Chapter I democracy and its mathematical discontents 1 Choosing an Electoral System



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    1. Which alternative is best if Error1 is the measure of fairness?

    2. Which alternative is best if Error2 is the measure of fairness?

    3. Which alternative is best if Error3 is the measure of fairness?




  1. Consider the results of the 2002 Legislative Elections in Pakistan28 provided below. The seats were allocated using the “First Past The Post” plurality system. The alternatives 1 & 2 were produced using two versions of proportional representation which will be discussed in chapters 2 & 3. Use the file injustice.xls to answer the following.




    1. Which alternative is best if Error1 is the measure of fairness?

    2. Which alternative is best if Error2 is the measure of fairness?

    3. Which alternative is best if Error3 is the measure of fairness?

    4. Is the current distribution of seats in Pakistan the “fairest”?




Party

Votes

Seats

Alternative 1

Alternative 2

Awami National Party

307,255

-

2

3

Baluchistan National Party

69,177

1

0

1

Jamhoori Watan Party

96,277

1

0

1

Mohajir Quami Movement

918,555

13

8

8

Muttahida Majilis-e-Amal

3,349,436

53

31

31

National Alliance

1,363,814

12

13

12

Awami Tehrik

204,349

1

1

2

Democratic Party

83,925

1

0

1

Muslim League (Functional)

328,137

4

3

3

Muslim League (Junejo)

212,749

2

2

2

Muslim League (Nawaz)

2,790,747

19

26

26

Muslim League (Quaid-e-Azam)

7,612,411

69

72

70

Muslim League (Ziau-ul-Huq)

87,394

1

0

1

People's Party Parliamentarians

7,632,708

71

72

70

People's Party (Sherpao)

98,638

2

0

1

Tehreek-e-Insaf

229,125

1

2

2

Independents

4,187,015

21

40

38


  1. Consider the following distribution of seats for the Pakistani Parliament obtained using proportional representation using the 2002 election results.29



Party

Votes

Seats

Awami National Party

307,255

3

Baluchistan National Party

69,177

1

Jamhoori Watan Party

96,277

1

Mohajir Quami Movement

918,555

8

Muttahida Majilis-e-Amal

3,349,436

31

National Alliance

1,363,814

12

Awami Tehrik

204,349

2

Democratic Party

83,925

1

Muslim League (Functional)

328,137

3

Muslim League (Junejo)

212,749

2

Muslim League (Nawaz)

2,790,747

26

Muslim League (Quaid-e-Azam)

7,612,411

70

Muslim League (Ziau-ul-Huq)

87,394

1

People's Party Parliamentarians

7,632,708

70

People's Party (Sherpao)

98,638

1

Tehreek-e-Insaf

229,125

2

Independents

4,187,015

38










  1. Calculate the natural quota of each party, qi.

  2. Calculate the error, qi-si, committed in assigning the seats to each party.

  3. Calculate the total sum of errors. What do you observe?



  1. Consider the apportionment of S seats between N parties. Let P1, P2, …, PN denote the number of votes each party obtained; q1, q2, …, qN, denote the exact portions of the parties and s1, s2, …, sN, the number of seats each party garnered. The total number of votes is denoted by P.

    1. Using the formula for qi, show that .

    2. Show that the sum of the errors




  1. The purpose of this problem is to prove the equivalence of the statements:

(I) For a given party i, the fraction of votes is equal to the fraction of seats:;

(II) The relative fraction of votes between two parties i and j is equal to the relative fraction of seats:


  1. Explain why.

  2. Use (I) and part (a) to show that (II) follows from (I).

  3. Explain why .

  4. Show that.

  5. Assuming (II) is true, show that .

  6. Simplify the expression.

  7. Use parts (c) through (f) to explain why (I) follows from (II).



Table 1.7. Seat Allocation Methods
Largest Remainders Methods (LR)


American Designation

European or Math Designation

Quota

Countries

Hamilton (1792)

Vinton Method of 1850



Hare (1859)

Method of Largest Remainders



P/S

Austria (lower), Belgium (lower), Denmark (higher), Germany (higher)




Droop (1868)

Hagenbach-Bischoff



P/(S+1)

Greece (lower)




Imperiali

P/(S+2)

Italy (lower)


Divisor Methods (D)


American Designation

European or Math Designation

nth divisor

Sequence of Divisors

Countries




Imperiali

(n+1)/2

1, 1.5, 2, 2.5, 3

Belgium (municipal)


Thomas Jefferson (1792)

d’Hondt (1882)

Hagenbach-Bischoff

Greatest Divisors


n

1, 2, 3, 4, 5

European Parliament, Belgium (higher), France (1986), Finland, Netherlands




Modified Sainte-Laguë

(10 n – 5)/7

1, 2.14, 3.57, 5, 6.43

Denmark, Norway, Sweden

Daniel Webster (1832);

W.F. Willcox (1910);

Owens (1920)


Sainte-Laguë (1910)

Major fractions

Arithmetic mean


2 n -1

1, 3, 5, 7, 9

Denmark (higher 45-53), Czech Republic, Poland

E.V. Huntington (1920)

Equal proportions

Geometric mean





0, 1.41, 2.45, 3.46, 4.47

USA (House of Representatives; adopted in 1941)




Danish

3 n -2

1, 4, 7, 10, 13

Denmark (within parties)

Adams

Smallest Divisors

n-1

0, 1, 2, 3, 4

--

Dean’s Method

Harmonic Mean



0, 1.33, 2.40, 3.43, 4.44

--


1.5 A Brief History of Democracy
The march of democracy30 is a “long, hard slog.”31 While the Magna Carta was promulgated in 1215 AD as a means to curb the powers of King John by his barons, it only became an important document in the great upheavals of mid-17th Century England.32 It took almost another hundred years from the “democracy of property-owners”33 which followed the revolt of Oliver Cromwell to the democracy of Parliamentary governments of the late 19th century.
Standard narrative has it that democracy is a pure Western construct34 which started with 5th-century Athens, then found traces in the “republics” of renaissance Italy, to fully materialize with the American revolution of 1776. This narrative of the history of democracy weeds out the factor of time and influence of environment.35
Germs of democracy can be found in various societies throughout history. Traces of “primitive” democracy can be found in Ancient Mesopotamia36 and among the ancient Israelite tribes.37 Prehistoric Mesopotamia was organized along democratic lines. Public affairs were handled by elders, while important affairs were brought before the general assembly or Puhrum,38 which comprised all the citizens of a given town or village.39 Thus warns the old Babylonian proverb,
Do not stand in the assembly;

Do not stray to the very place of strife;

It is precisely in strife that fate may overtake you;

Besides, you may be made a witness for them.



So that they take you along to testify in a lawsuit not your own.
According to Wolf, the terminology of the Old Testament suggests that originally the entire male population of Ancient Israel constituted the assembly. They always convened “at the gate of the city”, “before the tent” and “at the door of the tabernacle” to debate political and religious matters. Decisions were made by acclamation.40
The history of Africa offers similar examples. In his survey of pre-colonial political institutions in Africa over the last 7,000 years, Murdock found “primitive democracy” as “the first and simplest as well as the most widespread type of political system”. In this system, “decisions are reached through discussion and informal consensus.” 41 In Scandinavia, around 800 AD, free men met in allting (common assembly) in the various districts scattered around the country, where they discussed legal and political matters of general concern. During the Middle Ages, these tings, or assemblies, evolved into the local assemblies of rural districts and towns, and they acquired important functions in relations between the king and the common people. It was customary to pledge allegiance to a new king at one of these regional assemblies. The first lagtings (superior regional assemblies) came into existence when Norway united as a kingdom (900 – 1030 AD). These were representative assemblies at which delegates from the various districts in each region met to agree on legal judgments and pass laws. The years 1263 – 1660 were a remarkable period in Scandinavian history. A body of laws was codified (1263 – 80) and applied across the realm until Frederik III, king of the Danish-Norwegian union, declared absolute monarchy in 1660 and codified it as the King’s Act of 1665. This code—which became the constitution of Union of Denmark-Norway until 1814—ended Scandinavia’s early experiment in democracy.42
In medieval times representative government had an “occupational character”.43 The English merchant guild of the early Middle Ages held enormous sway over the government of the boroughs. In the years 1376-1384, the manufacturing guilds of London were able to place representatives in the common council chosen by the crafts guilds. The Lord Mayor of London as well as the four city representatives to Parliament were chosen by members of the livery companies. This was common practice down to the 19th century. In the medieval Italian republic of Florence, beginning in 1293, the Acti, a council of the twenty-one main federations of craft guilds chose the Priors and other ruling magistrates. There is even talk of the “Zunftrevolution” of 1368 when merchants and their followers forced a new constitution (Zunftverfassung) for Strasbourg giving certain guilds direct representation on the City Council. Delegates from twenty-five principal guilds (Zünfte) became part of the City Council and in fact held a majority over the patricians.44 Many cities across France elected guild members to City Councils. Representation to the Estates systems of medieval times also had an occupational basis. This system was revived by the Soviet Union in the early days of Communism. The parliaments of the late medieval period evolved from the Carolingian assemblies of knights. Matters of general policy and war were discussed by these assemblies with the king acting as a first among equals. The practice continued until the warrior assemblies emerged with the baronial Curia Regis45 and estate representations.46 Weber notes:
The basis of democratization is everywhere purely military in character: it lies in the rise of disciplined infantry, the hoplites of antiquity, the guild army of the middle ages…. Military discipline meant the triumph of democracy because the community wished and was compelled to secure cooperation of the non-aristocratic masses and hence put arms, and along with arms political power, in their hands. 47
Ibn Rushd’s Commentary on the Republic of Plato offers curious textual testimonies on “democracy” within an Islamic setting. The eminent Muslim philosopher and judge considered the application of “Platonic notions—conditioned by Greek concepts and institutions—as fully valid general principles, applicable to Muslim concepts and institutions.”48 In his discussion of the transformation of the timocratic49 state into the plutocratic50 state he writes,
“In general, the transformation of the timocratic into the hedonistic man is obvious, be it that he takes delight in money or in the other remaining pleasures. The same seems to apply to the timocratic and the hedonistic state. For the plutocratic and the hedonistic state belong to the same category. We often see kings becoming corrupted into such like men. Similarly, there is in our time the kingdom of the men known as Almoravids. At first they were imitating the constitution base on the Law… then they changed [it] under his son into the timocratic [constitution] together with an admixture in his of the love of money as well. Further, it changed under his grandson into the hedonistic [constitution] … and perished in his time.”
Rosenthal offers a relevant quote from Ibn Rushd on the propensity of democracy to turn into tyranny on the basis of the Almoravid revolution in the Maghreb:
“You can discern this from the democratic rule that exists in our time, for it frequently changes into tyranny. Take for example the rule existing in our own country, i.e. Cordova, after 500 AH. For it was almost completely democratic, [but] then after 540 AH it turned into tyranny.”51

This seems to be the oldest textual reference to a rule by democracy, apart from those from ancient Greek. Rosenthal notes that Ibn Rushd (known in the Europe of the Middle Ages as Averroes) uses timocracy and democracy interchangeably. “The contradiction can be resolved by assuming that ‘democratic’ refers to the Council of the Almoravids as representing the Jamā‘a of Almoravid Islam in the Maghreb.” 52


According to Muslim Sunni jurists, “an Islamic polity ought to be headed by a non-hereditary, elective sovereign, subject to but not above the law.” 53 This principle led some 19th and 20th century writers to describe the Islamic doctrine of Caliphate as “republican”.54 A favorite quote on accountability before the people—widely taught to children in the Islamic world—reads: One day, Omar ibn Al-Khattab, the second Caliph of Islam, stood on the pulpit addressing people: “O people! If you find that I have some crookedness, correct me.” One bedouin rose to his feet and said: “By Allah! If we find you crooked, we will correct you with our swords.” Yet Omar, who did not get angry or harbor malice towards him, raised his hands and said: “Praise be to Allah, Who has created among our people a person who is able to correct the crookedness of Omar.”

French revolutionaires and those who dumped tea in the Boston Harbor should feel vindicated. Their revolutions against the tyranny of kings have yielded their fruits. Today, no self-respecting state—with claims to democracy—would deny its people the right to say in their affairs through elected representatives. At the heart of the two grand revolutions of the 18th century were issues of taxation and representation. The French supported the rebellion of American colonists and needed new sources of revenue to help in the war. Tax burdens fell on the Third Estate (tiers état, i.e. the mob) with the Second Estate (nobility) paying nearly nothing. After 175 years without meeting, the Estates-General (États-Généraux, medieval French version of Parliament) was called to discuss the matter. The Estates-General voted by estates, with both Clergy (First Estate) and Nobility (Second Estate) outweighing the Third Estate, two to one. Thus the first spark: Third Estate delegates left the Estates-General and declared a ‘National Assembly.’ On the eve of the revolution of 1789, ninety-eight percent of the French were Third Estate—peasants, the working class and the bourgeoisie. It is not strange then that the earliest advocacy of the principle of representation has roots among the French and the Americans.
Among the French the names of the Marquis de Mirabeau, and the mathematicians Jean-Charles de Borda, and the Marquis de Condorcet stand out as the earliest men to concern themselves with electoral problems and practical issues regarding voting and representation. Addressing the French Assemblée Nationale, on the conditions of eligibility to the newly founded institution, wrote Condorcet55
“During the convocation of your Assembly, the deputies of the communes were nominated by the electors. However, in assemblies, the confection of records can give birth to parties and give populous eloquence a dangerous influence. In this era, two grand corporations, nobility and the clergy, have become almost completed isolated from the common citizens. These corporations were so little in number if compared with the totality of the inhabitants of the kingdom [of France]. However, they were many when compared with the people they served. … Why should the [Frenchman] be duped further, while elections to assemblies would be better organized, made easy, ..., when he can extend his choice to all the citizens, when his vote— thus far like blowing in the wind—will be guide for the conduct and opinion of those who are in public office... No, Sirs, you have nothing to fear from the legislature. You are now free from all the pecuniary conditions which degraded the dignity of man. The legislature will be as is your Assembly today: The elite of the nation. Enlightened peoples have established pecuniary conditions. However, in England, they are usually eluded, and they were never able to stop corruption. In the United States of America, pecuniary matters are of little importance, since it is easy to acquire property by law... the land is plenty... Thus, the inequality engendered by pecuniary conditions in England or America, are only sensed when close to elections... Moreover, in England, as well as in the United States, electors have no functional means to make choices solely on the basis of the public

conduct of the candidates.”


In his “ce que les citoyens ont droit d'attendre de leurs réprésentans,” [“What Citizens Should Expect from their Representatives”], he writes56
“In America, and even in England, one can say that individuals have civil rights. However, when it comes to political rights, they have none. In England, the principle of ‘sovereignty of the people’ is upheld. But, there is no mechanism to exercise it.”
In the words of Mirabeau (quoted in de Grazia, p. 47),
“[The] Assembly for a nation is nothing but a reduced map of its physical extent: whether small or large, the copy should always have the same proportions as the original.”
The earliest mathematical derivation of an election system seems to be Jean-Charles de Borda’s “Mémoire sur les Elections du Scrutin57 in 1781. De Borda (1733-1799), who later served in the French National Assembly, got interested in election schemes ten years earlier (see de Grazia). In 1770, he suggested to the French Academy that in an election by ballot between more than two candidates, the one who obtains the majority of the votes is not necessarily he whom the electors prefer to his competitors (Hart). His contribution was part of a popular trend among French mathematicians of the Enlightenment—such as Condorcet (who was elected to the French Academy of Sciences in 1769) and Laplace (1749-1823)—to produce “practical and usable knowledge”. Hoag & Hallett (1926) attribute the earliest record of a proportional system mathematical paper to another French mathematician by the name of Gergonne. However, his paper Arithmétique politique: Sur les élections et le système représentatif dates from 1820, and he suggested no specific method of implementation of his ideas. The first application of an actual proportional voting scheme at the level of a Society was done by English schoolmaster Thomas Wright Hill, father of Rowland Hill—the founder of the modern postal system, in 1821. His son Rowland was the first to apply it in the public sphere in Adelaide, South Australia, in 1839. He was then Secretary of the Colonization Commission of South Australia. According to Hart, Hill applied a version of Gergonne’s scheme rather than his father’s (p. 7).
In 1857, Thomas Hare, an inspector of charities published a 53-page pamphlet entitled The Machinery of Representation. In it, he advocated a new system of single transferable vote (STV). Hare devised his system after he saw the defeat of several prominent Liberal and Radical politicians who criticized the Crimean War, Palmerston’s bigotry, and ‘vengeance upon China’ (Hart, p. 26; Bromund). Hare’s pamphlet received much attention and went into second edition the same year. The pamphlet was enthusiastically expanded to a 370-pages long book entitled A Treatise on the Election of Representatives, Parliamentary and Municipal in January 1859. Hare’s book received much admiration from John Stuart Mill. Hare appeared “to have exactly, and for the first time, solved the difficulty of popular representation; and by doing so, to have raised up the cloud and gloom of uncertainty which hung over the future of representative government and therefore of civilisation.” (quoted in Hart, p. 38). Hare’s book inspired Mill “with new and more sanguine hopes respecting the prospects of human society.” (J.S. Mill, Autobiography). Hare had troubles with some aspects of his scheme. They were pointed out in 1868 by H.R. Droop, a British mathematician and lawyer. It was the correction by Droop that was been adopted by proportional representation advocates in latter years.
Hare’s scheme was received favorably on the Continent, but not much reception in England. It was discussed in Amsterdam in 1864 in a congress for the Progress of Social Sciences. A French newspaper publicized it. In fact, it was already in the stage of implementation on a national level in Denmark since 1855. The system was independently invited and introduced by Carl Christopher Georg Andrea, a mathematician and geodesist, then a Minister of Finance (Hoag & Hallett, Hart). In April, 1870, the Hare system of preferential voting was introduced by the Alumni of Harvard University for the nomination of Overseers of the College. The Proportional Representation Society (P.R.S.) was founded in January 1884 by Sir John Lubbock, a charismatic banker, naturalist and political economist. Lubbock hoped proportional representation would quell popular discontent. According to Bromund,
The leaders of the P.R.S. believed that proportional representation would help unify the nation and the empire by preventing Westminster from being dominated by organized parties, by allowing voluntary association of interests the fullest play, and by giving electors reasons to care about politics… The leaders of the P.R.S. believed the evils that would arise from unchecked democracy were foreshadowed in America, which contemporary commentators accepted was subject to levelling tendencies far more pronounced than those in Britain.
The list type of proportional representation was first introduced by Victor Considérant, a disciple of Fourier in 1834. Other names associated with the invention of this scheme are Thomas Gilpin of Philadelphia who published his pamphlet in 1844 and Swiss writer Antoine Morin who published a book on the list system in 1861. The year 1881 saw the birth, in Belgium—known then for divisions between Flanders and French, Catholics and Socialists, of the Association Reformiste pour l’Adoption de la Representation Proportionelle. A key founder was Victor d’Hondt (1841-1901) a professor of civil law at Ghent University. He formulated a new system of proportional representation the following year. In the summer of 1885, a major conference on Proportional Representation was convened in Antwerp. Delegates from key West European countries attended: Switzerland, France, Belgium, Italy, Germany, Holland and Denmark. A paper by Thomas Hare was read in his absence though none from the Proportional Representation Society of England attended. Later, a resolution adopted three key elements was passed. It rejected systems based on absolute majorities. It adopted ‘proportional representation [as] the only means of assuring power to the real majority of the country, an effective voice of minorities, and exact representation to all significant groups of the electorate.’ It also chose the d’Hondt “system of competing lists with divisors”. An electoral injustice in the Swiss canton of Ticino led the federal government to introduce proportional representation in 1889. Belgium and Serbia adopted it in 1899. Many other countries followed suit: Finland (1906), Cuba (1908), Sweden (1909), Portugal (1911), Bulgaria (1911). The adoption of proportional representation by Ireland in 1920 was achieved thanks to efforts by the Proportional Representation Society which sent Lord Courtney in 1911. He advocated PR as a solution to the home rule problem. It guaranteed representation for the Protestant minority58 (Carstairs). Thanks to the single-handed work of Catherine Helen Spence of Adelaide, Tasmania became the first English-speaking community to adopt PR in the Hare scheme of single transferable vote for public elections in 1896. In addition to John Stuart Mill, Alexis de Tocqueville, Henri Poincaré, and Lewis Carroll were among the prominent figures who lend support for PR.
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