Chapter I
Democracy and its mathematical discontents
1.1 Choosing an Electoral System
The last thirty years saw a worldwide movement towards democratic governance. This triggered a renewed search for effective methods of ensuring representative government. Of the world's 211 countries, 36 are “established democracies” according to the criteria of Lijphart.^{1} There are 77 democratic countries according to scales used by Blais & Massicotte.^{2} What distinguishes one democracy from another—apart from the particular form and history through which popular representation appeared—is the way its electoral system functions.
An electoral system is the method by which votes in national elections translate into seats in a Parliament or Lower House. This system may have emerged through a quirk of history, or through the impact of colonialism, or the influence of neighbors, by way of evolution, or through violent revolution, but it exerts a powerful influence on the political future and cohesion of a nation.
Three interdependent elements constitute democracy: political rights, civil rights and institutionalized checks and balances. Political rights comprise free speech and freedom for political participation. Civil rights protect people from practices such as unreasonable searches and seizures, slavery, and torture. Institutionalized checks and balances give citizens the right to limit the powers of the state. According to the maxim of Aristotle, to be "civilized" is to be "political": "Man is by nature a political animal". By this statement, Aristotle meant the obligation of civilized man to participate in the political affairs of the city.^{3} The 14^{th} century historian Ibn Khaldun considered this statement of Aristotle the cornerstone of society.^{4} It is interesting to note that the ancient Arabic translation of this statement does not include the word "animal", and simply reads, "Man is civic by nature".
Elections are the cornerstone of democracy. To quote Riker, "[T]he essential democratic institution is the ballot box and all goes with it."^{5} Huntington defines "a twentiethcentury political system as democratic to the extent that its most powerful decisionmakers are selected through fair, honest, and periodic elections in which candidates freely compete for votes and in which virtually all the adult population is eligible to vote."^{6} This is a minimal definition of democracy.^{7} The electoral system chosen by a country dictates the rules of the game under which that country’s democracy is practiced.
How are votes in a national election translated into seats in a parliament? And what constitutes "fairness"? Hundreds of electoral systems are currently used in the world, and, potentially, one can devise thousands more. They all fall into three broad families according to the way in which they aggregate votes:

PluralityMajority systems (PM);

SemiProportional Representation systems (semiPR) ; and

Proportional Representation systems (PR).
The following flowchart offers a summary of all systems.^{8}
Plurality and Majority systems are both winnertake all schemes; they differ in the number of representatives per district. PR reflects the proportions of votes received by competing parties as closely as possible. SemiPR systems are a mix of both.
While political scientists agree on the three categories, they sometimes disagree on how to classify particular countries.^{9} Table 1.1 shows that among the 208 countries and territories listed in the 2002 International Institute for Democracy and Electoral Assistance (IDEA) database, 76 countries have adopted PR, 73 chose plurality rule, 30 chose majority rule, while 21 countries opted for semiPR. Eight countries (including Saudi Arabia, Libya and China) cannot be classified. Countries with a British colonial legacy—such as Canada, India, and the United States of America—tend to use a plurality system. While there is a strong correlation between colonial background and the adoption of a specific electoral system, this is not universally the case. Former French colonies are not prone to adopt the majority rule which France has known for most of its history.^{10} Plurality is more common in North America, Africa, and Oceania. PR is more prevalent in Europe and South America. Countries struggling for a compromise between the “old” and the “new” tend to choose “semiPR” electoral formulas.
Table 1.1. Breakdown of Electoral Systems Worldwide (2002)
System

Number of countries

% of countries

Plurality

73

35

Majority

30

14

PR

76

37

SemiPR

21

15

Other

8

4

Total

208

100

The effect of different electoral systems is a matter of debate. The effect of pluralitymajority systems is generally to exclude extremist and fringe groups from parliament, unless the extremists are geographically concentrated. Critics have argued that pluralitymajority systems also tend to exclude minorities and women from "fair" representation.^{11} For example, in the U.S., African Americans comprised 12.3 % of the population at the last census. However, in the House of Representatives, which is elected by a PM system, they comprise only 9.0 %. The situation is worse for Hispanics, women and other minorities; compare Table 1.2 and Table 1.3. In the 2000 census, 12.6 % of the U.S. population reported being “Hispanic or Latino”^{12}; however, only 7.4 % of representatives in the House are Hispanic. Neither African Americans nor Hispanics currently have any representation in the Senate.
While women represent about 51 % of the U.S. population, only 14.3 % of the House is currently women and 13 % of the Senate. Nonminority men compose about 78 % of the representatives in the House, while nonminority women represent 8 %.^{13}
Table 1.2. Members of Congress (19812003)
Member of congress and year

Male

Female

Black

Hispanic

Other

REPRESENTATIVES












97th Cong. 1981

416

19

18

6

3

98th Cong. 1983

413

21

21

8

3

99th Cong. 1985

412

22

21

10

3

100th Cong. 1987

412

23

23

11

4

101st Cong. 1989

408

25

24

10

5

102nd Cong. 1991

407

28

26

11

3

103rd Cong. 1993

388

47

38

17

4

104th Cong. 1995

388

47

40

17

4

106th Cong. 1999

379

56

39

(NA)

(NA)

107th Cong. 2001

374

61

36

19

5

108th Cong. 2003

373

62

39

25

7







SENATORS












97th Cong. 1981

98

2





3

98th Cong. 1983

98

2





2

99th Cong. 1985

98

2





2

100th Cong. 1987

98

2





2

101st Cong. 1989

98

2





2

102nd Cong. 1991

98

2





2

103rd Cong. 1993

98

2





2

104th Cong. 1995

93

7

1



2

105th Cong. 1997

92

8

1



2

106th Cong. 1999

91

9



(NA)

(NA)

107th Cong. 2001

87

13







108th Cong. 2003

87

13





3

Data from across the world suggests women tend to be less represented in countries that use pluralitymajority systems than in those with PR. The 1995 “Women in Parliament” annual study commissioned by the InterParliamentary Union found that women made up to 11 % of the parliamentarians in established democracies using a particular form of plurality, but the figure is almost 20 % in countries using PR. This pattern of increased representation of women with PR is observed in emerging democracies, especially in Africa.^{14} Table 1.4 shows the average representation of women around the world in both lower and upper houses of parliament^{15} countries as of January 2004.^{16} Under PR, women’s representation in the lower house is, on average, about 18 %, in comparison to the 1112 % figures under plurality or majority. Ranked by the percentage of women, the U.S. was 17^{th} out of the 61 countries using plurality which were studied; the highest Rwanda with 48.8 % of women representation, and lowest being Kuwait and Korea with 0 %. Under the majority electoral formula, the highest was reported in Cuba (36%), followed by Vietnam (27.3 %); Australia ranked fourth with 25.3 %, while France was 9^{th} (12.2 %). Egypt appeared at the bottom of the majority systems when it comes to women’s representation (2.4 %).
Table 1.3. U.S. Population by Race and Sex, 18002000
Census Data

Total **

White

Black

Other

Sex Ratio *

1790

3,929

3,172

757



103.8

1800

5,308

4,306

1,002



104.0

1810

7,240

5,862

1,378



104.0

1820

9,639

7,867

1,772



103.3

1830

12,866

10,537

2,329



103.1

1840

17,070

14,196

2,874



103.7

1850

23,192

19,533

3,639



104.3

1860

31,443

26,923

4,442

79

104.7

1870

39,819

33,589

4,880

89

102.2

1880

50,156

43,403

6,581

172

103.6

1890

62,948

55,101

7,489

358

105.0

1900

75,994

66,809

8,834

351

104.4

1910

91,972

81,732

9,828

413

106.0

1920

106,711

94,821

10,463

427

104.0

1930

122,755

110,287

11,891

597

102.5

1940

131,669

118,215

12,866

589

100.7

1950

150,697

134,942

15,042

713

98.6

1960

179,823

158,832

18,872

1,620

97.1

1970

203,302

178,098

22,580

2,883

94.8

1980

226,546

194,713

26,683

5,150

94.5

1990

248,710

208,704

30,483

9,523

95.1

2000

281,422

211,460

34,658

35,304

96.3







* Males per 100 females





** in Thousands






Source: Dictionary of American History




Under PR, things are dramatically different. Scandinavian countries lead the way with figures as high as 45.3% in Sweden; the lowest were recorded in Turkey (4.4 %) and Madagascar (3.8%).
China contributed the bulk of the average appearing in Table 1.4 for countries which do not use the three electoral methods described, with 20.2 %. The others were Libya, with no data on women’s representation, Saudi Arabia and United Arab Emirates with no representation of women.
Table 1.4. Representation of Women Worldwide (2004)
EXERCISES

If the 435 seats of the U.S. House of Representatives were allocated according to the ethnic/racial divisions in the 2000 in Table 1.3, how many seats would each group get? What difficulties do you see? How would you resolve them?

The 435 seats in the U.S. House of Representatives were divided according to the gender breakdown in the U.S. population in 2000 (see Table 1.3), how many seats would go to women?

(a) Repeat Exercise1 to allocate the 100 seats in the U.S. Senate along ethnic/racial lines.
(b) Repeat Exercise 2 to allocate the Senate seats according to gender.

In 1814, the Stortinget, Norway’s national assembly, was formed at Eidsvoll.^{17} Of the 112 representatives, 25 represented the towns, 33 represented the army and navy, and 54 represented the rural districts. Norway’s first census was carried out in 1769; the population then was 723,618. In 1822 the population reached one million.^{18} For the purposes of this problem, assume the number of representatives in the Stortinget reflected the population distribution of Norway at the time.

Estimate upper and lower bounds for the proportions and numbers of urban and rural Norwegians in 1769.

Estimate the combined size of the Norwegian army and navy in 1769.

Broken down by occupation, the Norwegian Stortinget of 1814 included 37 landowning farmers, 13 merchants, 5 industrialists and 57 government officials. Assuming the Stortinget was representative of occupation, how large are the corresponding groups in the population? Is it likely that this assembly was representative?

Convicted felons are not allowed to vote in several states. Recently, the list used by Florida election officials to remove the names of felons from their voting roles was recently challenged. About 8 % of Florida voters identify themselves as Hispanic, while 11% identify as AfricanAmerican. Of nearly 48,000 felons on the list used by Florida officials—and thus scheduled to be removed from the list of eligible voters—61 were found to be Hispanic while 22,000 were AfricanAmericans.^{19} Hispanic Republicans in the state outnumber Hispanic Democrats by 100,000, while 90 percent of Florida’s 1,000,000 Black voters are Democrats.

Assuming the proportions of felons reflected those of voters, how many Hispanic and how many AfricanAmerican felons would you expect?

Assuming that the same proportions of AfricanAmerican felons are Republican and Democratic as of AfricanAmerican voters, how many AfricanAmerican Republicans and how many AfricanAmerican Democrats were on the list used by Florida officials?

How many Hispanic voters are there in Florida?

What percentage of the Hispanic voters are Republican?

Assuming the same percentage of Hispanic felons are Republican as of Hispanic voters, how many Hispanic Republicans and how many Hispanic Democrats were on the list used by Florida officials?

Find the total number of Hispanic and AfricanAmerican Republicans on the list. Find the total number of Hispanic and AfricanAmerican Democrats on the list.

Comment on the challenge to the list. What would have been the effect of using it?

Confirm the numbers in Table 1.1. Using “filter” or “countif” on the file worldvotingsystems.xls, filter into five categories: PR, SemiPR, Plurality, Majority, Other, and calculate the percentages of each. The first category (PR) has been filtered for you. (Instructions for these commands are in filteringdata.doc.)

To see how see women’s participation varies under different electoral systems, fill in the following table:
Average Percentage of Women in National Parliaments

Lower or single House

Upper House or Senate

Plurality



Majority



PR



SemiPR



Other




Do this using the data in the womeninparliament.xls file:

Filter into the five categories: PR, SemiPR, Plurality, Majority, Other. (Instructions for the “filter” command are in filteringdata.doc.)

Calculate the percentage of women in the Lower and Upper Houses of Parliament in each category.

What do you conclude from the numbers in the table?

For countries using the plurality system, which country had the largest percentage of women, and what was the percentage? Which had the smallest? How did the US rank?

For countries using the majority system, which country had the largest percentage of women, and what was the percentage? Which had the smallest?

For countries using PR, which country had the largest percentage of women, and what was the percentage? Which had the smallest?

South Africa uses a “closed” party list form of proportional representation (PR). Half of the parliament seats (200 seats) are filled by candidates elected from nine regional lists, while the other 200 are filled from national lists.^{20} We will focus on South Africa’s April 2004 Elections. The results are given in the file southafrica.xls. Since we do not have access to separate regional and national data, we will assume seats and votes distributions came from a single cumulative contest which does not distinguish the two.

Calculate the percent of votes garnered by each party.

Calculate the percents of seats won by each party.

Calculate the number seats to which each party is entitled, allowing decimals. Think of some sensible way of distributing the fractional parts.

If all the parties that did not receive a seat in the South African parliament ran as a coalition, to how many seats would they be entitled?

According to the census of 2001, South Africa’s population was 44.8 million; 79 % (35.4 million) identified as blacks, 9.6 % (4.3 million) were whites, four million are mixedrace and around one million Asians. How many seats would each ethnic group hold in South Africa’s parliament under PR?
1.2 Apportionment Schemes
In a federal system, proportional representation means a given state or canton receives a proportion of seats in the government commensurate with its population. For example, in the U.S., PR determines the number of seats each state has in the House. In a PRsystem such as in most European countries, proportional representation means each party contesting in a national election receives a number of representatives according to the votes receives. Mathematically, these two problems (calculating the number of seats or calculating the number of representatives) are the same. Apportionment problems of the same nature appear in many instances. A country wants to see how many divisions from each state it ought to mobilize to form an army with a prescribed size that is representative of its regions. A person dies and leaves cattle or property to be distributed fairly among his or her inheritors.
The central problem of PR is how to convert votes to seats. In its simplest form, the solution is to assign a fraction of seats equal to that party’s fraction of the votes. However the devil is in the details: how to assign the fractions of seat. This question is surprisingly difficult, and over the last 200 years has lead to a variety of methods, each with advantages and disadvantages. They fall in one of two categories:

Largest Remainder methods (LR), also called Quota Methods.

Divisor methods (D).
A summary description of eleven methods is illustrated in Table 1.5 (p. 17). It includes both European and American names for the schemes.
Largest Remainder Methods
The oldest and most natural method was first devised by British barrister Hare almost 150 years ago. Balinski & Young attribute the method to Hamilton, one of the American “founding fathers”. The methods in this category work by calculating a ‘quota’ representing the number of voters per seat, or the population per representative, so
Ideally, the seats would be apportioned according to
Of course, the number of seats assigned to Party A by this formula is likely to be a fraction. In Hare’s method, each party first gets the whole number of seats assigned to it by this formula. The unassigned seats are then distributed to the largest remainders first, until they are exhausted. You might feel this method is so straightforward that it must be the only one necessary. Chapter 4 shows, however, that this method leads to some surprising and undesirable consequences, which have led to the development of other methods. Two others, the Droop and Imperiali Methods, are similar to the Hare Method; the others are quite different.
Divisor Methods
Divisor methods are also known as “highest average” schemes. The eight most commonly used systems are described in Table 1.5; the most straightforward is the d’Hondt method. The best intuition as to how these methods work was offered by Michael Gallagher.^{21} Each party competes for each seat in a sequence of bids as if at an auction. To see how this system works, we consider the d’Hondt method with three parties. Suppose we arrange the votes in decreasing order:
Party A gets the first seat because it has the largest number of votes. To determine who gets the next seat, we divide each of the number of votes by 2 and compare
where Votes A is omitted because it has been “used up” since Party A got the first seat. Now there are two possibilities: Votes B is the largest (in which case Party B gets the next seat), or is largest (in which case Party A gets the second seat). To decide about the third seat, add, , and so on, to the table and omit the entry that led to the second seat and proceed. This is done until all seats have been exhausted.
The divisor methods differ in the sequence of divisors they employ. For example, the Huntington method used in the US to determine size of the Electoral College uses the divisors. The oldest methods go back to Belgian Professor d’Hondt and the French Mathematician A. SainteLaguë.
1.3 The Mathematical Problem: Measuring Injustice
Throughout these notes, whenever we refer to a party system, P_{1}, P_{2}, …, P_{N}, will denote the respective votes obtained by N parties contesting for seats in a national parliament. By s_{1}, s_{2}, …, s_{N}, we will denote the corresponding seats that these votes translate into. By P and S, we will designate the total number of votes and the total number of seats. Whenever we refer to a federal system, P_{1}, P_{2}, …, P_{N}, will denote the population of the N states in the federation, and s_{1}, s_{2}, … , s_{N} will be the number of representatives from each state. Then P and S are the total population of the country and the total number of representatives in its parliament, respectively. For bicameral systems, this is usually the lower house. We will carry our analysis for party systems. We have
,
and
If the sequence of votes is such that:
,
we expect the sequence of seats to satisfy:
.
In an ideal apportionment, the following should occur:

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

The relative fraction of votes between two parties i and j is equal to the relative fraction of seats:
However, the chance of either (a) or (b) to occur all at once for all parties in a given election is almost zero.^{22} In practice, one attempts to make the difference between the left hand side and right hand side of either (a) or (b) as small as possible for all parties, or all pairs of parties.
Two numbers a and b can be compared using either their absolute difference or their relative difference If we use absolute differences, we attempt to:

Minimize the in the case of (a).

Minimize the in the case of (b).
In an ideal apportionment, both these minima will be zero. In practice, these minima are not zero, and their values give a measure of the injustice of the apportionment. The larger the minimum value obtained, the less fair the apportionment.
Alternatively, using relative differences, in the case of (b), we would minimize
.
In an ideal apportionment, this minimum will be zero. In practice, its value is a measure of injustice. Deciding which measure of injustice to minimize leads to different electoral systems.
Another measure of injustice is given by the following argument. Dividing the S seats among the N parties gives the following exact proportions:
, , …, .
Summing all these exact proportions gives S:
.
In an ideal apportionment where all the q_{i}’s are integers,
, , …, .
In practice, the q_{i} are not whole numbers. In a proportional apportionment, the party should get either or +1 seats (here denotes the result from rounding x down to the nearest integer). These are called, respectively, the Hare minimum and Hare maximum for that given party. The fractions left in assigning seats are
, , …, .
Since each party should ideally get a number of seats number of seats equal to either its Hare minimum or its Hare maximum, these errors are either positive or negative fractions between and 1. The sum of all these errors is zero:^{23}
.
Thus,
,
Since by the definition the q_{i}s,
,
we have
.
Approximating the ideal case, we attempt to minimize the following weighted error:
.
We could also drop the weights, namely the votes for each party, and minimize the standard error:
.
In fact, the methods we will describe in the following chapters stem from these interpretations as to what constitutes a measure of injustice. Other interpretations lead to more complications which will be addressed later.
For the purposes of illustration, we will use three of the four measures of error described above:
In these definitions, Error_{2} and Error_{3} are scaled relative errors, while Error_{1} is an absolute error. None of these errors have units. A seat allocation under one system is better than another one if the error it generates is smaller.
Let us illustrate these measures by looking at the Antigua and Barbuda Legislative Elections of March 23, 2004. The House of Representatives of these Caribbean Islands is composed of 17 seats.^{24} However, in the last elections, one seat remained undecided. The actual distribution of the remaining 16 seats was decided using a majority system; the results are listed in the third column of Table 1.5. Alternative seat distributions, calculated using proportional representation, are in the fourth, fifth columns, and sixth columns termed “Alternative 1”, “Alternative 2”, and “Alternative 3”.^{25} We have used the Excel file injustice.xls to calculate the three proposed error measures for the Antigua and Barbuda example.^{26}
Table 1.5. Antigua & Barbuda Legislative Elections, March 23, 2004
Party

Votes

Seats

Alternative 1

Alternative 2

Alternative 3

United Progressive Party

21,892

12

10

9

8

Antigua Labour Party

16,544

4

4

7

6

Barbuda People's Movement

492

0

1

0

1

Others

791

0

1

0

1

Undecided Seat


1

1

1

1

The calculations are summarized in Table 1.6. The current distribution of seats is better than “Alternative 1” and “Alternative 3” if Error_{1} is our measure of what constitutes a “fair” election. “Alternative 2” is obtained under the Hare scheme discussed earlier. It constitutes the best seat allocation if Error_{2} is the measure of a fair election. It is also the best alternative if Error_{3} is the measure of fairness.
Table 1.6. Antigua & Barbuda Injustice Measures

Current Distribution

Alternative 1

Alternative 2

Alternative 3

Error_{1}

4.03x10^{4}

1.63x10^{3}

4.04x10^{4}

1.63x10^{3}

Error_{2}

8.35x10^{7}

1.23x10^{6}

3.70x10^{7}

1.11x10^{6}

Error_{3}

6.11x10^{4}

1.85x10^{3}

5.70x10^{4}

1.84x10^{3}

1.4 Assumptions
In this section, we list some of the underlying assumptions for a “reasonable” apportionment. In a perfect proportional scheme:

No party should lose seats if the total number of seats is increased.

Each party should stay within one seat of its quota q_{i}_{, }that is .

A large party should not be favored at the expense of a small party artificially or viceversa.
In a federal system, two additional assumptions are required:

No state should lose a representative as a result of the addition of a new state to the union, provided the total number of seats increases.

Each state should have at least one representative.
EXERCISES

Consider the results of the 1999 Legislative Elections in Antigua & Barbuda^{27} provided in the table below. Use the file injustice.xls to answer the following:
Party

Votes

Seats

Alternative 1

Alternative 2

Alternative 3

United Progressive Party

14,713

4

12

8

6

Antigua Labour Party

17,521

12

4

9

8

Barbuda People's Movement

418

1

1

0

2

Others

445

0

0

0

1
 