Introduction to the Mathematics of Voting

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Election Theory: Guided Notes

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 a mathematical theory about 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 situation is as 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) that there 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 ballot always gives 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). Voters 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 6 PREFERENCE 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 that 3 voters (Al, Carrie, and Dave) cast identical preference ballots (G, H, J, I), 2 other voters (Betty and Frank) cast the same preference ballot (I, H, J, G) and 1 voter (Edith) cast the preference ballot (J, G, H, I). A preference SCHEDULE summarizes the election results as follows: # of Voters

Place 3 2 1 1st G I J 2nd H H G 3rd J J H 4th I G I

THIS IS A PREFERENCE 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 by fair? 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 four Fairness 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 the Marquis 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, and Pat's Peach Pie third; 24 members ranked Pat's Peach Pie first, Chris' Cream Pie second first, and Al's Apple Pie third second; 2 members ranked Chris' Cream Pie first, Pat's Peach Pie second, and Al's Apple Pie third second. 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 a preference 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 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

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

The preference 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 an N candidate election, the Method of Plurality with Elimination requires N-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. The preference 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 to N points for each first place vote (where N is 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.  Determine the number of first place votes for each candidate and place these numbers appropriately in the 'calculation template'.answer  Determine the number of second place votes for each candidate and place these numbers appropriately in the 'calculation template'. answer  Determine the number of third place votes for each candidate and place these numbers appropriately in the 'calculation template'.answer  Determine the number of fourth place votes for each candidate and place these numbers appropriately in the 'calculation templete'.answer DO THE ARITHMETIC. answer Who wins the election using the Borda Count Method? answer The preference 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 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: George: 4( ) + 3( ) + 2( ) + 1( ) = Holly: 4( ) + 3( ) + 2( ) + 1( ) = James: 4( ) + 3( ) + 2( ) + 1( ) = Inez: 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). Aadding 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 this Borda Count election. FILL IN THE BLANKS IN THE CALCULATION TEMPLATE.  Determine the number of first place votes for each candidate and place these numbers appropriately in the 'calculation template'. answer  Determine the number of second place votes for each candidate and place these numbers appropriately in the 'calculation template'. answer  Determine the number of third place votes for each candidate and place these numbers appropriately in the 'calculation template'. answer  Determine the number of fourth place votes for each candidate and place these numbers appropriately in the 'calculation templete'. answer DO THE ARITHMETIC. answer Who wins the election using the Borda Count Method? answer Back The Method of Pairwise Comparisons For the Method of Pairwise Comparisons, each candidate (or alternative) is matched head-to-head (one-on-one) with each of the other candidates. Each candidate (alternative) gets 1 point for a one-on- one win and a half a point for a tie. The candidate (alternative) with the most total points is the winner. For instance, in a 4 candidate election, suppose candidate A beats candidates B and D head-to-head, candidate B beats candidate D and ties candidate C head-to-head, candidate C beats candidate A and ties candidate B head-to-head, and candidate D beats candidate C head-to- head. Candidate A would get 2 points (1 each for beating B and D), candidate B would get 1.5 points (1 point for beating D and half a point for the tie with C), candidate C would get 1.5 points (1 point for beating A and half a point for the tie with B), and candidate D would get 1 point (for beating C). Since candidate A has the highest point total, candidate A is the winner by the Method of Pairwise Comparisons. The Method of Pairwise Comparisons was explicitly designed to satisfy the fairness criterion called the Condorcet Criterion. The Condorcet Criterion addresses the fairness of declaring a candidate the winner even though some other candidate won all possible head-to-head matchups. With the Method of Pairwise Comparisons, any candidate who wins all possible head-to-head matchups always has a higher point total than any other candidate and thus is declared the winner. The mayor of Smallville is being chosen in an election using the Method of Pairwise Comparisons. 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 The Method of Pairwise Comparisons revolves around head-to-head matchups so, to begin, we need an organized way of displaying all possible head-to-head matchups. Perhaps the simplest way to do this is with a table giving each candidate a row and a column. Paul Rita Sarah Tim

Tim This table gives us a place to list the winner of each head-to-head matchup. For instance, to record the winner of the head-to-head matchup between Paul and Rita, just go to the Paul row and move over to the Rita column. Paul Rita Sarah Tim

Paul *

Tim Of course, there are certain cells in the table that are not needed. For example, the winner of the head-to-head matchup between Paul and Rita could also have been recorded by going to the Rita row and moving over to the Paul column. Paul Rita Sarah Tim

Rita *

Tim Since the cell in the Paul row and Rita column represents the same matchup as the cell in the Rita row and Paul column, we can just cross one of these cells out. And, since this is true for every row, column pair, we can just cross out all the cells below (or above) the main diagonal (the cells where the row and column names are the same: Paul row, Paul column; Rita row, Rita column, etc.). Paul Rita Sarah Tim

------Rita --

------Sarah -- --

------Tim ------We can also eliminate all the cells on the main diagonal since none of the candidates compete against themselves (no Sarah versus Sarah matchup).

Paul Rita Sarah Tim

------Paul --

------Rita -- --

------Sarah ------

------Tim ------The remaining 6 cells (for the 4 candidate case) can be used to record all the possible head-to-head winners. As a bonus, we can also look at the number of cells above the main diagonal to find out how many head-to-head matchups are possible. Since there are 6 cells above the main diagonal, we know that there are 6 possible head-to-head matchups. Formula for the total number of head-to-head matchups Now let's fill in the 6 empty cells in the table above. The first empty cell is for the winner of the head-to-head matchup between Paul and Rita. To determine this winner, we must go back to the preference schedule and see how many voters prefer Paul to Rita and how many prefer Rita to Paul. Since there are 500 voters, a candidate must get at least 251 votes (a majority) to win a head-to-head matchup. Remember, we are dealing with the head-to-head matchup between Paul and Rita so we only need to look at Paul and Rita. # 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 Who wins the head-to-head matchup between Paul and Rita? answer

Now we move to the head-to-head matchup between Paul and Sarah. # 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 Who wins the head-to-head matchup between Paul and Sarah? answer

Now we move to the head-to-head matchup between Paul and Tim. # 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 Who wins the head-to-head matchup between Paul and Tim? answer

Next we move to the head-to-head matchup between Rita and Sarah. # 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 Who wins the head-to-head matchup between Rita and Sarah? answer

Next we move to the head-to-head matchup between Rita and Tim. # 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 Who wins the head-to-head matchup between Rita and Tim? answer

Finally, we move to the head-to-head matchup between Sarah and Tim. # 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 Who wins the head-to-head matchup between Sarah and Tim? answer

The completed head-to-head matchup table is shown below. Paul Rita Sarah Tim

------Paul RITA SARAH PAUL ----

------Rita RITA RITA ------

------Sarah ------SARAH ------

------Tim ------

Now that we know the 6 head-to-head winners, we can total up points and declare the winner of the election. Remember, 1 point for a win, half a point for a tie. What is the point total for Rita? answer What is the point total for Sarah? answer What is the point total for Paul? answer What is the point total for Tim? answer Who wins the election using the Method of Pairwise Comparisons? answer The preference 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 The Method of Pairwise Comparisons revolves around head-to-head matchups so, to begin, we need an organized way of displaying all possible head-to-head matchups. Perhaps the simplest way to do this is with a table giving each candidate a row and a column. Feorge Holly James Inez

George

Inez This table gives us a place to list the winner of each head-to-head matchup. For instance, to record the winner of the head-to-head matchup between Holly and James, just go to the Holly row and move over to the James column. George Holly James Inez

George

Holly *

Inez Of course, there are certain cells in the table that are not needed. For example, the winner of the head-to-head matchup between Holly and James could also have been recorded by going to the James row and moving over to the Holly column. George Holly James Inez

George Holly

James *

Inez Since the cell in the Holly row and James column represents the same matchup as the cell in the James row and Holly column, we can just cross one of these cells out. And, since this is true for every row, column pair, we can just cross out all the cells below (or above) the main diagonal (the cells where the row and column names are the same:George row, George column; Inez row, Inez column, etc.). George Holly James Inez

George

------Holly -

------James - ---

------Inez ------We can also eliminate all the cells on the main diagonal since none of the candidates compete against themselves (no Holly versus Holly matchup). George Holly James Inez

------George -

------Holly - ---

------James ------

------Inez ------The remaining 6 cells (for the 4 candidate case) can be used to record all the possible head-to-head winners. As a bonus, we can also look at the number of cells above the main diagonal to find out how many head-to-head matchups are possible. Since there are 6 cells above the main diagonal, we know that there are 6 possible head-to-head matchups. Formula for the total number of head-to-head matchups Now let's fill in the 6 empty cells in the table above. The first empty cell is for the winner of the head-to-head matchup between George and Holly. To determine this winner, we must go back to the preference schedule and see how many voters prefer George to Holly and how many prefer Holly to George. Since there are 35 voters, a candidate must get at least 18 votes (a majority) to win a head-to-head matchup. Remember, we are dealing with the head-to-head matchup between George and Holly so we only need to look at George and Holly. # of Voters Place 13 12 10 1st G I J 2nd H H G 3rd J J H 4th I G I Who wins the head-to-head matchup between George and Holly? answer

Now we move to the head-to-head matchup between George and James. # of Voters Place 13 12 10 1st G I J 2nd H H G 3rd J J H 4th I G I Who wins the head-to-head matchup between George and James? answer

Now we move to the head-to-head matchup between George and Inez. # of Voters Place 13 12 10 1st G I J 2nd H H G 3rd J J H 4th I G I Who wins the head-to-head matchup between George and Inez? answer

Next we move to the head-to-head matchup between Holly and James. # of Voters Place 13 12 10 1st G I J 2nd H H G 3rd J J H 4th I G I Who wins the head-to-head matchup between Holly and James? answer

Next we move to the head-to-head matchup between Holly and Inez. # of Voters Place 13 12 10 1st G I J 2nd H H G 3rd J J H 4th I G I Who wins the head-to-head matchup between Holly and Inez? answer

Finally, we move to the head-to-head matchup between James and Inez. # of Voters Place 13 12 10 1st G I J 2nd H H G 3rd J J H 4th I G I Who wins the head-to-head matchup between James and Inez? answer

The completed head-to-head matchup table is shown below. George Holly James Inez

------George GEORGE JAMES GEORGE ------Holly ------HOLLY HOLLY -

------James ------JAMES -

------Inez ------

Now that we know the 6 head-to-head winners, we can total up points and declare the winner of the election. Remember, 1 point for a win, half a point for a tie. What is the point total for George? answer What is the point total for James? answer What is the point total for Holly? answer What is the point total for Inez? answer Who wins the election using the Method of Pairwise Comparisons? answer

The examples above deal only with 4 candidate elections and require 6 head-to-head matchups to determine a Pairwise Comparisons winner. Using the formula for the total number of head-to-head matchups, N(N- 1)/2 (where N is the number of candidates), we can see that the number of candidates in an election tells us immediately how many head-to- head matchups we will need to do in order to determine the winner of a Pairwise Comparisons election. For instance, a 3 candidate Pairwise Comparisons election requires 3 [3(3-1)/2 = 3(2)/2 = 6/2 = 3] head-to- head matchups but a 5 candidate Pairwise Comparisons election requires 10 [5(5-1)/2 = 5(4)/2 = 20/2 = 10] head-to-head matchups. List all the head-to-head matchups in a Pairwise Comparisons election among candidates Abbott, Barnes, and Collins. answer List all the head-to-head matchups in a Pairwise Comparisons election among candidates Devers, Evans, Franklin, and Grant. answer List all the head-to-head matchups in a Pairwise Comparisons election among candidates Henry, Iverson, Jenkins, Kent, and Lawson.answer List all the head-to-head matchups in a Pairwise Comparisons election among candidates May, Nash, Overton, Pickens, Queen, and Rogers. answer The Method of Pairwise Comparisons uses all the information from the preference table BUT not all at once. As each pair of candidates is matched up, the information about the remaining candidates is ignored. In this regard, the Method of Pairwise Comparisons is similar to the Method of Plurality with Elimination. With the Method of Plurality with Elimination, only the first place votes are used in each round. However, since lower order preferences (2nd, 3rd, etc.) can make it onto the first place row in later rounds, the Plurality with Elimination Method uses much more of the information contained in the preference schedule than does thePlurality Method. When using the Plurality Method, all of the information in the preference schedule not related to first place is ignored. This is a powerful theoretical argument in favor of the Method of Plurality with Elimination and the Method of Pairwise Comparisons over the Plurality Method. Of course, of the 4 preferential voting methods covered here, the Borda Count Method is the only one that uses all the information in the preference table simultaneously

Approval Voting V. Vitols, Anaheim Hills, Calif. posed a question to Scientific American Magazine: Approval Voting is a voting procedure in which voters can vote for as many candidates as they wish. Each candidate approved of receives one vote and the candidate with the most votes wins. It was independently proposed by several people in the 1970s. In the United States, the case for Approval Voting seems particularly strong in primary and nonpartisan elections which often draw large fields of candidates. Unlike more complicated ranking systems, Approval Voting is simple for voters to understand and use. Approval Voting is used today by various governments and organizations around the world (including its use by the United Nations to elect the secretary-general). And, lest you get lost in the details, keep in mind that almost everyone agrees that the traditional single-vote plurality system does the worst job of picking the best candidate. Approval Voting has several compelling advantages over other voting procedures:  It gives voters more flexible options  It helps elect the strongest candidate  It reduces negative campaigning  It increases voter turnout  It gives minority candidates their proper due  It is very practical First period high school physical education class has decided to take a vote to determine which activity they will take up next in PE. Their choices are aerobics (A), badminton (B), football (F), or softball (S). The winning activity will be determined using Approval Voting. Suppose the following summarizes the responses of the class: 12 students vote for Aerobics and Badminton 5 students vote for Badminton, Football, and Softball 10 students vote for Football and Badminton 13 students vote for Softball and Badminton To determine the winning activity, we must calculate the number of votes received by each. The activity with the most votes wins. How many total votes did AEROBICS receive? answer How many total votes did BADMINTON receive? answer How many total votes did FOOTBALL receive? answer How many total votes did SOFTBALL recieve? answer Which activity is selected by first period using Approval Voting? answer

Second period class votes to choose from among the same 4 activities (AEROBICS, BADMINTON, FOOTBALL, SOFTBALL) but decides to use the Borda Count method rather than Approval Voting to select their next activity. Suppose the preference schedule below summarizes the preferences of the second period PE class.

10 16 14

To determine the winning activity, we must find the Borda Count total for each activity. The activity with the largest Borda Count total is the winner. What is the Borda Count total for Aerobics? answer What is the Borda Count total for Badminton? answer What is the Borda Count total for Football? answer What is the Borda Count total for Softball? answer Which activity is selected by second period using the Borda Count Method? answer

What might have happened if Approval Voting had been used? (Suppose the class was surveyed and all students indicated that they would approve their first two choices and disapprove their last two choices.)

10 16 14

1st S S F (APPROVED)

2nd B B B (APPROVED)

3rd (NOT F A S APPROVED)

4th (NOT A F A APPROVED)

To determine the winning activity, we must calculate the number of votes received by each. The activity with the most votes wins. Notice that each activity in bold above receives an approval vote (and it does not matter whether it was first place or second place). How many total votes did AEROBICS receive? answer How many total votes did BADMINTON receive? answer How many total votes did FOOTBALL receive? answer How many total votes did SOFTBALL recieve? answer Which activity is selected by second period using Approval Voting (when everyone approves of their first two choices from the previous preference schedule)? answer

What might have happened with Approval Voting for the second period class if each student approved their first two choices (as above) EXCEPT the students that made Softball their first choice (who only approve Softball)?

10 16 14

1st S S F (ALL APPROVED)

2nd (ONLY 14 VOTES IN B B B LAST COLUMN APPROVED)

3rd (NOT F A S APPROVED)

4th (NOT A F A APPROVED)

Looking at the preference schedule above, this means that 26 (10 + 16) students who chose Badminton as their second choice would no longer vote for it under this Approval Voting scheme. So... How many total votes did AEROBICS receive? answer How many total votes did BADMINTON receive? answer How many total votes did FOOTBALL receive? answer How many total votes did SOFTBALL recieve? answer Which activity is selected by second period using Approval Voting (when everyone approves of their first two choices from the previous preference schedule)? answer As you can see from the above examples, Approval Voting is similar in some respects to the Borda Count method. Both methods involve computing a total and the winner is the candidate or choice with the largest total. However, the entire process is much simpler with Approval Voting. In addition, Approval Voting is perfectly suited to situations where candidates or choices are added or removed after a vote has already been taken. example A Nice Property of Approval Voting In an election using Approval Voting, adding or removing candidates or alternatives does not change the point totals of the other candidates or alternatives. (If candidates/alternatives drop out, simply remove then from the list. If candidates/alternatives are added, vote totals for the original candidates/alternatives stay the same and voters only have to give their approval or disapproval of the candidates/alternatives that were added.)