Chapter 9 : VOTING
Elections with 2 alternatives
Suppose we poll the class to see whether you prefer the pizza at Domino‟s or Pizza Hut. How should we
determine the winner? This is called majority rule.
E.g. If there are 49 voters, how many are needed to constitute a majority? ___________________________
E.g. If there are 50 voters, how many are needed to constitute a majority? ___________________________
Elections with more than 2 alternatives
Suppose we poll the class to see whether you prefer the pizza at Domino‟s, Pizza Hut, Papa John‟s,
Godfather‟s or Little Caesars. Suppose there are 55 students who vote as follows:
Domino‟s 18 Little Caesars 9
Papa John‟s 12 Godfather‟s 6
Pizza Hut 10
A. The plurality method says to take the option with the most votes. This would be _____________.
B. But often elections require a majority of candidates to favor the winning option. If this is the case, we
could have a runoff between the Domino‟s and Papa John‟s. To do this, we would need to know how
those who voted for Pizza Hut, Godfather‟s and Little Caesars rank the top two finishers. Suppose we ask
for even more information. We ask each of the 55 voters to rank all 5 chains in order of preference.
After gathering all 55 rankings we see that many of the voters turned in the same ranking. In fact, there
are only six distinct rankings:
18 votes 12 votes 10 votes 9 votes 4 votes 2 votes
1st Domino‟s Papa John‟s Pizza Hut Little Caesars Godfather‟s Godfather‟s
2nd Little Caesars Godfather‟s Papa John‟s Pizza Hut Papa John‟s Pizza Hut
3rd Godfather‟s Little Caesars Godfather‟s Godfather‟s Little Caesars Little Caesars
4th Pizza Hut Pizza Hut Little Caesars Papa John‟s Pizza Hut Papa John‟s
5th Papa John‟s Domino‟s Domino‟s Domino‟s Domino‟s Domino‟s
Each of the five rankings above is called a preference list (or preference schedule).
In a two-way runoff between Domino‟s and Papa John‟s,
Domino‟s votes = _______________________________________________________________
Papa John‟s votes = _______________________________________________________________
So ____________________ would win. (On a test, show who makes runoff and vote totals in runoff.)
So we see the methods of plurality and plurality with runoff yield different winners. In the plurality with
runoff method, if an alternative receives a majority of the first place votes, that alternative wins without a
runoff.
C. HARE SYSTEM: Suppose instead of eliminating all but the top two finishers, we eliminate one alternative
at a time until we are left with a winner. The contender eliminated at each stage is the one with the fewest
first place votes.
First stage: Godfather‟s is eliminated by virtue of having the fewest (6) first place votes. Since Godfather‟s
is no longer an option, we revise our preference list to exclude it.
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The second and fifth columns are now the same so we combine them into a single preference list with 16
votes.
18 votes 16 votes 10 votes 9 votes 2 votes
st
1 Domino‟s Papa John‟s Pizza Hut Little Caesars Pizza Hut
2nd Little Caesars Little Caesars Papa John‟s Pizza Hut Little Caesars
3rd Pizza Hut Pizza Hut Little Caesars Papa John‟s Papa John‟s
th
4 Papa John‟s Domino‟s Domino‟s Domino‟s Domino‟s
Now we again eliminate the alternative with the fewest first place votes: this time Little Caesars. The last
three columns are now identical so we combine them into a single preference list with 21 votes
18 votes 16 votes 21 votes
1st Domino‟s Papa John‟s Pizza Hut
2nd Pizza Hut Pizza Hut Papa John‟s
3rd Papa John‟s Domino‟s Domino‟s
Papa John‟s now has the fewest first place votes and is eliminated. The last two columns now become
the same leaving us with:
18 votes 37 votes
1st Domino‟s Pizza Hut
2nd Pizza Hut Dominos
So Pizza Hut wins. This is called the Hare System. (See Prof. Rosenthal‟s web site for biographical
information on Thomas Hare.) If there are only 3 alternatives in an election, the Hare System is the same
as Plurality with runoff.
.
D. BORDA COUNT METHOD: Here is another way we will study to determine the winner of our election.
Suppose we assign 4 points for each first place vote, 3 points for each second place vote, 2 points for
each third place vote, one point for each fourth place vote, and no points for each last place vote. Then
we total the points and the candidate with the most votes wins.
Domino‟s: ___________________________________________________________________
Little Caesars: ___________________________________________________________________
Papa John‟s: ___________________________________________________________________
Pizza Hut: ___________________________________________________________________
Godfather‟s: ___________________________________________________________________
So _________________ is the winner. This procedure is called the Borda Count method and sports
fans may recognize it as the method the Associated Press uses for ranking college football teams. (See
Prof. Rosenthal‟s web site for biographical information on Jean-Charles de Borda.) It is also used to choose
the winner of the Heisman Trophy, the Country Music Vocalist of the Year and in many hiring decisions.
E. SEQUENTIAL PAIRWISE VOTING: The final method we will study pairs up two contenders in a head-to-
head comparison and eliminates the loser. It then pairs the winner with the next contender and keeps
repeating the process until all but one option is eliminated. For example, let‟s start by comparing Domino‟s
with Little Caesars. The winner will then face Papa John‟s, and that winner will then face Pizza Hut. Finally,
the survivor goes against Godfather‟s.
Little Caesars vs Domino:
Little Caesars vs. Papa John‟s:
Little Caesars vs. Pizza Hut:
Little Caesars vs. Godfather‟s:
So _______________________ is the winner using the method called sequential pairwise voting.
Hopefully, you are now convinced that the voting system used is crucial to determining the winner. We
have applied five different voting systems to our election and gotten five different winners! In sequential
pairwise voting, the order in which we consider the five alternatives is called an agenda. The agenda we
just considered is D, LC, PJ, PH, G (which is the same as the agenda LC, D, PJ, PH, G). We will always
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list agendas horizontally to avoid confusing them with preference lists, which we will continue to list
vertically.
Examples depicting problems, paradoxes and desirable properties:
A. Condorcet winner:: Let‟s repeat the problem using the agenda PJ, G, D, PH, LC:
Solution: Godfather‟s vs. Papa John‟s: _____________
Godfather‟s vs. Domino‟s: _____________
Godfather‟s vs. Pizza Hut: _____________
Godfather‟s vs. Little Ceasars: _____________
This agenda shows that ________________ will win regardless of the agenda we use. When one
alternative beats every other alternative in head-to-head votes, that alternative is called the Condorcet
winner. (See Prof. Rosenthal‟s web site for biographical information on Marquis de Condorcet,
pronounced KON-door-say.) Not every election has a Condorcet winner, but when a Condorcet winner
occurs, that alternative would win under sequential pairwise voting.
B. Example: Sequential pairwise voting arises naturally in the legislative process. When the U.S. House of
Representatives considers a bill, an amendment to the bill can be offered. Two votes are then taken. The
first vote is whether to accept the amendment. If accepted, the second vote is between the amended bill
and no bill at all. If the amendment is defeated, the second vote is between the original bill and no bill at
all. In 1956 the House was considering a bill that would provide federal funding for the construction of
schools. An amendment was offered that would only provide the federal funding to states with integrated
school systems. The House could more or less be divided into three groups: Republicans, northern
Democrats, and southern Democrats. The Republicans generally opposed federal aid, but favored
integration. So their first choice was no bill, but they preferred the amended bill to the original bill.
Northern Democrats favored the federal aid and integration so their first choice was the amended bill, but
preferred the original bill to no bill. Southern Democrats, who came from states with segregated school
systems, wanted the aid, but abhorred the amendment. The three preference lists looked like this (with
the breakdown of Democrats an estimate):
Republicans (203) Northern Democrats (116) Southern Democrats (116)
1. No bill 1. Amended bill 1. Original bill
2. Amended bill 2. Original bill 2. No bill
3. Original bill 3. No bill 3. Amended bill
What was the outcome of the two votes?
Solution: On the vote on the amended bill vs. the original bill, __________________________________
______________________________________________. On the vote between the amended bill and no
bill, _____________________________________________________________________________.
What would have happened if no amendment had been introduced? _____________________________
Of what significance is this? Say you were a Republican and through polling done by your staff, you were
aware of the preference lists above. You could introduce the amendment, even though it is not truly your
first preference, to sway the outcome your way. Or, if you were a northern Democrat and were aware of
the preference lists, you would not introduce the amendment even though it is your first choice because it
would lead to your least preferred outcome!
C. Example: Is there a Condorcet winner in this election?
Solution: _________________________________________________________________________
_________________________________________________________________________
________________________________________________________________________
D. Condorcet Paradox: Since no alternative wins every pairwise match up, there is no Condorcet winner. In
mathematics, if a > b, and b > c, we can conclude that a > c. This is called the transitive property of
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inequality. One might think that the same holds true for voting. In fact, while we assume it is true for
individual preferences, is it true for society? Namely: If A is preferred to B and B is preferred to C, then A
would be preferred to C. The last example shows us this is not true. A paradox is a conclusion that is
counterintuitive. The fact that voting preferences are not transitive is called Condorcet’s voting paradox.
E. Pareto Condition: The three members of FIU‟s Student Entertainment Committee vote on which of four
movies will be shown in the Graham Center that week. The alternatives are Star Wars (SW), Dances with
Wolves (DW), Pearl Harbor (PH), and Jurassic Park (JP). The preference lists are as follows:
Juan Maria Sally
1. SW 1. JP 1. DW
2. DW 2. SW 2. PH
3. PH 3. DW 3. JP
4. JP 4. PH 4. SW
Use sequential pairwise voting with the following agendas:
a) PH, JP, DW, SW b) DW, SW, PH, JP c) JP, SW, DW, PH d) SW, DW, JP, PH
Solution: a) ________ b) ________ c) ________ d) ________
This example illustrates that the agenda used can often determine the winner in sequential pairwise
voting. In general, the later you bring up an item in an agenda, the better its chance of winning. The last
agenda we used shows another troubling characteristic of sequential pairwise voting. Pearl Harbor won
although every voter preferred Dances with Wolves to Pearl Harbor.
Here is our first desirable property of a voting system:
The Pareto Condition: If every voter prefers alternative A to alternative B, then alternative
B should not be the winner.
(See Prof. Rosenthal‟s web site for biographical information on Vilfredo Pareto, pronounced pare-ETT-o.)
Sequential pairwise voting does not satisfy the Pareto condition.
F. Condorcet Winner Criterion: A college has just received a donation of $1,000,000 with the only condition
being that it is all used for the same purpose. At a meeting of the Board of Trustees, four possible uses
are proposed: increasing financial aid (IFA), construction (C), increasing salaries (IS), and hiring new
faculty (HNF). The 26 trustees have the following preferences:
8 votes 5 votes 6 votes 7 votes
IFA C IS HNF
C IS C C
IS HNF HNF IS
HNF IFA IFA IFA
Which alternative prevails under the plurality method? Solution:_IFA with 8 votes
Recall we said earlier that a Condorcet winner is an alternative that beats every other winner in head-to-
head competition. Although, not every election has a Condorcet winner, this one does and it is
Construction. Here is our second desirable property of a voting system:
The Condorcet winner criterion: Either there is no Condorcet winner or, if there is, the
Condorcet winner wins the election.
Our example shows that Plurality voting does not satisfy the Condorcet winner criterion.
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G. Monotonicity: Three candidates run for president of the local chapter of their labor union. There are 100
voters with the following preference lists:
38 votes 30 votes 25 votes 7 votes
A C B B
B A C A
C B A C
The voting method is plurality with runoff. Who wins?
Solution: A wins with 68 votes.
Suppose, in her acceptance speech, A announces a change in one of her positions. This causes the 7 voters
in the last column to now prefer A to B. The first and last columns being the same, we combine them.
45 votes 30 votes 25 votes
A C B
B A C
C B A
Who would win now with the same voting method? A and C make the runoff.
Solution: C wins with 55 votes.
Here is our third desirable property of a voting system:
Monotonicity: If an election is held in which alternative A is the winner, and a second
ballot is held after a change that is favorable only to A, then A is still the winner.
Even though A won the first ballot, and the only change made was favorable to A, A now loses!
The method of plurality with runoff does not satisfy monotonicity. In mathematics, the word monotonic
generally means “same direction.” Our example showed plurality with runoff is not monotonic because
when A‟s vote went up, A‟s final ranking went down.
H. Majority Criterion:: The five members of a city commission have to determine where to build a new
baseball stadium. There are three proposed sites: next to the park (P), along the river (R), and at the
intersection (I) of the two main thoroughfares. The preference lists are:
3 votes 2 votes
P R
R I
I P
A Borda count is used. Who wins?
Solution: P: 2(3) + 0 = 6
R: 2(2) + 1(3) = 7
I: 1(2) = 2
So the River with 7 points is the winner. Do you have an objection to this outcome? (The park received
a majority of the votes and didn‟t win.)
Here is our fourth desirable property of a voting system:
Majority criterion: If an alternative receives a majority of the first-place votes, then
that alternative should win.
The Borda Count does not always satisfy the majority criterion.
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I. Independence of Irrelevant Alternatives (IIA): The Hare system was used by the International Olympic
Committee to choose the site of the 2000 Summer Olympic games. Eighty-eight voters took part in the
election that was won by Sydney, Australia. Suppose 88 voters have the following preference lists:
44 votes 22 votes 22 votes
Beijing Istanbul Manchester
Istanbul Manchester Istanbul
Manchester Beijing Beijing
Who wins under the Hare system?
Solution: Beijing wins with 44 votes
Suppose Manchester and Istanbul are switched on the last preference list. Who wins now under the Hare
system?
Solution: The preference lists now look like: 44 votes 44 votes
Beijing Istanbul
Istanbul Manchester
Manchester Beijing
Now the Hare system yields a tie. Here Istanbul went from losing to a tie even though no voter reversed
his or her ranking of Istanbul and Beijing. Here is our fifth desirable property of a voting system:
Independence of irrelevant alternatives (IIA): If alternative A goes from being a
loser of an election to one of the winners of a new election, then at least one voter
had to reverse his or her ranking of A and the first winner.
The Hare system does not satisfy IIA. An irrelevant alternative refers to non-winning alternatives.
Independence means „not affected by.” So IIA means the outcome should not be affected by a change
involving non-winners.
J. Arrow‟s Impossibility Theorem: So each of the five voting methods we have studied has at least one
shortcoming. That raises the question: Is there a voting system that possesses all of the desirable
properties? In 1951, economist Kenneth Arrow proved that the answer to this question is no. This result,
known as Arrow’s Impossibility Theorem, was one of the reasons he won the Nobel Prize in Economics in
1972. (See my web site for biographical information on Arrow.) One of the reasons that Arrow‟s Theorem
is true is that it is very hard to satisfy IIA. In fact, none of the voting systems we have studied satisfy IIA.
Mathematical problem involving Borda Count and Condorcet: Consider the following preference lists:
12 votes 10 votes 8 votes x votes
A B C C
D D B D
C A D B
B C A A
A. Find all values of x that make D the Borda count winner but not the Condorcet winner. X x > 14
D vs. A: _______________________ D vs. B: __________________________________________
D vs. C: ________________________
Final answer: x = 1 , 2, 3, 4, 5, 15
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B. In an election with 5 alternatives and 20 voters, what is the sum of the Borda points?
Solution: (4 + 3 + 2 +1 +0)20 = (10)(20) = 200
Explain why certain methods satisfy certain conditions:
A. Plurality vs. Pareto Condition: In a few sentences, explain why plurality with runoff satisfies the Pareto
condition. Any time you are asked to show that a voting system satisfies the Pareto condition, you will
start your answer by assuming everyone prefers A to B. Then you will have to explain why B can‟t win.
Solution: Assume that everyone prefers A to B. B cannot have any first place votes; therefore, B
cannot make the runoff. Thus, B cannot win.
B. Borda vs. Monotonicity: In a few sentences, explain why the Borda count satisfies monotonicity.
Any time you are asked to show that a voting system satisfies monotonicity, start by assuming A wins.
make one change, moving A above a losing alternative. Then explain why this losing alternative still
loses.
Solution: Suppose A is a winner under Borda count. Suppose the only change is that one voter
puts A above B. This will increase A’s points and decrease B’s. All other alternatives remain the
same. So A still wins.
Strategic or Insincere Voting: Suppose 100 voters are asked to rank 3 football teams: Miami (UM), Florida
State (FSU), and Nebraska (N). Here are the preference lists:
52 votes 38 votes 10 votes
UM FSU N
FSU UM FSU
N N UM
Who wins using a Borda count?
Solution: UM: 2(52) + 1(38) +0 = 142
FSU: ___________________________________________________________________
N: ___________________________________________________________________
Suppose the 38 voters in the middle decide to vote against their true beliefs and flip UM and N. Now who
wins using the Borda count?
Solution: UM: ___________________________________________________________________
FSU: ___________________________________________________________________
N: ___________________________________________________________________
This is an example of strategic (or insincere) voting. By voting in a manner different to their true beliefs, the
voters who favored FSU were able to manipulate the voting so that their team won. Sincere voting is when all
voters vote according to their actual preferences. Insincere voters can manipulate all of the voting methods we
have studied.
Approval Voting: Arrow proved that there is no perfect voting system. That doesn‟t mean that there might
not be better ones. Some political scientists are advocating approval voting. Rather than ranking the
candidates, voters may vote for as many alternatives as they wish. For every candidate, the voter either marks
yes, he or she approves of the candidate or no, he or she does not approve of the candidate. The candidate
with the most yes votes wins.
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A. Example: Six candidates (A, B, C, D, E, and F) are competing for a spot on the board of directors of a
company. The current nine members each vote for as many candidates as they wish using approval
voting. An X indicates a vote of approval.
A X X X X X X
B X X X
C X X
D X X
E X X X X X X X X
F X X
Which candidate gets the vacant board seat?
Solution: ____________________________________________________________________________
B. Discussion: Approval voting is most effective in races with many candidates. For that reason, some
political scientists advocate approval voting for presidential primaries. They feel voters are more likely to
vote when the decision to be made is easier. Expressing whether or not you approve of a candidate is
often simpler than having to rank candidates. Ranking candidates requires more extensive knowledge on
the part of the voter, knowledge voters are often lacking. An advantage of approval voting is that it usually
results in a winner acceptable to a large proportion of the electorate. A disadvantage is that it doesn‟t
allow the voter to distinguish between enthusiastic support for a candidate and grudging approval.
C. Example: The Baseball Writers Association of America polls its membership each year to elect new
members to the baseball Hall of Fame. In 1999, 497 writers cast a ballot. Writers can vote for as many
players as they feel are deserving of enshrinement. To be elected, a player must be listed on at least 75% of
the ballots. Here are the top six finishers in the 1999 balloting along with the number of votes each
received.
Nolan Ryan 491
George Brett 488
Robin Yount 385
Carlton Fisk 330
Tony Perez 302
Gary Carter 168
a) Which ones were elected to the Hall?
b) What percent of voters who did not vote for Carter would have had to change and vote for him in order
for Carter to be elected?
Solution: a) __________________________________________________________________________
____________________________________________________________________________________
____________________________________________________________________________________
b) __________________________________________________________________________________
____________________________________________________________________________________
____________________________________________________________________________________
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