The Mathematics ofviii CONTENTS 10 Proportional (Mis)representation 191 The U.S. House of Representatives 192 Hamilton's Apportionment Method 194 Jefferson's Apportionment Method 197

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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10 9 8 7 6 5 4 3 2 1 0 0 9

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



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



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



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



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



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



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



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



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



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]


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.


Bibliography

[1] Alle n Ange l an d Stuar t Porter . A Survey of Mathematics with Appli-cations. Addiso n Wesley , Boston , 6t h edition , 2001.

[2] Kennet h J . Arrow . Social Choice and Individual Values. Joh n Wile y and Sons , New York , 1951.

[3] Kennet h J . Arrow . Social Choice and Individual Values. Joh n Wile y and Sons , Ne w York, 2n d edition , 1963 .

[4] Richar d Aufman , Joann e Lockwood, Richar d Nation , an d Daniel Clegg. Mathematical Excursions. Houghto n Mifflin , Boston , 2004 .

[5] Miche l Balinsk i an d H . Peyto n Young . Fair Representation. Yal e Uni -versity Press , Ne w Haven , 1982 .

[6] Joh n Banzhaf. Weighted voting doesn't work: A mathematical analysis . Rutgers Law Review, 13:317-343 , 1965.

[7] Jeffre y Bennet t an d Willia m Briggs . Using and Understanding Mathe-matics: A Quantitative Reasoning Approach. Addiso n Wesley , Boston , 2nd edition , 2002 .

[8] Rober t Blitzer . Thinking Mathematically. Prentic e Hall , Uppe r Saddl e River, NJ , 3r d edition , 2005 .

[9] Jame s Bradley an d D . Marc Kilgour . Ca n sequentia l election s solve the non-separability problem ? Preprint , 1997 .

[10] Steve n J . Brams , D . Mar c Kilgour , an d Willia m S . Zwicker . Votin g o n referenda: Th e separabilit y proble m an d possibl e solutions . Electoral Studies, 16(3):359-377 , 1997 .

[11] Steve n J . Brams , D . Mar c Kilgour , an d Willia m S . Zwicker . Th e para -dox of multiple elections . Social Choice and Welfare, 15:211-236 , 1998 .

217

218 BIBLIOGRAPHY

[12] Steve n J . Brams and Alan D. Taylor. Fair Division: From Cake-Cutting to Dispute Resolution. Cambridg e Universit y Press , Cambridge , 1996 .

[13] Mar k Buchanan . Ho w strategists desig n the perfec t candidate . Science, 306:799-800, 2004 .

[14] Mauree n T . Carroll , Ely n K . Rykken , an d Jod y M . Sorenson . Th e Canadians shoul d hav e won!? Math Horizons, 5-9 , Februar y 2003 .

[15] Th e Cente r fo r Votin g an d Democracy , h t tp : / /www.fa i rvote .org/ .

[16] Certifie d Lis t o f Candidates : Octobe r 7 , 200 3 Statewid e Specia l Elec -tion, h t tp : / /www.ss .ca .gov/e lec t ions /2003_cer t_ l i s t .pdf .

[17] J . Deega n an d Edwar d Packel . A new index for simple n-person games . International Journal of Game Theory, 7:113-123 , 1978 .

[18] Jame s M . Elenow and Melvi n J . Hinich . The Spatial Theory of Voting. Cambridge Universit y Press , Cambridge , 1984 .

[19] C.C . Elgot. Trut h functions realizabl e by single threshold organs . AIE E Conference Pape r 60-1311 , 1960 .

[20] COMA P (Consortiu m fo r Mathematic s an d it s Applications) . For All Practical Purposes. Freeman , Ne w York , 6t h edition , 2003 .

[21] Joh n Geanakoplos . Thre e brief proofs of Arrow's impossibility theorem , h t tp : / / cowles .econ .ya le .edu /P /cd /d l l a /d l123- r .pdf .

[22] W . M. Gorman. Th e structure o f utility functions . Review of Economic Studies, 35:367-390 , 1968 .

[23] Georg e Herve y Hallett . Proportional Representation - The Key to Democracy. Nationa l Municipa l League , Ne w York , 1940 .

[24] Th e Heisma n Trophy , h t t p : //www.heisman. com/.

[25] Melvi n J . Hinic h an d Michae l C . Munger . Analytical Politics. Cam -bridge Universit y Press , Cambridge , 1997 .

[26] Jonatha n K . Hodge. Separable Preference Orders. Ph.D. thesis, Western Michigan University , Kalamazoo , MI , Augus t 2002 .

[27] R.J . Johnston . O n the measurement o f power: Som e reactions to Laver . Environment and Planning, 10A:907-914 , 1978 .

BIBLIOGRAPHY 21 9

[28] D . Mar c Kilgour . A forma l analysi s o f th e amendin g formul a o f Canada's Constitutio n Act . Canadian Journal of Political Science, 16:771-777, 1983 .

[29] Dea n Lac y an d Emerso n M.S . Niou . A proble m wit h referendums . Journal of Theoretical Politics, 12(1):5-31 , 2000.

[30] Dav e Leip' s Atla s o f U.S . Presidentia l Elections . http://www . u s e l e c t i o n a t l a s . o r g / .

[31] Kennet h May . A set o f independent necessar y an d sufficien t condition s for simpl e majorit y decisions . Econometrica, 20:680-684 , 1952 .

[32] Joh n Stuar t Mill . Considerations on Representative Government. Bobbs-Merrill, Indianapolis , 1958 . Edited with an introduction b y Cur -rin V . Shields .

[33] Minnesot a Stat e Genera l Election : Officia l Results , http:/ /www.sos . s ta te .mn.us /e lec t ion /gens ta te .pdf .

[34] Joh n Parker . R.L. Moore: Mathematician and Teacher. Spectru m Series. Mathematica l Associatio n o f America , Washington , 2005 .

[35] Donal d G. Saari . Basic Geometry of Voting. Springer-Verlag , Berlin , 1995.

[36] Donal d G . Saari . Chaotic Elections: A Mathematician Looks at Voting. American Mathematica l Society , Providence , 2001.

[37] Donal d G . Saari . Decisions and Elections: Explaining the Unexpected. Cambridge Universit y Press , Cambridge , 2001.

[38] Lloy d Shaple y an d Marti n Shubik . A metho d fo r evaluatin g th e dis -tribution o f powe r i n a committe e system . American Political Science Review, 48:787-792 , 1954 .

[39] Kennet h A . Shepsl e an d Mar k S . Bonchek . Analyzing Politics: Ratio-nality, Behaviors, and Institutions. W.W . Norto n an d Company , Ne w York, 1997 .

[40] Kar l Smith . The Nature of Problem Solving in Algebra. Brook s Cole , Belmont, CA , 2004 .

[41] Ronal d Staszko w an d Rober t Bradshaw . The Mathematical Palette. Brooks Cole , Pacifi c Grove , CA , 3r d edition , 2004 .

220 BIBLIOGRAPHY

[42] Statemen t o f Vote : 200 3 Statewid e Specia l Election , h t t p : / / www.ss.ca.gov/elections/sov/2003_special/contents.htm.

[43] Statistic s i n Sports , h t tp : / /www.amsta t .o rg / sec t ions / s i s / .

[44] Phili p D . Straffin . Game Theory and Strategy. Ne w Mathematica l Library. Mathematica l Associatio n o f America , Washington , 2001 .

[45] Pete r Tannenbaum . Excursions in Modern Mathematics. Prentic e Hall , Upper Saddl e River , NJ , 5t h edition , 2003 .

[46] Pete r Tannenbau m an d Rober t Arnold . Excursions in Modern Mathe-matics. Prentic e Hall , Uppe r Saddl e River , NJ , 4t h edition , 2000 .

[47] Ala n D . Taylor . Mathematics and Politics: Strategy, Voting, Power and Proof. Springer-Verlag, Ne w York , 1995 .

[48] Tw o Cousin s Colleg e Footbal l Emporium , http://www.2cuz.com/ .

[49] U.S . Censu s Bureau , h t t p : //www. census.gov/.

[50] U.S . Federa l Electio n Commission , http://www.fec.gov/ .

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


INDEX 223


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



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



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

The Mathematics ofviii CONTENTS 10 Proportional (Mis)representation 191 The U.S. House of Representatives 192 Hamilton's Apportionment Method 194 Jefferson's Apportionment Method 197

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