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The Mathematics of Voting and Apportionment

An Introduction

  • Textbook
  • © 2019

Overview

Authors:
  1. Sherif El-Helaly

    1. Department of Mathematics, The Catholic University of America, Washington, DC, USA

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  • Provides students with an accessible overview of the mathematical concepts central to voting systems
  • Allows readers to engage with the material in a variety of ways, with chapters that can be approached in any order
  • Includes a chapter on the mathematics of apportionment, a topic not always covered in adequate detail

Part of the book series: Compact Textbooks in Mathematics (CTM)

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Table of contents (3 chapters)

  1. Front Matter

    Pages i-xv
    Download chapter PDF

  2. Social Choice

    • Sherif El-Helaly
    Pages 1-114

  3. Yes-No Voting

    • Sherif El-Helaly
    Pages 115-157

  4. Apportionment

    • Sherif El-Helaly
    Pages 159-250

  5. Back Matter

    Pages 251-264
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Keywords

About this book

This textbook contains a rigorous exposition of the mathematical foundations of two of the most important topics in politics and economics: voting and apportionment, at the level of upper undergraduate and beginning graduate students. It stands out among comparable books by providing, in one volume, an extensive and mathematically rigorous treatment of these two topics.


The text’s three chapters cover social choice, yes-no voting, and apportionment, respectively, and can be covered in any order, allowing teachers ample flexibility. Each chapter begins with an elementary introduction and several examples to motivate the concepts and to gradually lead to more advanced material. Landmark theorems are presented with detailed and streamlined proofs; those requiring more complex proofs, such as Arrow’s theorems on dictatorship, Gibbard’s theorem on oligarchy, and Gärdenfors’ theorem on manipulation, are broken down into propositions and lemmas in order to make them easier to grasp. Simple and intuitive notations are emphasized over non-standard, overly complicated symbols. Additionally, each chapter ends with exercises that vary from computational to “prove or disprove” types.


The Mathematics of Voting and Apportionment will be particularly well-suited for a course in the mathematics of voting and apportionment for upper-level undergraduate and beginning graduate students in economics, political science, or philosophy, or for an elective course for math majors. In addition, this book will be a suitable read for to any curious mathematician looking for an exposition to these unpublicized mathematical applications.


No political science prerequisites are needed. Mathematical prerequisites (included in the book) are minimal: elementary concepts in combinatorics, graph theory, order relations, and the harmonic  and geometric means. What is needed most is the level of maturity thatenables the student to think logically, derive results from axioms and hypotheses, and intuitively grasp logical notions such as “contrapositive” and “counterexample.”

Reviews

“The mathematics of voting and apportionment is a well-written, thorough introduction to social choice and social welfare theory. … This book is suitable as the main text for a one-semester introduction to voting theory and apportionment. I would also recommend this book to anyone interested in independently studying these topics. It is a clearly written, entry-level text into voting theory and apportionment.” (Brittany Shelton, Mathematical Reviews, February, 2020)

“As intended audience the author mentions students in economics, political science, philosophy and (applied) mathematics, but I think the book is also to be recommended to law students … . I like to suggest to include this topic in a next edition of this admirable book.” (H. C. M. de Swart, zbMATH 1426.91001, 2020)

“This is a nicely written book, with clear explanations that are supported by a number of useful, fully worked out, examples. … more appropriate audience would be math majors in a proof-based course who already have some experience in reading precisely stated definitions and statements of theorems, and a willingness to track through the details of a proof.” (Mark Hunacek, MAA Reviews, August 11, 2019)

“This book, is a textbook that is clearly addressing an audience of social science students, in particular in the US … . It is not only a textbook for his students, but it brings together a lot of material that is not easily found in this compact form and as such it will be of interest to any politician or anyone who is generally interested in the subject.” (Adhemar Bultheel, European Mathematical Society, euro-math-soc.eu, July 01, 2019)


Authors and Affiliations

  • Department of Mathematics, The Catholic University of America, Washington, DC, USA

    Sherif El-Helaly

About the author

Sherif El-Helaly is an associate professor at The Catholic University of America in Washington, DC, where he has been teaching a course on mathematical topics in the social sciences for twenty years.

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