The Mathematics of Voting and Apportionment - an introduction

275 Pages·2019·2.52 MB·English

by  Sherif El-Helaly

Compact Textbooks in Mathematics
Sherif El-Helaly
The
Mathematics
of Voting and
Apportionment
An Introduction
CompactTextbooksinMathematics
Compact Textbooks in Mathematics
This textbook series presents concise introductions to current topics
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11225
SherifEl-Helaly
The Mathematics
of Voting and
Apportionment
An Introduction
SherifEl-Helaly
DepartmentofMathematics
TheCatholicUniversityofAmerica
Washington,DC,USA
ISSN2296-4568 ISSN2296-455X (electronic)
CompactTextbooksinMathematics
ISBN978-3-030-14767-9 ISBN978-3-030-14768-6 (eBook)
https://doi.org/10.1007/978-3-030-14768-6
LibraryofCongressControlNumber:2019935503
MathematicsSubjectClassification:91-01,91B12,91F10
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vii
Preface
Untilrecently,themathematicalcontentofthesocialsciencesundergraduate
curriculum was mostly limited to computations and routine manipulation
of symbols. The undergraduate social sciences student was left with little
or no exposure to the more sophisticated and elegant conceptual side of
mathematicsthatinvolvesthelogicaldeductionofresultsfromaxiomsand
hypotheses. In the 1990s, mathematical applications involving deductive
reasoning started to find their way into the social sciences undergraduate
curriculum. The main topics introduced included social choice, yes-no
voting,apportionment,gametheory,andfairdivision.Mostofthetextbooks
thatcoveredthisnewmaterialtargetedthefreshmanandsophomorelevels,
whilejuniorandseniorstudentsreceivedtheattentionofonlyafewauthors.
Audience Basedontwodecadesofteachingtheabove-mentionedtopicsat
theCatholicUniversityofAmerica,Iwrotethisbooktoservetheupper-level
audienceasfollows:
1. The book can be used as a textbook for a rigorous one-semester
course in the mathematics of voting and apportionment for upper-level
undergraduateandbeginninggraduatestudentsmajoringineconomics,
politicalscience,andphilosophy.
2. It can be used for self-study, especially by graduate students in eco-
nomics and political science who need a straightforward and self-
contained introduction to the topics therein before they get started on
theirdissertations.
3. It can also be used in an elective course for mathematics majors in the
applicationsofmathematicsinthesocialsciences.
4. The book is also a suitable read for any curious mathematician who
wishes to get acquainted with these unpublicized applications of
mathematics.
Developed by economists and political scientists with contributions by
mathematicians,thetheoriesofvotingandapportionment(socialchoiceand
socialwelfaretheoryinparticular)aregenuinebranchesofadvancedmathe-
matics.Forexample,Arrow’stheoremondictatorships(consideredbymany
tobethecornerstoneofsocialwelfaretheory)andthesubsequenttheorem
byGibbardonoligarchiesaredeepandoriginalmathematicalresultswhose
only prerequisite is an elementary knowledge of the mathematical concept
of“ranking”(or“orderrelation”)whichIintroducein(cid:2)Chap.1alongwith
completeanddetailedproofsofthesetwotheorems.
Prerequisites No political science prerequisites are needed to use this
book.Themathematicalprerequisitesareminimal,andtheydonotinclude
elementary algebra, calculus, statistics, or geometry. There is no need for
viii Preface
computerskills;asimplecalculatorthatcanperformthebasicarithmetical
operations and compute square roots and includes a memory function will
suffice.
The only mathematical requirement (included in the book in order to
make it self-contained) is some acquaintance with elementary concepts in
combinatorics,graphtheory,andorderrelations,inadditiontothegeometric
andharmonicmeans.Amoreimportantrequirementisthelevelofmaturity
that enables the student to think logically, derive results from axioms and
hypotheses, and intuitively grasp logical notions such as “contrapositive”
and“counterexample.”Thislevelofmaturitycanbeexpectedofjuniorand
senior undergraduate students in economics,political sciences,philosophy,
andmathematics.
Book Contents The book consists of three mutually independent chapters
whichcanbecoveredinanyofthesixpossibleorders,withonecaveat:An
instructorwhowishestostartwith(cid:2)Chap.1shouldcover(cid:2)Sect.2.3(on
combinatorics)beforestarting(cid:2)Sects.1.5(onsocialwelfarefunctions)and
1.7 (on the manipulation of social choice functions). I included a detailed
Table of Contents that should give the reader a good idea about the topics
coveredineachchapter.Thefollowingaresomehighlights:
(cid:2) Chapter 1, on social choice, starts with a discussion of the short-
comingsofthewidelyusedpluralityprocedureofelectionsin(cid:2)Sect.1.1.
Eliminationproceduresareintroducedin(cid:2)Sect.1.2asattemptstoavertthe
shortcomings of the plurality procedure. The Condorcet tournament (com-
monlyknownastheround-robintournament)withitsgraphrepresentation
and related concepts are introduced in (cid:2) Sect. 1.3; and special attention is
given to the Condorcet paradox and how it can be used in a parliamentary
proceduretomanipulatethevotingoutcomebycreatingakilleramendment
(akapoisonpillamendment).IendSect.3withthemathematicallyelegant
but often overlooked theorem proved in 1953 by David McGarvey. This
theorem is useful in the construction of examples and counterexamples as
it guarantees the existence of a scenario of voting that can be represented
by any given complete directed graph. In (cid:2) Sect. 1.4, the vulnerability of
Bordacounttotheinfluenceofirrelevantalternativesanditsfrequentfailure
to elect an existing Condorcet winner are demonstrated. It is interesting
thatbringinganeliminationprincipletotheBordacount(similartothatof
Hare’sprocedure)resultsinaprocedurethatsatisfiestheCondorcetwinner
criterion (the Hare-Borda procedure). (cid:2) Section 1.5 is a brief introduction
tosocialwelfaretheory,includingthetheoremsofDuncanBlack,Kenneth
May, Kenneth Arrow, and Allen Gibbard. In (cid:2) Sect. 1.6, I provide natural
generalizationsofsomestandardsocialchoiceproceduresandCondorcet’s
conceptstothecasewhereindifferenceinthevoter’srankingsandtiesare
allowed.Thefocusin(cid:2)Sect.1.7isonthemanipulabilityofsocialchoice
functions that satisfy the conditions of anonymity and neutrality and allow
indifference and ties. I also present the 1976 theorem of Peter Gärdenfors
ix
Preface
on the manipulability of social choice functions that satisfy the Condorcet
winnercriterion,inadditiontotheaforementionedconditions.
(cid:2) Chapter 2, on yes-no voting, starts with an elementary view of the
Banzhaf and Shapley-Shubik indices of power in (cid:2) Sects. 2.1 and 2.2,
including a comparison between the concepts and the computations of the
twoindices.(cid:2)Section2.3includesthecombinatoricsneededtounderstand
the self-checking method of computing both indices in the same setting in
(cid:2)Sect.2.4.Thefocusin(cid:2)Sect.2.5isonthequestionofwhenagivenyes-
novotingsystemcanberepresentedbyweightsandaquota.Thetheoremof
TaylorandZwickerispartiallyproven,andtheirnovelexampleofthemagic
squarevotingsystemispresented.
(cid:2) Chapter 3, on apportionment, starts with two introductory sec-
tions that demonstrate the complexity of the apportionment problem. In
(cid:2) Sect. 3.3, I present quota procedures with a focus on the procedures of
HamiltonandLowndes,theAlabamaparadox,thepopulationparadox,and
theBalinski-Youngtheorem.Divisorproceduresarethendiscussedindetail
in (cid:2) Sect. 3.4, together with the process of using the threshold divisors to
find the interval of proper divisors for each of the procedures of Jefferson,
Adams,Webster,andHill-Huntington.Theequitycriteriaandthemeasures
of inequity associatedwith the procedures of Webster and Hill-Huntington
are discussedin (cid:2) Sect. 3.5. Motivatedby a different measure of inequity,
Dean’s procedure and the associated harmonic mean are introduced, and
thenthebiasesofthedifferentdivisorproceduresareheuristicallydiscussed.
(cid:2)Section3.6comparesthequotaproceduresandthedivisorprocedureswith
regardtotheapportionmentparadoxes.(cid:2)Section3.7isdevotedtothequota-
divisorproceduresintroducedbyBalinskiandYoung,whichreconcilehouse
monotonicitywiththequotarule.
Pedagogy The pedagogical approach can be highlighted in the following
points:
1. Each chapter begins with an elementary introduction and several
examples, gradually leading to the more advanced material. (This is
particularlyusefultoreaderswithnoearlierexposuretothematerialat
thefreshmanlevel.)
2. New concepts and definitions are motivated by examples before being
formallydefined.
3. The topics treated in this book are plagued by highly complicated and
nonstandard notations. To help overcome this problem, I tried to make
notationsassimpleandintuitiveaspossible,and,insomecases,Iopted
to use words (instead of subscripts and superscripts) in order to keep
symbols simple and uncluttered. I also avoided using acronyms and
abbreviations to denote mathematical objects and concepts. (The use
of the standard acronym IIA instead of “Independence of Irrelevant
Alternatives”isoneofthefewexceptions.)
x Preface
4. Landmark theorems are presented in this book, with fully detailed,
rigorous, and streamlined proofs. Complex proofs, such as those of
Arrow’s theorems, Gibbard’s theorem, and Gärdenfors’ theorem, are
brokendownintopropositionsandlemmasinordertomakethemeasier
tograsp.
5. Definitionsoftheform:“AscenarioofvotingforasetV ofvotersand
a set A of alternatives is a member the set RV where R is the set of
all order relations on A” and “A social welfare function is a function
λ : RV → R” would be too alien to most of the audience of this
book.Nevertheless,itispossibletoconveytheexactmeaningsofsuch
definitions,inaworkablemannerthatcanbeusedinrigorousproofs,in
amorelucidandlessformalmanner;thisiswhatIoptedtodo.
6. Computationsarekeptassimpleaspossibleinordertohelpthestudent
focusontheconcepts.
7. Eachchapterendswithasetofexercisesthatvaryfromroutineexercises
(tohelpthestudentmasterconceptsandtechniques)toexercisesinwhich
thestudentisaskedtoproveordisproveagivenstatement.
8. Answerstomostexercisesaregivenattheendofthebook.
Washington,DC,USA SherifEl-Helaly