria for determining which allocation is fairest, including the seat-transfer approach and optimization. We compare these methods to delegate apportionment methods based on two indices of disproportionality. We also discuss how the apportionment problem has been solved in the European Union Parliament using the concept of degressive proportionality, and use this concept to motivate the idea of regressive proportionality. This mirror image of degressive proportionality, used in conjunction with divisor methods, offers a potential solution to the delegate apportionment problem which avoids some of the inconsistency and paradoxical behavior exhibited by existing delegate allocation methods. We conclude by discussing the factors that make delegate apportionment unique, and how the apportionment methods used by each state could be better tailored to where the election falls in the primary calendar.
Michael A. Jones, David McCune, Jennifer M. Wilson
##### Backmatter
Title
Delegate Apportionment in the US Presidential Primaries
Authors
Michael A. Jones
David McCune
Jennifer M. Wilson