Chapter 11: VOTING SYSTEMS

Arthur: I am your King.
Peasant woman: Well I didn't vote for you.
Arthur: You don't vote for kings.
Woman: Well how'd you become king, then?
Arthur: The Lady of the Lake, her arm clad in the purest shimmering samite, held aloft Excalibur from the bosom of the water, signifying by divine providence that I, Arthur, was to carry Excalibur.  That is why I'm your king.
Dennis: Listen. Strange women lying in ponds distributing swords is no basis for a system of government. Supreme executive power derives from a mandate from the masses, not from some farcical aquatic ceremony.
Arthur: Be quiet!
Dennis: You can't expect to wield supreme executive power just because some watery tart threw a sword at you!
Arthur: Shut up!
Dennis: If I went 'round sayin' I was an emperor just because some moistened bint had lobbed a scimitar at me, they'd put me away!.

From the Monty Python and the Holy Grail

 

In most political elections each of the citizens with a single vote for a political candidate (president, state representative, mayor …). The candidate with the highest number of votes wins the election. We call that “majority rule”. In this chapter we will discuss different procedures of counting votes for candidates. When only two candidates are in a race, “majority rule” is a good system of choosing the winner. However, when three or more candidates are running, there is no perfect way to decide an election.

 

Majority Rule

When choosing only between two alternatives, the obvious voting procedure is as follows: each voter votes for one or the other choice and the alternative with the most votes wins. We call this procedure majority rule. Note that every vote is treated equally. That is, no single vote is more important than any other vote. If a single voter changes the vote from a losing alternative to a winning one, the outcome of the election would stay the same. And if every voter reversed his or her vote, then the outcome would be reversed as well.

The only potential problem with majority rule is a tie. In a two-way race ties are usually broken in some prearranged way prescribed by election procedures. For example, the following news story is an example how a tie was broken in a small town in New Mexico in 1998:

 

Quick game of poker settles New Mexico mayoral contest

Copyright © 1998 Reuters News Service

ESTANCIA, N.M. (March 5, 1998 11:18 p.m. EST) - Two candidates who tied in a vote for mayor of this small New Mexico town settled their contest Thursday with a flip of a coin and a quick game of five-card draw.

Incumbent James Farrington and JoAnn Carlson each received 68 votes in the Estancia mayoral race in central New Mexico Tuesday. According to the town's provision, the two had to decide on a game of chance to break the tie.

"I wanted five card draw, she wanted to roll the dice," said Farrington, a sandwich shop owner who was appointed mayor 2 1/2 years ago. "So we flipped a coin to see what game would be played. She lost the flip, we played five-card draw and I won with ace high."

"I lost to a dumb deck of cards," said Carlson. "It doesn't reflect the mandate of the people, but that's what the state rules call for so that's how it was done."

Carlson says she will contest the result in district court Friday, saying Farrington is not a resident of Estancia, but only has a business there.


However, the perils of the majority rule occur when more than two alternatives are in a race, or when not every option is represented as an alternative with a fair chance. For example:

·         In South Korea's 1987 presidential election, 2 liberals faced the heir of a military dictatorship. The liberals got a majority of the votes but split their supporters, so the conservative won under a majority vote-counting rule. These rules elect whoever gets the most votes; 50% is not required. The militarist party claimed a mandate to continue its repressive policies. Defeated at the next election years later, its leaders were convicted of treason for ordering the tragic shooting of pro-democracy demonstrators.

·         In North Carolina, the plurality rules effectively deny representation to African-Americans. They have enough voters to totally fill 2 election districts. However, they are a 25% minority scattered over 8 districts. So for 100 years they won no federal representation. In the absence of fair representation many felt invisible as voters.

U.S. Presidential Elections[1]

At the time the founding fathers wrote the U. S. Constitution, they had two choices: a direct election and a representative election. One of the concerns about the direct representation (an election in which the people elect the president directly by majority vote) was mostly technological: ability of presidential candidates to travel and establish presence in remote areas of the country. One of the logical outcomes of direct election could be a fragmentation of the country in regions, each with their own candidate. On the other hand, their concern with representative election (an election in which the representatives of the people elect the president) was the ability by the members of Congress to both accurately assess the desire of the people of their states and to actually vote accordingly. This could have led to elections that better reflected the opinions and political agendas of the members of Congress than the actual will of the people.

The Electoral College was established by the founding fathers as a compromise between election of the president by Congress and election by popular vote. The electors are a popularly elected body chosen by the States and the District of Columbia on the Tuesday after the first Monday in November (November 7, 2000). The Electoral College consists of 538 electors (one for each of 435 members of the House of Representatives and 100 Senators; and 3 for the District of Columbia by virtue of the 23rd Amendment). Each State's allotment of electors is equal to the number of House members to which it is entitled plus two Senators.

                The electors meet in each State on the first Monday after the second Wednesday in December (December 18, 2000). A majority of 270 electoral votes is required to elect the President and Vice President. No Constitutional provision or Federal law requires electors to vote in accordance with the popular vote in their State.

If no presidential candidate wins a majority of electoral votes, the 12th Amendment to the Constitution provides for the presidential election to be decided by the House of Representatives. The House would select the President by majority vote, choosing from the three candidates who received the greatest number of electoral votes. The vote would be taken by State, with each State delegation having one vote.

How to Lose the Election and Win the Electoral College Vote

A little math will tell you that the Electoral College system makes it possible for a candidate to actually lose the nationwide popular vote, but be elected president by the Electoral College.

In fact, it is possible for a candidate to not get a single person's vote – not one – in 39 states or the District of Columbia, yet be elected president by wining the popular vote in just 11 of these 12 states:

 

State

Electoral

Votes

State

Electoral
Votes

California

54

Ohio

21

New York

33

Michigan

18

Texas

32

New Jersey

15

Florida

25

North Carolina

14

Pennsylvania

23

Georgia

13*

Illinois

22

Virginia

13*

* Either state could be the 11th state

 

A presidential candidate must win a majority – 270 – electoral votes to be elected. Since 11 of the 12 states in the chart above account for exactly 270 votes, a candidate could win these states, lose the other 39, and still be elected. Of course, a candidate popular enough to win California and New York will almost certainly win some smaller states, as well.

Has it Ever Happened?

Has a presidential candidate ever lost the nationwide popular vote but been elected president in the Electoral College? Yes, twice.

In 1876 there were a total of 369 electoral votes available with 185 needed to win. Republican Rutherford B. Hayes, with 4,036,298 popular votes won 185 electoral votes. His main opponent, Democrat Samuel J. Tilden, won the popular vote with 4,300,590 votes, but won only 184 electoral votes. Hayes was elected president.

In 1888 there were a total of 401 electoral votes available with 201 needed to win. Republican Benjamin Harrison, with 5,439,853 popular votes won 233 electoral votes. His main opponent, Democrat Grover Cleveland, won the popular vote with 5,540,309 votes, but won only 168 electoral votes. Harrison was elected president.

Multi-Candidate Voting Procedures

In this section we will discuss different techniques selecting a winner of an election.

Plurality Voting

In many situations there are more than two alternatives in an election. In those situations, the most common voting procedure is to simply count the number of first places for each candidate (alternative). The candidate with the most first place votes wins the election. This voting procedure is called plurality voting, and is in effect in counting the plurality vote in U. S. Presidential elections. For example:

Example 1: 1968 Presidential Election

In 1968 presidential election, the race was between a Republican candidate Richard Nixon, a Democrat Hubert Humphrey and an Independent George Wallace. Even though none of the candidates won a majority (more than 50% of the votes cast), the winner was Richard Nixon who won 43.56% of the votes.

 

Richard Nixon

Hubert Humphrey

George Wallace

     31,785,480

        31,275,166

            9,906,473

43.56%

42.86%

13.58%

 

Example 2: 1992 Presidential Election

In 1992 presidential election, the race was between a Democrat Bill Clinton, Republican George Bush and a Reform party candidate Ross Perot. The results of the election were as follows:

 

Total Votes

Clinton

Bush

Perot

Cast

Democrat

Republican

Independent

104,425,014

44,909,326

43.01%

39,103,882

37.45%

19,741,657

18.91%

 

Therefore, Bill Clinton was elected president in 1992 with 43% of the vote.

An alternative way to select the winner would be to hold a runoff election between the top two candidates. In the case of 1992 election, the runoff would have been between Clinton and Bush, with Perot being eliminated. If we assume that Perot voters preferred Bush to Clinton by a two-to-one margin, as the polls at that time indicated, the voters would have chosen in the following way: Clinton 51,489,878 and Bush 52,264,987. In other words, Bush would have won the 1992 election.

In this example, George Bush was Condorcet[2] winner: he would have defeated both Clinton and Perot in one-on-one election.

Some countries (like France and Australia) employ the majority rule when determining the winner of an election. However, a critical difficulty with the majority formula is that, in a multiparty political system, the formula may produce an electoral deadlock if no candidate secures 50 percent of the total vote. In order to break such deadlocks a runoff election is required. This system thus ensures that the elected representative has the support of the majority of the voters. The majority formula is employed in Australia and France:

If no candidate secures a majority in the first round of the French National Assembly elections, another round of elections is required. In this second round, the candidate securing a plurality of the popular vote is declared winner.

 

In Australia voters rank the candidates on a preference ballot. A preference ballot is an ordering of the candidates from the best to the worst according to voter preference. If a majority is not achieved on the first round of elections, the weakest candidate is eliminated, and the votes of the weakest candidate are distributed to the other candidates according to the second preference on the ballot. This redistribution process is repeated until one candidate collects a majority (more than 50%) of the votes. This voting procedure is called sequential runoff or the Hare[3] method.

Hare Method

Example 3: Math Club

The Math club at Indiana University is holding an election for president. The candidates are Abe, Becky, Charlie and Diane. Each of the 17 voters ranks the four candidates in order of preference. The summary of this election is given in the preference table below. Each column shows a different ranking among the candidates, and the number at the bottom of the column specifies the number of voters who chose that particular ranking.

 

 

 

First

Abe

Charlie

Diane

Becky

Becky

Second

Becky

Diane

Charlie

Diane

Charlie

Third

Charlie

Becky

Becky

Charlie

Abe

Fourth

Diane

Abe

Abe

Abe

Diane

 

6

4

4

2

1

 

Using the plurality voting procedure, the winner of this election would be Abe with 6 = 35% first place votes. But is it really fair when 59% of the voters ranked Abe last on their preference ranking?

An alternative is to use the Hare method:

 

In the first round of election, the weakest candidate is Becky, with only 3 first place votes. She is therefore eliminated from the preference table:

 

First

Abe

Charlie

Diane

Diane

Charlie

Second

Charlie

Diane

Charlie

Charlie

Abe

Third

Diane

Abe

Abe

Abe

Diane

 

6

4

4

2

1

 

Combining the third and fourth column (which are now same), we have:

 

First

Abe

Charlie

Diane

Charlie

Second

Charlie

Diane

Charlie

Abe

Third

Diane

Abe

Abe

Diane

 

6

4

6

1

 

In the second round of elections, the weakest candidate is Charlie, with only 5 first place votes, and is therefore eliminated:

 

First

Abe

Diane

Diane

Abe

Second

Diane

Abe

Abe

Diane

 

6

4

6

1

 

Combining the identical columns, we obtain the following preference table:

 

First

Abe

Diane

Second

Diane

Abe

 

7

10

 

Therefore, Diane wins this election with more than half of the first place votes in the last round.

 

Borda Count

 

Example 4: AP Top 25 College Football Poll

Every year, the Associated Press polls seventy (70) sportswriters on a weekly basis beginning with the preseason poll, and progressing through the season. Each of the sportswriters ranks the teams according to her/his preference. Each first place team receives 25 points; the second place team receives 24; the third 23, and so on, while the 25th place wins 1 point. The team with the most points after all 70 rankings are counted is on top if the AP Top 25 College Football Poll. At the end of the 1999-2000 NCAA football season, the rankings were the following:

 

 

 

No.

School

Record

Pts

1

Florida St (70)

12-0

1,750

2

Virginia Tech

11-1

1,647

3

Nebraska

12-1

1,634

4

Wisconsin

10-2

1,519

5

Michigan

10-2

1,406

6

Kansas St

11-1

1,402

7

Michigan St

10-2

1,357

8

Alabama

10-3

1,236

9

Tennessee

9-3

1,168

10

Marshall

13-0

1,136

11

Penn St

10-3

1,038

12

Florida

9-4

941

13

Mississippi St

10-2

923

14

So Mississippi

9-3

788

15

Miami Fla

9-4

678

16

Georgia

8-4

640

17

Arkansas

8-4

575

18

Minnesota

8-4

452

19

Oregon

9-3

358

20

Georgia Tech

8-4

345

21

Texas

9-5

340

22

Mississippi

8-4

281

23

Texas A&M

8-4

272

24

Illinois

8-4

201

25

Purdue

7-5

198

 

The sportswriters were unanimous at the end 1999-2000 football season, giving Florida State the maximum: 70@25 = 1,750 points. This voting procedure is called Borda count.

 

For example, in the Math Club example above, the Borda count would produce the following result of the election:

First

4

Abe

Charlie

Diane

Becky

Becky

Second

3

Becky

Diane

Charlie

Diane

Charlie

Third

2

Charlie

Becky

Becky

Charlie

Abe

Fourth

1

Diane

Abe

Abe

Abe

Diane

 

 

6

4

4

2

1

 

Therefore, Abe has 6@4 + (4 + 4 + 2) @1 + 1@2 = 36 points. Becky has 6@3 + (4 + 4)@2 + (2 + 1)@4 = 46 points. Charlie has 6@2 + 4@4 + 4@3 + 2@2 + 1@3 = 47, and Diane has 6@1 + 4@3 + 4@4 + 2@3 + 1@1 = 41 points. Thus, according to Borda count, the winner is Charlie.

 

Example 5: More on 1968 Presidential Election

In 1968 presidential election, the race was between a Republican candidate Richard Nixon, a Democrat Hubert Humphrey and an Independent George Wallace. Even though none of the candidates won a majority (more than 50% of the votes cast), the winner was Richard Nixon who won 43.56% of the votes.

 

Richard Nixon

Hubert Humphrey

George Wallace

     31,785,480

        31,275,166

            9,906,473

 

Unfortunately, we don’t know how the voters would rank these candidates, given a chance to do so, but for the simplicity’s sake we will assume that all Nixon supporters prefer Wallace to Humphrey, all Humphrey supporters prefer Wallace to Nixon, and Wallace supporters prefer Humphrey to Nixon in a 2:1 ratio. Therefore, we obtain the following preference table:

 

First

3

Nixon

Humphrey

Wallace

Wallace

Second

2

Wallace

Wallace

Humphrey

Nixon

Third

1

Humphrey

Nixon

Nixon

Humphrey

 

 

31,785,480

31,275,166

6,604,315

3,302,158

 

Using Borda count, Nixon would have 139,840,237 points; Humphrey would have 142,121,766 points, and Wallace would have 155,840,711 points. Therefore, according to these assumptions, George Wallace would have won the presidential election in 1968.

 

Pairwise Comparisons

So far, we have seen several voting procedures, which can sometimes be contradictory, i.e., which can produce different winners for the same election preferences of the voters. In our Math club example:

 

First

Abe

Charlie

Diane

Becky

Becky

Second

Becky

Diane

Charlie

Diane

Charlie

Third

Charlie

Becky

Becky

Charlie

Abe

Fourth

Diane

Abe

Abe

Abe

Diane

 

6

4

4

2

1

 

Abe wins this election under the Plurality vote; Charlie wins it if we use Borda count, while Diane wins this election if we use Hare system. On the other hand, Becky’s supporters can point out the following: If compare Becky to any other candidate in a one-on-one election race, Becky would win:

Becky – Abe:        11 – 6

Becky – Charlie:   9 – 8

Becky – Diane:     9 – 8

 

This method of election is called Pairwise Comparisons, since we compare the candidates in pairs. It was first invented by Marquis de Condorcet.

 

 

The Condorcet Voting Paradox

Born: Sept. 17, 1743, Ribemont, France

Died: March 29, 1794, Bourg-la-Reine

French philosopher of the Enlightenment and advocate of educational reform. He was one of the major Revolutionary formulators of the ideas of progress, or the indefinite perfectibility of mankind.

Condorcet was the friend of almost all the distinguished men of his time and a zealous propagator of the progressive views then current among French men of letters. A protégé of the French philosopher and mathematician Jean Le Rond d'Alembert, he took an active part in the preparation of the Encyclopédie.

 

 

Suppose we have three candidates, A, B, and C running in an election, and:

 

If Candidate A is the winner, 66.66% of the people are unhappy with the outcome and think that Candidate C is better.

If Candidate B is the winner, 66.66% of the people are unhappy with the outcome and think that Candidate A is better.

If Candidate C is the winner, 66.66% of the people are unhappy with the outcome and think that Candidate B is better.

 

Thus, whichever candidate wins this election two-thirds of the voters will be justifiably unhappy with the selection of the winner of this election.

 

While it is impossible to resolve the paradox above without convincing ?????????????????????? criteria of fairness.

 

 

Criteria of Fairness

In recent chapters, we have investigated several different voting procedures and have seen several examples of imperfect election outcomes. We are primarily interested in procedures that will select alternatives in a way that reflects, in some sense, the will of the people. A meaningful comparison of different procedures will require our haying at hand some properties that are at least intuitively desirable.

 

In the next section we will discuss which of the voting procedures satisfy these conditions. We will Now we will write down (but not proof) which procedures satisfy which properties and which not:

 

Pareto

CWC

Monotonicity

IIA

Plurality

Yes

No

Yes

No

Borda

Yes

No

Yes

No

Hare

Yes

No

No

No

SPV

No

Yes

Yes

No

Dictatorship

Yes

No

Yes

Yes

 

 

 

 

 

We can see that none of our five social choice procedures satisfies all four desirable properties of social choice procedures.

 

 

 

Counterexamples

 

 

 

Arrow´s impossibility Theorem:

In the following we will assume, that we have a fixed set A of three or more alternatives and a fixed finite set P of people.

Definition: A social welfare function is called weakly reasonable if it satifies the following three conditions:

The Question if there is any social welfare function, that ist weakly reasonable can be answered, as we saw above, with yes because of the dictatorship. But because dictatorship is not very popular and democratic, there ist the question if there exists a ny other social welfare function, that ist weakly reasonable. The answer to this question gives the next theorem:

Theorem of Arrow: If A has at least three elements and the set P of individuals is finite, then the only social welfare function for A and P satisfying the Pareto condition, independence of irrelevant alternatives, and monotonicity is a d ictatorship.

Definition: If there is no individual preference list guarantees that x will be over y in the social choice preference list we say that the non-dictatorship is satisfied.

With this Definition we can formulate the Theorem of Arrow in the following way:

Restatement of Arrow´s Theorem: If A has at least three elements and the set P of individuals is finite, then it is impossible to find a social welfare function for A satisfying the Pareto condition, independence of irrelevant alternative s and non-dictatorship.

 

In Chapter 12 we will revisit some of these voting procedures and investigate in more detail how to evaluate their effectiveness.

Extra Exercises: Chapter 11

1 Twelve voters are asked to rank four candidates: A, B, C, and D. The twelve voters turn in the following ballots:

                ABCD                    CADB                    CADB                    CDBA                    ABCD                    ABCD

                BACD                    DABC                    ABCD                    DABC                    ABCD                    CADB

                Make a preference table for these ballots.

 

2 Twenty voters are asked to rank four candidates: A, B, C, and D. The twenty voters turn in the following ballots:

                CDBA                    ABCD                    ABCD                    CADB                    CADB                    CDBA

                BACD                    DABC                    ABCD                    ABCD                    CADB                    CADB

                ABCD                    DABC                    ABCD                    DABC                    ABCD                    DABC

                ABCD                    DABC

                Make a preference table for these ballots.



3 Ten voters are asked to rank three kinds of cookies: Chips Ahoy (C), Thin Mints (T), and Shortbread (S). The ten voters turn in the following ballots:

                CTS                        CST                        CTS                        TCS                        TCS

                TSC                        CTS                        TCS                        CST                        TCS

                Make a preference table for these ballots.

 

4 Indiana Senator Evan Bayh (Democrat) want to pass a bill in Congress this year (2000). He has already spoken to President Clinton and knows that the president will support the bill. In year 2000, the House of Representatives consists of 222 Republicans, 211 Democrats, and 2 Independents. The Senate consists of 45 Democrats and 55 Republicans. Assuming that all of the Democrats will vote along the party lines, how many Republicans does Senator Bayh needs in order to pass the bill?

 

5 In Alabama, funding education has been a hot issue for years. The lottery has been discussed as a way to provide funding. Although many states do have lotteries, many Alabamians are concerned about the social and moral implications of introducing gambling into the community. Since the state does need to increase funding, if the lottery is not implemented, a tax increase would be necessary to improve training. Assume that

Choose the winner by:

(a) Plurality

(b) Hare method

(c) Borda count

(d) Pairwise comparison

 

6 Choose the winner from the preference table in Problem 1 by:

(a) Plurality

(b) Hare method

(c) Borda count

(d) Pairwise comparison

 

7 Choose the winner from the preference table in Problem 2 by:

(a) Plurality

(b) Hare method

(c) Borda count

(d) Pairwise comparison

 

 



[1] The following Web sites: http://usgovinfo.about.com/newsissues/usgovinfo/blelect2000.htm and http://www.nara.gov/fedreg/proced.html have been used to provide some of the information in this section.

[2] The Marquis de Condorcet was born in 1743, and became an eminent mathematician, elected secretary of the Academy of Sciences and a member of the French Academy. During the French Revolution he was elected to represent Paris in the Legislative Assembly and became its secretary. He advocated women's suffrage, the separation of church and state, state-supported education, and the abolition of slavery. In 1794 he was arrested and died that night from poison.

[3] After Englishman Thomas Hare, who developed this method in 1861.