Introduction to the Mathematics of Voting
Name:
Period:
Date:
Election Theory: Guided Notes
Introduction to the Mathematics of Voting
Voting theory is the mathematical treatment of the process by which democratic societies or groups resolve the many and conflicting opinions of the members of the group into a single choice of the group. A vote is an expression of a voter's preference about the outcome of an election.
Why do we need amathematical theoryabout something so simple as voting?
How difficult could it be to find a simple, fair, and consistent procedure for determining the outcome of an election?
Actually, when an election involves only 2 candidates (or alternatives), then the situationisas simple as you might have imagined.
For instance, suppose there is an election between Janice and Betty for senior class president.
How should the election be set up so that the result fairly expresses the wishes of the senior class?Click here to see.
The situation is very different, however, when an election involves more than two candidates or alternatives and we wish to rank each of them in order of preference (preferential voting). Mathematical economist Kenneth Arrow proved (in 1952) thatthere is NO consistent method of making a fair choice among three or more candidates with preferential voting. This remarkable result assures us that there is no single preferential election procedure that can always fairly decide the outcome of an election that involves more than two candidates or alternatives.
Fundamental Terms and Ideas
Ballot
______
______
Majority
______
______
Plurality
______
In a 3 candidate election, candidate A gets 14 votes, candidate B gets 20 votes, and candidate C gets 16 votes (total of 50 voters). Candidate B has a plurality of the votes (but, in this case, not a majority of the votes).
Preference Ballot
______Below is the preference ballot of an individual voter who voted for candidate B for 1st place, candidate C for 2nd place, candidate A for 3rd place, and candidate D for 4th place.
1st : B
2nd: C
3rd: A
4th: D
Notice that a preference ballot is different from a typical U.S. election ballot. On a typical U.S. election ballot, the voter only votes for his/her first choice. However, a preference ballotalwaysgives a complete ranking of the candidates from first to last place.
Suppose a 10 person committee elects a chairman by first casting preference ballots. (Actually only the 6 members who are not running for chairman get to vote.) Following is a listing of the 6 preference ballots (G=George, H=Holly, I=Inez, J=James):
Each of the 6 columns in the body of the table (the unshaded part) represents a preference ballot for a voter. For instance, Carrie's preference ballot is G, H, J, I (meaning that Carrie ranked George first, Holly second, James third, and Inez fourth).
Place / Al / Betty / Carrie / Dave / Edith / Frank
1st / G / I / G / G / J / I
2nd / H / H / H / H / G / H
3rd / J / J / J / J / H / J
4th / I / G / I / I / I / G
THIS IS A SET OF 6PREFERENCE BALLOTS.
Preference Schedule
A preference schedule is a table which summarizes the results of all the individual preference ballots for an election.
Look back at the 6 preference ballots in the example just above. Notice that3voters (Al, Carrie, and Dave) cast identical preference ballots (G, H, J, I),2other voters (Betty and Frank) cast the same preference ballot (I, H, J, G) and1voter (Edith) cast the preference ballot (J, G, H, I).
A preference SCHEDULE summarizes the election results as follows:
Place / 3 / 2 / 1
1st / G / I / J
2nd / H / H / G
3rd / J / J / H
4th / I / G / I
THIS IS APREFERENCE SCHEDULE.
Notice that the preference schedule compactly summarizes the votes in the election by grouping identical preference ballots together along with a count of how many voters cast each identical preference ballot.
Fairness Criteria
What do we mean byfair?
Over the years, those who study voting theory have proposed numerous criteria which most people would expect a 'fair' preferential election method to satisfy. In this course, we will consider fourFairness Criteria. (There are others but we will only consider these four.)
4 Fairness Criteria
The Majority Criterion
The Condorcet Criterion
The Monotonicity Criterion
The Independence of Irrelevant Alternatives Criterion.
For many years, mathematicians and others interested in voting theory searched for a preferential voting procedure that would satisfy a reasonable set of fairness criteria (such as the set of 4 above). Arrow's work in 1952 (mentioned previously) abruptly ended these efforts when he showed that such a search was in vain. In terms of the 4 fairness criteria above, Arrow's result means that there is NO consisent preferential voting method that can satisfy all four of them.
Thus, for example, a preferential procedure designed to satisfy the Majority Criterion will always violate at least one of the other criteria. This, in turn, means that there is no 'perfect' preferential voting procedure and the decision about the procedure to be used is, by necessity, subjective. The best one can hope for is to be able to objectively analyze the strengths and weaknesses of various preferential procedures and to apply that knowledge to the task of selecting a 'good' one for a particular situation.
The Majority Criterion
______
In other words, it would seem unfair to most people if Candidate A got 51 first place votes, Candidate B got 40 first place votes, and Candidate C got 9 first place votes (100 votes in all) but Candidate B was declared the winner of the election. Such an outcome would violate the Majority Criterion.
Back to the 4 Fairness Criteria
The Condorcet Criterion
(named after theMarquis de Condorcet)
______
Suppose 4 candidates A, B, C, and D run for mayor of a small town (a very small town!). There are 20 registered voters. The local newspaper performed a post-election survey of each of the 20 registered voters. Among other things, the survey asked the voters who they preferred in a two-way race between candidate C (the one endorsed by the paper's editorial staff) and each of the other candidates. Here are the results:
11 voters preferred candidate C over candidate A
11 voters preferred candidate C over candidate B
17 voters preferred candidate C over candidate D
So, in head-to-head competition, candidate C won against each of the other candidates.
Wouldn't it seem unfair if candidate C was not declared the winner?
When the actual votes were tabulated, candidate A got 9 first place votes, candidate B got no first place votes, candidate C got 8 first place votes, and candidate D got 3 first place votes. If candidate C is not declared the winner, this would be a violation of the Condorcet Criterion.
Back to the 4 Fairness Criteria
The Monotonicity Criterion
______
Three students, Al, Bob, and Carrie are running for class president. The class will vote in rounds. The student with the fewest votes in the first round will drop out and a new vote will be taken between the two remaining candidates. The student with the most votes in this final round will be declared the winner of the election. In the first round, Al gets 11 first place votes, Bob gets 8 first place votes, and Carrie gets 10 first place votes. Bob drops out (since he had the fewest votes in the first round). In the final round, Al gets 11 first place votes and Carrie gets 18 first place votes. Carrie wins the election!
But wait! The chairman of the election oversight committee destroyed the ballots before the results had been officially certified by the administration. You guessed it! The election had to be repeated. In the first round of the repeated election everyone voted exactly as in the first round of the original election except for 4 voters who decided to jump on the bandwagon and vote for Carrie instead of Al. As a result, Al gets 7 first place votes, Bob gets 8 first place votes, and Carrie gets 14 first place votes. This causes Al to drop out instead of Bob so that the final round of the repeated election is between Bob and Carrie. BUT the 7 students who originally voted for Al prefer Bob over Carrie. So, all 7 of them cast their votes for Bob in the final round. This gives Bob 15 votes and Carrie 14 votes. Bob wins the repeated election EVEN THOUGH THE ONLY CHANGES IN VOTER PREFERENCE WERE THE 4 VOTES THAT CHANGED FROM AL TO CARRIE (THE ORIGINAL WINNER). This illustrates a violation of the Monotonicity Criterion.
Back to the 4 Fairness Criteria
The Independence of Irrelevant Alternatives Criterion
______
As a publicity stunt for its soon-to-be-published cookbook, the Culinary Club of Smallville decided to have a 'Best Pie' contest. The entries were narrowed down to three for the final round of the contest. In this final round, each club member ranked the 3 pies. Each first place vote is worth 3 points, each second place vote is worth 2 points, and each third place vote is worth 1 point. Here is a summary of the results:
27 members ranked Al's Apple Pie first, Chris' Cream Pie second, and Pat's Peach Pie third;
24 members ranked Pat's Peach Pie first, Chris' Cream Pie second, and Al's Apple Pie third;
2 members ranked Chris' Cream Pie first, Pat's Peach Pie second, and Al's Apple Pie third.
Based on the rules above,
Al's Apple Pie got 27 first place votes at 3 points each (81 points) and 26 third place votes at 1 point each (26 points) for a grand total of 107 points.
Chris' Cream Pie got 2 first place votes at 3 points each (6 points) and 51 second place votes at 2 points each (102 points) for a grand total of 108 points.
Pat's Peach Pie got 24 first place votes at 3 points each (72 points), 2 second place votes at 2 points each (4 points), and 27 third place votes at 1 point each (27 points) for a grand total of 103 points
Thus, Chris' Cream Pie gets first place (108 points), Al's Apple Pie gets second place (107 points), and Pat's Peach Pie gets third place (103 points).
Before the results can be publicized, Pat (who is upset about a third place finish) demands that her Peach Pie entry be retroactively withdrawn from the contest. Bowing to her wishes, the club removes the Peach Pie and recalculates points (2 points for each first place, 1 point for each second place).
Since Pat's Peach Pie is now out, the rankings are:
27 members ranked Al's Apple Pie first, Chris' Cream Pie second, andPat's Peach Pie third;
24 members rankedPat's Peach Pie first, Chris' Cream Piesecondfirst, and Al's Apple Piethirdsecond;
2 members ranked Chris' Cream Pie first,Pat's Peach Pie second, and Al's Apple Piethirdsecond.
The new results are,
Al's Apple Pie gets 27 first place votes at 2 points each (54 points) and 26 second place votes at 1 point each (26 points) for a grand total of 80 points;
Chris' Cream Pie gets 26 first place votes at 2 points each (52 points) and 27 second place votes at 1 point each (27 points) for a grand total of 79 points.
Thus, Al's Apple Pie gets first place (80 points), and Chris' Cream Pie gets second place (79 points).
Notice what happened. Because a loser (Pat) dropped out, the winner changed (from Chris to Al). This is a violation of the the Independence of Irrelevant Alternatives Criterion.
Voting Methods
4 Preferential Voting Methods
Preferential voting methods are those methods that use information from apreference schedule(a ranking of candidates in order of preference).
The Plurality Method
The Method of Plurality with Elimination
The Borda Count Method
The Method of Pairwise Comparisons
The Plurality Method
For the Plurality Method, the candidate with the most first place votes wins. The winner does NOT have to receive a majority of the first place votes!
The Plurality Method is the election method with which many of us are most familiar.
The mayor of Smallville is being chosen in an election using the Plurality Method. The four candidates are Paul (the town barber), Rita (head of the town council), Sarah (Superintendent of Education), and Tim (former District Attorney).
500 registered voters cast their preference ballots. The results are summarized in the preference schedule below.
# of / VotersPlace / 130 / 120 / 100 / 150
1st / P / T / T / S
2nd / R / R / R / R
3rd / S / S / P / P
4th / T / P / S / T
How many voters cast preference ballots in this election?answer
How many first place votes are needed for a majority?answer
Did any candidate receive a majority of first place votes?answer
Who is the winner by the Plurality Method?answer
Thepreference schedule below represents an election among George (G), Holly (H), James (J), and Inez (I).# of Voters
Place / 13 / 12 / 10
1st / G / I / J
2nd / H / H / G
3rd / J / J / H
4th / I / G / I
How many voters cast preference ballots in this election?answer
How many first place votes are needed for a majority?answer
Did any candidate receive a majority of first place votes?answer
Who is the winner by the Plurality Method?answer
Back
The Method of Plurality with Elimination
This method is a preferential voting method and is a variation of the Plurality Method.Plurality with Elimination is carried out in rounds. After each round of voting the candidate (or alternative) with the fewest first place votes is eliminated and a new round of voting is done with the remaining candidates. When only two candidates remain in a round, the candidate with the most votes wins the election. For anNcandidate election, the Method of Plurality with Elimination requiresN-1rounds.
The mayor of Smallville is being chosen in a Plurality with Elimination election. The four candidates are Paul (the town barber), Rita (head of the town council), Sarah (Superintendent of Education), and Tim (former District Attorney).
How many rounds will it take to determine a winner?answer
500 registered voters cast their preference ballots. The results are summarized in the preference schedule below.
# of / Voters
Place / 130 / 120 / 100 / 150
1st / P / T / T / S
2nd / R / R / R / R
3rd / S / S / P / P
4th / T / P / S / T
Using the Method of Plurality with Elimination, which candidate is eliminated in the first round?answer
Since Round 1 eliminated a candidate, Round 2 begins with the 3 remaining candidates.
IMPORTANT POINT!!
Since each voter completed a preference ballot (all of which are summarized in the preference schedule above), we do not have to hold a new election. We simply remove the eliminated candidate from the preference schedule and adjust the remaining candidates appropriately. For instance, if a column of a preference schedule was B (1st), A (2nd), C (3rd), D (4th) and candidate A got eliminated, the column would be B (1st), C (2nd), D (3rd) in the preference schedule for the next round. In other words, for each column of a preference schedule, each candidate below an eliminated candidate moves up one place while the positions of candidates above an eliminated candidate remain unchanged.
Thepreference schedule below represents an election among George (G), Holly (H), James (J), and Inez (I).
How many rounds will it take to determine a winner using the Method of Plurality with Elimination?answer
# of Voters
Place / 13 / 12 / 10
1st / G / I / J
2nd / H / H / G
3rd / J / J / H
4th / I / G / I
Using the Method of Plurality with Elimination, which candidate is eliminated in the first round?answer
Since Round 1 eliminated a candidate, Round 2 begins with the 3 remaining candidates.
IMPORTANT POINT!!
Since each voter completed a preference ballot (all of which are summarized in the preference schedule above), we do not have to hold a new election. We simply remove the eliminated candidate from the preference schedule and adjust the remaining candidates appropriately. For instance, if a column of a preference schedule was B (1st), A (2nd), C (3rd), D (4th) and candidate A got eliminated, the column would be B (1st), C (2nd), D (3rd) in the preference schedule for the next round. In other words, for each column of a preference schedule, each candidate below an eliminated candidate moves up one place while the positions of candidates above an eliminated candidate remain unchanged.
Write out the preference schedule for Round 2 (the round with the 3 candidates left after Round 1).answer
Which candidate is eliminated in the second round?answer
Write out the preference schedule for Round 3 (the round with the 2 candidates left after Round 2).answer
Which candidate is eliminated in the third (and final) round?answer
Who wins the election using the Method of Plurality with Elimination?answer
The Borda Count Method
For the Borda Count Method, each candidate (or alternative) gets 1 point for each last place vote received, 2 points for each next-to-last point vote, etc., all the way up toNpoints for each first place vote (whereNis the number of candidates/alternatives). The candidate with the largest point total wins the election.For instance, in a 4 candidate election, each 4th place vote is worth 1 point, each 3rd place vote is worth 2 points, each 2nd place vote is worth 3 points, and each 1st place vote is worth 4 points.
The Borda Count Method, or some variation of it, is often used for things like polls which rank sporting teams or academic institutions.
The mayor of Smallville is being chosen in an election using the Borda Count Method. The four candidates are Paul (the town barber), Rita (head of the town council), Sarah (Superintendent of Education), and Tim (former District Attorney).
500 registered voters cast their preference ballots. The results are summarized in the preference schedule below.
# of / Voters
Place / 130 / 120 / 100 / 150
1st / P / T / T / S
2nd / R / R / R / R
3rd / S / S / P / P
4th / T / P / S / T
There are various ways of performing the calculations needed to determine the winner of a Borda Count election. Perhaps the most straightforward way is to set up a 'calculation template' and then insert the appropriate numbers from the preference schedule. For instance, the 'calculation template' for the election above would be:
Paul: / 4( / ) + 3( / ) + 2( / ) + 1( / ) =
Rita: / 4( / ) + 3( / ) + 2( / ) + 1( / ) =
Sarah: / 4( / ) + 3( / ) + 2( / ) + 1( / ) =
Tim: / 4( / ) + 3( / ) + 2( / ) + 1( / ) =
To see why this works, notice that:
the column with '4( ___ )' is where each candidate's 1st place votes are entered
(for a 4 candidate election, each 1st place vote is worth 4 pts);
the column with '3( ___ )' is where each candidate's 2nd place votes are entered
(for a 4 candidate election, each 1st place vote is worth 3 pts);
the column with '2( ___ )' is where each candidate's 3rd place votes are entered
(for a 4 candidate election, each 3rd place vote is worth 2 pts);
the column with '1( ___ )' is where each candidate's 4th place votes are entered
(for a 4 candidate election, each 4th place vote is worth 1 pt).
Adding all the resulting products gives us the Borda Count total for each candidate.
Usng this 'template' technique, the first step is to write out the 'calculation template' based on the number of candidates in the election. Next, fill in the blanks and, finally, do the arithmetic.
Let's complete the work for the Borda Count election of Smallville's mayor.
FILL IN THE BLANKS IN THE CALCULATION TEMPLATE.
- Determinethe number of first place votes for each candidate and place these numbers appropriately in the 'calculation template'.answer
- Determinethe number of second place votes for each candidate and place these numbers appropriately in the 'calculation template'.answer
- Determinethe number of third place votes for each candidate and place these numbers appropriately in the 'calculation template'.answer
- Determinethe number of fourth place votes for each candidate and place these numbers appropriately in the 'calculation templete'.answer
Who wins the election using the Borda Count Method?answer
Thepreference schedule below represents an election among George (G), Holly (H), James (J), and Inez (I).