The Mathematics of Elections and Voting

103 Pages·2014·0.601 MB·English

by  W.D. Wallis (auth.)

W.D. Wallis
The Mathematics
of Elections
and Voting
The Mathematics of Elections and Voting

W.D. Wallis
The Mathematics of
Elections and Voting
123
W.D.Wallis
DepartmentofMathematics
SouthernIllinoisUniversity
Evansville,IN,USA
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ISBN978-3-319-09809-8 ISBN978-3-319-09810-4(eBook)
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forMadelinMarie

Preface
Inrecentyears,therehasbeenanincreaseinthenumberofcollegecoursesdesigned
tointroducemathematicstonon-mathstudents.Oneofthetopicsusuallyincluded
is a very cursory introduction to the theory of voting. This is a topic that involves
some interesting mathematics, although there have not been very many advanced
papersonthesubject.
IthoughtIwouldliketointroducevotingtomoreadvancedstudentsofmathe-
matics. This volume is designed for such students. I have introduced a number of
electoralmethodsandincludedanumberofexercises.However,Iassumethosewho
arereallyinterestedinthisareaofmathematicswillexplorethetopicsfurtheronline
andinothersources,suchas[28]and[31].
Thereareseveralworkedexamplesinthetext,andsomearefollowedbypractice
exercises.Iwouldsuggestthatreadersgothroughtheseastheyencounterthem.The
exercisesareattheendofthechapters,andanswerstotheodd-numberedquestions
will be found at the end of the book. I have supplied more complete solutions to
many of them than is usual in a text. But please try them before you look at the
answers.Acompletesolutionsmanualisalsoavailable.
I would like to express my thanks to the staff of Springer, and in particular to
RaziaAmzadforherhelpandencouragement.
Evansville,IN,USA W.D.Wallis
vii

Contents
Preface ....................................................... vii
1 Introduction................................................... 1
1.1 SummaryoftheText........................................ 1
2 SimpleElectionsI .............................................. 3
2.1 ElectionswithTwoCandidates ............................... 3
2.2 MajorityandPlurality....................................... 3
2.3 SequentialVoting .......................................... 4
2.4 TheHareMethod .......................................... 7
2.5 TheCoombsRule .......................................... 10
2.6 PointMethods ............................................. 12
2.7 ElectingtheAmericanPresident .............................. 13
3 SimpleElectionsII:Condorcet’sMethod ......................... 19
3.1 TheCondorcetMethod...................................... 19
3.2 Condorcet’sExtendedMethod................................ 20
3.3 CondorcetWinnerCriterion.................................. 22
3.4 CondorcetLoserCriterion ................................... 23
3.5 TheBucklinMethod........................................ 23
3.6 CondorcetVotingSystems ................................... 25
3.7 SequentialPairwiseVoting................................... 26
ix