The Mathematics ofviii CONTENTS 10 Proportional (Mis)representation 191 The U.S. House of...
Transcript of The Mathematics ofviii CONTENTS 10 Proportional (Mis)representation 191 The U.S. House of...
The Mathematics of Voting and Elections:
A Hands-On Approach
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Mathematical World
Volum e 22
The Mathematics of Voting and Elections:
A Hands-On Approach
Jonathan K. Hodge Richard E. Klima
>AMS AMERICAN MATHEMATICA L SOCIET Y
http://dx.doi.org/10.1090/mawrld/022
2000 Mathematics Subject Classification. P r i m a r y 9 1 - 0 1 ; Secondar y 91B12 .
For addi t iona l informatio n an d upda te s o n thi s book , visi t
www.ams.org/bookpages/mawrld-22
Library o f Congres s Cataloging-in-Publicatio n Dat a
Hodge, Jonatha n K. , 1980 -The mathematic s o f votin g an d election s : a hands-o n approac h / Jonatha n K . Hodge ,
Richard E . Klima . p. cm . — (Mathematica l world , ISS N 1055-942 6 ; v. 22 )
Includes bibliographica l reference s an d index . ISBN 0-8218-3T98- 2 (alk . paper ) 1. Voting—Mathematica l models . 2 . Elections—Mathematica l models . 3 . Socia l choice .
4. Gam e theory . 5 . Socia l sciences—Mathematica l models . I . Klima , Richar d E . II . Title . III. Series .
JF1001.H63 200 5 324.9/001/5195—dc22 200504103 4
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Contents
Preface i x
Acknowledgments xii i
1 What' s S o Goo d abou t Majorit y Rule ? 1
The Mayo r o f Stickeyvill e 1
Anonymity, Neutrality , an d Monotonicit y 3
Majority Rul e an d May' s Theore m 5
Quota System s 6
Back t o May' s Theore m 1 0
Questions fo r Furthe r Stud y 1 1
Answers t o Starre d Question s 1 3
2 Perot , Nader , an d Othe r Inconvenience s 1 7
The Pluralit y Metho d 1 8
The Bord a Coun t 2 0
Preference Order s 2 2
Back t o Bord a 2 4
May's Theore m Revisite d 2 6
Questions fo r Furthe r Stud y 2 8
Answers t o Starre d Question s 3 3
3 Bac k int o th e Rin g 3 7
Condorcet Winner s an d Loser s 3 9
Sequential Pairwis e Votin g 4 3
v
vi CONTENTS
Instant Runof f 4 8
Putting I t Al l Togethe r 5 1
Questions fo r Furthe r Stud y 5 2
Answers t o Starre d Question s 5 5
4 Troubl e i n Democrac y 5 9
Independence o f Irrelevant Alternative s 6 0
Arrow's Theore m 6 5
What i s a Voting System ? 6 6
Arrow's Condition s 6 8
The Punchlin e 7 0
Pareto's Unanimit y Conditio n 7 1
Questions fo r Furthe r Stud y 7 3
Answers t o Starre d Question s 7 6
5 Explainin g th e Impossibl e 7 9
Proving Arrow' s Theore m 8 0
Potential Solution s 8 9
Weakening th e Paret o Conditio n 8 9
Approval Votin g 9 0
Intensity o f Binary Independenc e 9 4
Concluding Remark s 9 6
Questions fo r Furthe r Stud y 9 7
Answers t o Starre d Question s 9 9
6 On e Person , On e Vote ? 10 3
Weighted Votin g System s 10 5
Dictators, Dummies , an d Vet o Powe r 10 8
Swap Robustnes s 10 9
Trade Robustnes s 11 3
Questions fo r Furthe r Stud y 11 6
Answers t o Starre d Question s 11 8
7 Calculatin g Corruptio n 12 1
CONTENTS vi i
The Banzha f Powe r Inde x 12 3
The Shapley-Shubi k Inde x 12 6
Banzhaf Powe r i n Psykozi a 13 0
A Splas h o f Combinatoric s 13 2
Shapley-Shubik Powe r i n Psykozi a 13 5
Questions fo r Furthe r Stud y 13 7
Answers t o Starre d Question s 14 0
8 Th e Ultimat e Colleg e Experienc e 14 7
The Electora l Colleg e 14 9
The Winner-Take-Al l Rul e 15 0
Some Histor y 15 2
Power i n the Electora l Colleg e 15 4
Swing Votes an d Pervers e Outcome s 15 7
Alternatives t o th e Electora l Colleg e 16 2
Questions fo r Furthe r Stud y 16 3
Answers t o Starre d Question s 16 6
9 Troubl e i n Direc t Democrac y 16 9
Even Mor e Troubl e 17 1
The Separabilit y Proble m 17 3
Binary Preferenc e Matrice s 17 6
Testing fo r Separabilit y 17 7
Tool # 1 : Symmetr y 17 7
Tool #2 : Union s an d Intersection s 17 8
Some Potentia l Solution s 18 0
Solution # 1 : Avoi d Nonseparabl e Preference s 18 1
Solution #2 : Set-wis e Votin g 18 2
Solution # 3 : Sequentia l Votin g 18 3
Solution #4 : Contingen t Ballot s 18 6
Solution #5 : T o Be Determined 18 6
Questions fo r Furthe r Stud y 18 6
Answers t o Starre d Question s 18 9
viii CONTENTS
10 Proportiona l (Mis)representatio n 19 1
The U.S . House o f Representatives 19 2
Hamilton's Apportionmen t Metho d 19 4
Jefferson's Apportionmen t Metho d 19 7
Webster's Apportionmen t Metho d 20 2
Three Apportionmen t Paradoxe s 20 4
Hill's Apportionmen t Metho d 20 7
Another Impossibilit y Theore m 20 9
Concluding Remark s 21 0
Questions fo r Furthe r Stud y 21 1
Answers t o Starre d Question s 21 4
Bibliography 21 7
Index 22 1
Preface
Over th e pas t decad e o r so , topic s fro m th e socia l science s hav e graduall y made thei r wa y into a number o f mathematics texts , bot h a t th e secondar y and collegiat e levels . I n college , thes e topic s ar e ofte n taugh t a s a uni t i n a liberal arts mathematics cours e intende d fo r non-mathematic s majors . I n high school, they are used as an exercise in mathematical modeling and prob-lem solving , typicall y i n a wa y tha t directl y addresse s th e NCT M proces s standards o f reasonin g an d proof , communication , connections , an d repre -sentation. Som e college s an d universitie s hav e eve n begu n offerin g entir e semester-length course s devote d t o th e mathematic s o f politic s an d socia l choice. Gran d Valle y Stat e Universit y recentl y adde d suc h a cours e t o it s curriculum, an d thi s book was written i n response to a need create d b y tha t course.
Grand Valley' s course , entitle d The Mathematics of Voting and Elec-tions^ i s a junior/senio r leve l cours e aime d a t student s fro m a variet y o f mathematical backgrounds . I t ca n b e take n a s par t o f a student' s gen -eral educatio n requirements , an d it s onl y mathematica l prerequisit e i s th e completion o f a course i n the university' s mathematic s foundatio n category , which includes college algebra, libera l arts mathematics, introductory statis -tics, logi c (taugh t b y th e philosoph y department) , an d eve n Visua l Basi c programming. Thi s bein g the case , the audienc e fo r th e cours e i s highly di -verse. I n its initial offering, student s came from a variety of major programs , including accounting , business , compute r science , economics , engineering , English, geography , history , mathematics , philosophy , an d politica l scienc e (all i n a clas s o f onl y 1 7 students!) . A t th e sam e time , a simila r course , but wit h a n audienc e consistin g almos t entirel y o f mathematics majors , wa s taught a t Appalachia n Stat e University .
We believe that thi s book is appropriate fo r bot h settings . Student s wit h more mathematical training will approach the problems from a different per -spective tha n thos e wh o ar e no t a s mathematicall y inclined . Furthermore ,
IX
X PREFACE
the instructo r ca n modify hi s or he r approac h an d expectation s t o mee t th e needs o f both o f these groups . W e also believ e tha t thi s boo k i s wel l suite d for independen t stud y primaril y becaus e o f it s hands-on , problem-base d approach.
Prom a pedagogica l standpoint , thi s boo k wa s inspire d b y ou r involve -ment i n th e Legac y o f R.L . Moor e Project , a n initiativ e base d ou t o f th e University o f Texa s a t Austi n tha t aim s t o promot e th e discovery-base d teaching method s pioneere d b y th e lat e Dr . R.L . Moore . Moor e wa s a topologist whos e teachin g styl e revolve d aroun d carefull y constructe d se -quences of problems, which students would solve , present, an d critique . Th e approach take n b y thi s boo k coul d bes t b e describe d a s a modified Moor e method, perhap s mos t significantl y (an d ironically ) becaus e Moor e neve r used a textbook i n hi s own classroom .
When w e set ou t t o writ e thi s book , w e wanted t o captur e th e spiri t o f a Moor e method course , but w e also wanted t o make sure that th e resultin g text wa s accessibl e t o a non-mathematica l audience . T o d o so , w e mad e a poin t o f writin g i n a casua l an d non-threatenin g tone . W e als o trie d t o place eac h topi c o f study i n it s appropriat e historica l contex t an d t o tel l a n interesting an d engagin g stor y throug h ou r investigations .
If yo u ar e accustome d t o workin g wit h mor e traditiona l mathematic s texts, yo u ma y notic e severa l commo n feature s tha t ar e missin g fro m thi s one. Fo r on e thing , w e have no t include d an y worked-ou t example s withi n the bod y o f the text . Instead , w e have provide d "starred " question s whos e answers appea r i n ful l o r i n par t a t th e en d o f eac h chapter . Thes e ques -tions ar e intende d t o hel p th e reade r gaug e hi s o r he r ow n understandin g of foundationa l definition s an d concept s befor e movin g o n t o mor e difficul t material. Thus , ou r starre d question s pla y th e sam e rol e a s example s i n other texts , bu t the y d o so in a way tha t force s th e reade r t o mor e activel y engage the idea s bein g developed .
We hav e no t include d an y repetitiou s o r skill-and-dril l typ e problems , but hav e instea d focuse d o n askin g question s tha t requir e in-dept h analysi s and critica l thinkin g skills . I n fact , w e us e thes e question s no t onl y t o supplement materia l presente d i n th e text , bu t als o a s a n essentia l par t of th e developmen t o f thi s material . Fo r thi s reason , i t i s absolutel y essential fo r reader s t o approac h thi s boo k wit h penci l an d pape r close a t hand , an d t o carefull y wor k throug h eac h questio n withi n the mai n bod y o f th e tex t befor e movin g on . Th e onl y exception s t o this rul e ar e th e Question s fo r Furthe r Stud y provide d a t th e en d o f eac h chapter, which , thoug h recommended , are , strictl y speaking , optional .
PREFACE XI
It woul d b e difficul t t o cove r al l o f th e materia l i n thi s boo k i n a one -semester cours e o n votin g theory . Certai n section s an d chapters , however , can b e omitte d withou t los s of continuity . Specifically :
• Chapter s 1 - 4 introduc e th e basic s o f mathematical votin g theory u p to Arrow' s theorem , an d the y shoul d b e covere d i n order . However , the proof o f May's theorem (beginnin g on page 8) can be omitted fro m Chapter 1 without causin g an y difficultie s late r on .
• Chapte r 5 walks th e reade r throug h a proo f o f Arrow' s theore m an d then discusse s thre e potentia l option s fo r resolvin g th e difficultie s re -vealed b y th e theorem . Thi s entir e chapte r ca n b e omitted , althoug h it would be worthwhile to a t leas t cove r the section on approval votin g (beginning o n pag e 90) .
• Chapter s 6 an d 7 go togethe r an d shoul d b e covere d i n order . The y rely onl y casuall y o n the materia l i n the first fou r chapters .
• Chapter s 8 , 9 , an d 1 0 are essentiall y independen t fro m th e res t o f th e text an d from eac h other; they can be covered in any order, o r omitted . Chapter 8 uses a smal l amoun t o f terminology fro m Chapter s 6 and 7 (specifically, th e language of coalitions and power indices), but require s only a surface-leve l understandin g o f these ideas .
Finally, despit e th e fac t tha t thi s boo k wa s designe d t o b e use d i n a junior/senior leve l course o n voting theory , w e believe tha t part s o f i t coul d also b e use d i n a standar d libera l art s mathematic s course , o r a s a sup -plement t o existin g secondar y curricula . Furthermore , althoug h ou r ow n approach t o teaching wit h thi s book involve s group work , studen t presenta -tions, discussions , debates , an d virtuall y n o lecturing whatsoever , w e would encourage instructors to experiment wit h other techniques and clas s format s as well. W e hope that thi s book serves as a useful startin g point fo r whateve r your instructiona l goal s might be , an d w e hope you'l l fee l fre e t o contac t u s if you hav e an y comments , questions , o r suggestions .
- Jo n Hodg e [email protected]
- Ric k Klim a [email protected]
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Acknowledgments
From Jo n
This projec t woul d no t hav e been possibl e withou t th e suppor t an d encour -agement o f friends, family , an d colleagues .
I am particularl y gratefu l t o Ji m Bradle y fo r introducin g m e to th e fiel d of voting theory , an d t o Ar t Whit e an d Alle n Schwen k fo r helpin g m e tur n my interes t i n mathematic s int o somethin g mor e tha n jus t a hobby .
I a m als o grateful t o m y colleagues a t Gran d Valle y Stat e Universit y fo r their persona l an d professiona l suppor t throughou t th e las t fe w years. The y have taught m e a great dea l about wha t i t means to be a good teacher an d a good mathematician , an d I fee l fortunat e t o belon g t o suc h a n outstandin g group o f teachers an d scholars .
Writing a book is hard work, but having a great co-author makes the task seem less formidable. I truly appreciat e Rick's creativity, hi s hard work , an d the fact tha t h e has handily compensated fo r my complete lack of knowledge about anythin g sports-related .
I hav e been blesse d wit h a wonderful family , grea t friends , an d brother s and sister s i n Chris t wh o hav e bot h encourage d an d challenge d me . I ow e an enormou s deb t o f gratitud e t o m y wife , wh o ha s sacrifice d mor e fo r m e than an y man coul d ever reasonably expect . Melissa , you are the love of my life an d I look forwar d t o al l the memorie s tha t w e have ye t t o make .
Finally, thi s boo k woul d hav e neve r com e to b e ha d Go d no t see n fi t t o make i t so . Thoug h H e has been kin d enoug h to allo w me to tak e credi t fo r some of His ideas , they ar e i n fac t Hi s and no t mine . Thi s bein g the case , I hope tha t i n some way H e wil l be glorified throug h thi s boo k an d whateve r may com e o f it .
xin
XIV ACKNOWLEDGMENTS
From Ric k
I woul d lik e to exten d a specia l thank s t o Jo n fo r offerin g m e suc h a signif -icant rol e i n th e writin g o f thi s book . Jon' s interes t i n votin g an d electio n theory i s bot h professiona l an d recreational , wherea s min e i s primaril y th e latter. A s such , I originall y signe d o n t o b e a n edito r an d t o suppl y Jo n with som e historica l an d biographica l informatio n an d question s fo r furthe r study fo r his book . B y th e end , I ha d don e muc h mor e tha n bot h h e an d I originall y envisioned , includin g writin g th e complet e firs t draft s o f tw o chapters, an d contributin g extensivel y t o eac h o f the others . Thi s mad e th e book ours instea d o f just hi s (and , fo r tw o chapter s anyways , gav e u s eac h a tast e o f the other' s rol e in the project) . However , ever y book begin s a s a n idea in on e person' s mind , an d I would b e remis s i f I failed t o mentio n tha t for thi s boo k tha t perso n wa s Jon .
From Jo n an d Ric k
We would like to offer specia l thanks to the Educational Advancement Foun -dation, Gran d Valle y State University , an d Appalachian Stat e University fo r generously fundin g th e projec t tha t le d to thi s book . W e also wish to than k Harry Lucas , Jr . fo r hi s vision and generosity ; Gre g Foley for introducin g u s to eac h other ; an d Stev e Schlicker , Bil l Bauldry , an d Catherin e Frerich s fo r reviewing ou r gran t application s an d offerin g thei r support .
We are also especially grateful t o Sergei Gelfand an d the American Math -ematical Societ y fo r enthusiasticall y supportin g ou r effort s an d fo r makin g the publication process smooth and efficient. Alon g these same lines, we wish to than k Elain e Becker , Mat t Boelkins , and Geral d Klim a for reviewin g ou r manuscript an d offerin g a numbe r o f helpfu l comment s an d suggestions .
Finally, durin g th e summe r o f 2004 , w e ha d th e pleasur e o f workin g with thre e outstandin g studen t assistants : Mik e Cheyne , Pet e Schwallier , and Dav e Wils . Thei r insight s an d perspective s hav e bee n invaluable , an d we can' t imagin e havin g writte n thi s boo k withou t them . I n fact , w e fee l compelled t o offe r a bi t o f friendly advic e to an y prospective employer s wh o may someday hav e the opportunity t o work with Mike , Pete, o r Dave : Hir e them befor e somebod y els e does ! Seriousl y - thes e guy s ar e smart , hardworking, an d jus t plai n nic e to b e around . We'v e been blesse d b y thei r involvement i n thi s projec t an d w e wis h the m th e bes t i n al l thei r futur e endeavors.
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Index
y, 22 «,45 fc,81 n!, 12 9 (2). 133
Academy awards , 5 4 Acton, Lord , 12 1 Adams' apportionment method , 20 1 Adams, Joh n Quincy , 20 1 Agenda, 4 6 Alabama paradox , 20 5 American Idol, 76 American Mathematical Society , 91 American Statistica l Association ,
91 Anonymity
in a n electio n wit h mor e tha n two candidates , 2 6
in a n electio n wit h tw o candi -dates, 4
Apportionment, 19 2 Apportionment metho d
Adams', 20 1 Dean's, 21 1 Hamilton's, 19 4 Hill's, 20 7 Jefferson's, 19 7 Lowndes', 21 1 of harmonic means , 21 1 Vinton's, 20 3 Webster's, 20 2
Approval voting , 90 , 16 2 Arithmetic mean , 20 7 Arrow's conditions , 6 8 Arrow's theorem , 7 0
proof of , 80-8 9 strong form , 72 , 8 1
Arrow, Kenneth , 6 5 Associated Pres s colleg e footbal l
poll in 1968 , 31 in 1971 , 20 in 1994 , 2 1
Australia House o f Representatives , 4 8 national votin g system , 118 ,
139
Balinski and Young's theorem, 209 Balinski, Michel , 20 9 Banzhaf
index, 12 3 power, 12 3
percentages i n th e 200 4 Electoral College , 15 6
Banzhaf, Joh n F . Ill , 12 3 Baseball
hall o f fame , 9 8 Most Valuabl e Playe r award ,
32 Beedham, Brian , 17 1 Binary preferenc e matrix , 17 6 Binomial coefficient , 13 8
221
222 INDEX
Bitwise complement , 17 7 Black's votin g system , 5 9 Black, Duncan , 5 9 Board o f Supervisor s i n Nassa u
County, Ne w York in 1965,12 4 in 1994 , 12 5
Borda coun t votin g system, 20 , 25 Borda, Jean-Charle s de , 20 , 40 Brams, Steven , 9 7 Browne, Harry , 14 7 Buchanan, Pat , 14 7 Burr, Aaron , 16 5 Bush, Georg e W. , 17 , 14 7
California Fresno cit y council , 117 , 13 9 gubernatorial recal l election in
2003, 20 , 30 mayor o f Sa n Francisco , 4 8
Canada population distribution in 2001,
115 procedure to amen d Constitu -
tion, 11 5 Captain Ahab's Fish & Chips, 103,
121 Catholic church , 1 3 Center fo r Voting and Democracy ,
48 web address , 5 3
Churchill, Winston , 7 9 Citizen sovereignty , 6 8 City counci l o f Fresno , California ,
117, 13 9 CLC, 4 0 Clinton, Bill , 1 8 CNN /USA Today colleg e footbal l
poll i n 1993 , 31 Coalition, 10 6
losing, 10 6 minimal winning , 10 6 winning, 10 6
Coleman, Norm , 3 8 College footbal l pol l
Associated Pres s in 1968 , 31 in 1971 , 20 in 1994 , 21
CNN/USA Today i n 1993 , 31 United Pres s Internationa l i n
1991, 3 2 Combinatorics, 13 2 Condorcet
completion system , 7 4 loser, 4 0
criterion, 4 0 the Marqui s de , 39 , 21 1 winner, 4 0
criterion, 4 0 Condorcet's paradox , 45 , 66 Contingent ballot , 18 6 Conventional rounding , 19 2 Coombs votin g system , 5 4 Corrupt bargain , 16 5 Critical voter , 12 3 CVAAB, 2 3 CWC, 4 0 Cyclic societa l preferences , 6 7
Davis, Gray , 20 , 30 Dean's apportionmen t method ,
211 Deegan-Packel index , 14 0 Dictator
in a weighte d votin g system , 108
in a n electio n wit h mor e tha n two candidates , 6 8
INDEX 223
in a n electio n wit h tw o candi -dates, 2
Dictatorship, 2 Direct democracy , 17 0 Divisor method, fo r apportionment ,
197 Dummy, 10 8
Edwards, John , 15 1 Elector, 14 9 Electoral College , 18 , 104 , 14 9
Banzhaf powe r percentage s i n 2004, 15 6
Electoral vot e total s b y stat e in 2000,15 3 in 2004 , 15 3
Elgot, C.C. , 11 5 European Economi c Community ,
139 European Union , 11 8
Factorial, 12 9 Florida
proposal t o asses s a ne w ta x or fee , 7
U.S. presidentia l electio n i n 2000, 17 , 150
vote totals , 18 , 15 1 France, presidentia l electio n i n
2002, 5 3 Fresno, California, cit y council, 117,
139 Function, 6 6
Geometric mean , 20 7 Gerrymandering, 21 4 Gore, Al , 17 , 14 7 Gorman, W.M. , 17 9 Gubernatorial electio n
California recal l i n 2003 , 20 , 30
Louisiana i n 1991 , 53 Minnesota i n 1998 , 38
Hamilton's apportionmen t meth -od, 19 4
Hamilton, Alexander , 165 , 19 3 Hare, Thomas , 4 8 Harmonic mean , 21 1 Hayes, Rutherford B. , 204 Heisman Memoria l Trophy , 3 3 Hill's apportionmen t method , 20 7 Hill, Joseph , 20 7 Hitler, Adolf , 9 9 House o f Representatives , U.S. ,
192 Humphrey, Skip , 3 8 Huntington, Edward , 20 7
IBI, 9 5 IIA, 6 2 Imposed rule , 2 Independence o f irrelevan t alter -
natives criterion , 6 2 Instant runof f votin g system , 48 ,
49 Institute fo r Operation s Researc h
and Management Science , 91
Institute o f Electrica l an d Elec -tronics Engineers , 9 1
Intensity o f binar y independenc e criterion, 9 5
Intensity o f voters' preferences , 9 5 Intersection of referendum electio n
proposals, 18 0 Ireland, Presiden t of , 4 8 Isomorphic, 10 7
Jefferson's apportionmen t method , 197
Jefferson, Thomas , 19 5
224 INDEX
Johnston index , 14 0
Kerry, John , 30 , 16 4
Lacy, Dean , 17 0 Last Comic Standing, 7 6 Lehman, John , 12 1 Lemma, 8 2 Little Valle y College , 16 9 London, mayo r of , 4 8 Losing coalition , 10 6 Louisiana, gubernatoria l electio n
in 1991 , 53 Lowndes' apportionmen t method ,
211
Major leagu e basebal l hall o f fame , 9 8 Most Valuabl e Playe r award ,
32 Majority, 1 7
criterion, 21 , 24 rule
in a n electio n wit h mor e than tw o candidates , 1 9
in a n electio n wit h two can -didates, 6
Matrix, 17 6 May's theorem , 6 , 2 6
proof of , 1 0 May, Kenneth , 6 McCain, John , 3 0 Mean
arithmetic, 20 7 geometric, 20 7
Median vote r theorem , 16 6 Mill, Joh n Stuart , 4 8 Minimal winnin g coalition , 10 6 Minnesota, gubernatoria l electio n
in 1998 , 38 Minority rule , 3
Modified Paret o condition , 8 9 Monotonicity
in a n electio n wit h mor e tha n two candidates , 2 6
in a n electio n wit h tw o candi -dates, 4
Montana v . United State s Depart -ment o f Commerce , 21 3
Moore, R.L. , x Motion, 10 5
Nader, Ralph , 18 , 61, 148 web address , 7 5
Nassau County , Ne w York , Boar d of Supervisor s
in 1965,12 4 in 1994 , 12 5
National Academ y o f Sciences, 91, 208
National Baseball Hall of Fame, 98 Neutrality
in a n electio n wit h mor e tha n two candidates , 2 6
in a n electio n wit h tw o candi -dates, 4
New York , Nassa u Count y Boar d of Supervisor s
in 1965,12 4 in 1994,12 5
New-states paradox , 20 7 Niou, Emerson , 17 0 Nobel Priz e i n economi c science ,
1972, 66
Ohio, U.S . presidentia l electio n i n 2004, 16 4
Olympic games , 53 , 74 2000 Summer , 5 4 2004 Summer , 5 4
Oscars, 5 4
INDEX 225
Pairwise comparisons , metho d of , 75
Pareto condition , 7 1 modified, 8 9
Pareto, Vilfredo , 7 1 Pascal's Triangle , 13 4 Pascal, Blaise , 13 4 Perot, H . Ross , 18 , 61, 152 Pivotal vote r
in a weighte d votin g system , 126
in th e proo f o f Arrow' s theo -rem, 8 4
Plurality, 1 9 Podunk University , 4 9 Population estimates by state, U.S.
in 1790 , 19 4 in 2004 , 15 9
Population paradox , 20 6 Power index , 12 2 Preference
ballot, 2 2 order, 2 2 schedule, 2 3
Presidential electio n France i n 2002 , 5 3 U.S. i n 1800 , 15 0 U.S. i n 1824 , 15 0 U.S. i n 1876 , 12 , 204 U.S. i n 1992 , 18 , 61, 152 U.S. i n 2000 , 30 , 61 , 76, 14 7
in Florida , 17 , 15 0 vote total s b y candidate ,
148 vote total s b y state , 15 8 vote total s i n Florida , 18 ,
151 U.S. i n 2004 , 30
in Ohio , 16 4 Psykozia, 10 9
Quota in a weighte d votin g system ,
105 in an electio n wit h tw o candi -
dates, 6 rule, 20 2 system, 6
Reagan, Ronald , 3 1 Referendum election , 17 0 Republican Leadershi p Council ,
30 Roman Catholi c church , 1 3 Roosevelt, Frankli n D. , 20 9 Rounding, conventional , 19 2
Saari, Donald , 9 4 San Francisco , mayo r of , 4 8 Schwarzenegger, Arnold , 20 , 30 Secretary-General o f th e Unite d
Nations, 9 1 Separability problem , 17 3 Separable
preferences o f a voter , 17 4 proposals wit h respec t t o a
voter, 17 4 Sequential pairwise voting system ,
44 Sequential votin g i n a referendu m
election, 18 3 Set-wise voting , 18 2 Shapley, Lloyd , 12 6 Shapley-Shubik
index, 12 6 power, 12 6
Shubik, Martin , 12 6 Single transferabl e vot e votin g
system, 4 8 Societal preferenc e order , 2 3 Spoiler candidate , 18 , 61
226 INDEX
Standard divisor , 19 8 Standard quota , 19 4 Starvation Island, 9 5 Stickeyville, 1 Strong for m o f Arrow' s theorem ,
72, 8 1 Survivor, 5 5 Swap, 11 1 Swap robust , 11 1 Symmetric binar y preferenc e ma -
trix, 17 7
Taylor, Alan , 11 4 Tilden, Samuel , 20 4 Total powe r
Banzhaf, 12 3 Shapley-Shubik, 12 6
Trade, 11 3 Trade robust , 11 3 Transitivity, 6 6
U.S. Federa l Electio n Commissio n web address , 14 7
U.S. House of Representatives, 19 2 U.S. population estimate s by stat e
in 1790 , 19 4 in 2004 , 15 9
U.S. presidentia l electio n in 1800 , 15 0 in 1824 , 15 0 in 1876 , 12 , 204 in 1992 , 18 , 61, 152 in 2000 , 30 , 61 , 76, 14 7
in Florida , 17 , 150 vote total s b y candidate ,
148 vote total s b y state , 15 8 vote total s i n Florida , 18 ,
151 in 2004 , 30
in Ohio , 16 4 Unanimity, 7 1 Union o f referendu m electio n pro -
posals, 17 9 United Nation s
Secretary-General, 9 1 Security Council , 109 , 13 9
United Pres s Internationa l colleg e football pol l i n 1991 , 32
Universality, 6 8
Ventura, Jesse , 3 8 Veto power , 10 8 Vinton's apportionmen t method ,
203 Vinton, Samuel , 20 3 Voting system , 3 , 68
weighted, 10 5
Washington, George , 19 3 Webster's apportionmen t method ,
202 Webster, Daniel , 20 1 Weight
of a coalition , 10 6 of a voter , 10 5
Weighted votin g system , 10 5 Willcox, Walter , 20 7 Winning coalition , 10 6
minimal, 10 6 World's Sexies t Ma n contest , 6 0 World's Witties t Woma n contest ,
63
Young, Peyton , 20 9
Zwicker, William , 11 4