Voting Methods President Barack Obama votes in the 2012 election. (credit: Pete Souza/White House, Public Domain) Learning Objectives After completing this section, you should be able to: Apply plurality voting to determine a winner. Apply runoff voting to determine a winner. Apply ranked-choice voting to determine a winner. Apply Borda count voting to determine a winner. Apply pairwise comparison and Condorcet voting to determine a winner. Apply approval voting to determine a winner. Compare and contrast voting methods to identify flaws. Today is the day that you begin your quest to collaborate on the constitution of Imaginaria! Let’s begin by thinking about the selection of a leader who can serve as president. It seems straightforward; if the majority of citizens prefer a particular candidate, that candidate should win. But not all votes are decided by a simple majority. Why not? What are the options? Majority versus Plurality Voting When an election involves only two options, a simple majority is a reasonable way to determine a winner. A majority is a number equaling more than half, or greater than 50 percent of the total. Let’s take a look at the outcomes of U.S. presidential elections to understand more. displays the results of the 2000 U.S. presidential election. Like most presidential elections, this election involved more than two options. If that is the case, is it possible that no single candidate will receive more than half of the votes cast? Candidate (Party Label) Popular Vote Total Al Gore (Democrat) 50,999,897 George W. Bush (Republican) 50,456,002 Ralph Nader (Green) 2,882,955 Patrick J. Buchanan (Reform/Independent) 448,895 Harry Browne (Libertarian) 384,431 Howard Phillips (Constitution) 98,020 Other 134,900 Total: 105,405,100 Results of the Popular Vote for the 2000 U.S. Presidential Election (source: https://www.fec.gov/introduction-campaign-finance/election-and-voting-information/federal-elections-2000/president2000/) Majority of Popular Vote in the 2000 U.S. Presidential Election Refer back to . Did any single candidate secure the majority of popular votes? Step 1: Calculate 50 percent of 105,405,100 by multiplying the decimal form of 50 percent, which is 0.50, by 158,394,605: 0.50 ( 105,405,100 ) = 52,702,550 Step 2: Determine the minimum number of votes needed to have a majority. The minimum number of votes required is the lowest counting number that is larger than 50 percent of the votes. To have a majority, an individual candidate must have more than 52,702,550; so, a majority candidate must have 52,702,551 votes or more. Step 3: Compare the number of votes each candidate received to 52,702,551. According to the data in , none of the candidates secured a majority. Unlike in the 2000 U.S. presidential election, a candidate won the majority of votes in the 2020 election (see ). It is a common occurrence for no single candidate to receive a majority of the votes in an election with more than two candidates. When this occurs, the candidate with the largest portion of the votes is said to have a plurality . Plurality of Popular Vote in the 2000 U.S. Presidential Election Refer again to . In the 2000 U.S. presidential election, which candidate had a plurality of popular votes? Al Gore secured 50,999,897 votes which was more than any other single candidate. Therefore, he had a plurality of the popular votes. Your plans for Imaginarian elections will likely include primary elections, or preliminary elections to select candidates for a principal or general election. displays the results of the 2018 U.S. Senate Republican primary for Maryland. Top Four Republican Candidates Votes Percentage of Party Votes Cambell, Tony 51,426 29.22% Chaffee, Chris 42,328 24.05% Grigorian, Christina J. 30,756 17.48% Graziani, John R. 15,435 8.77% Total Votes 175,981 100% (source: https://ballotpedia.org/United_States_Senate_election_in_Maryland_(June_26,_2018_Republican_primary)) Consider how election by plurality, not majority, is the most common method of selecting candidates for public office. The U.S. Electorial College: Winning the Presidency without a Plurality In the 2000 U.S. presidential election, Al Gore had a plurality of the popular votes, but he did not win the election. Why? This occurred because the U.S. president and vice president are elected by electors rather than a direct vote by the citizens. The electors are part of the Electoral College, a body of people representing the states. Why was the Electoral College created? The Electoral College was created as a compromise between those authors of the U.S. Constitution who believed Congress should elect the president, and those who believed the citizens should vote directly. The popular vote was not recorded until the presidential election of 1824. Since then, only five presidents have been elected without winning a plurality of the popular vote: John Quincy Adams in 1824, Rutherford B. Hayes in 1876, Benjamin Harrison in 1888, George Bush in 2000, and Donald Trump in 2016. Runoff Voting Has your family ever debated what to have for dinner? Suppose your family is deciding on a restaurant and exactly half of you want to have pizza but the other half want hamburgers. How do you decide when the result is a tie? You need a tiebreaker! Will the new democracy of Imaginaria need tiebreakers? When no candidate satisfies the requirements to win the election, a runoff election , or second election, is held to determine a winner. How would runoff voting work in Imaginaria? There are many types of runoff voting systems , which are voting systems that utilize a runoff election when the first round does not result in a winner. The method for implementing a runoff election can vary widely, particularly in the criteria used to determine whether a candidate will be on the ballot in the second election. For example, a two-round system is a runoff voting system in which only the top candidates advance to the runoff election. In some two-round systems, only the top two candidates are on the second ballot, or it may be any candidate who secures a certain percentage of the vote will advance. The Hare Method is another runoff voting system in which only the candidate(s) with the very least votes are eliminated. This can potentially result in several rounds of runoff elections. Runoff Election for Condominium Association President A condominium association elects a new president every two years by a two-round system of voting. If none of the candidates receive a majority, the association charter states that the top two candidates will be eligible to participate in a runoff election. In a particular year, five residents were nominated. The results of the first round are given in the table below. Candidate Votes in First Round Abou 18 Baiocchi 10 Campana 5 Dali 11 Eugene 4 Is there a winner based on the first round? Why or why not? If there is a winner, who won? If there should be a runoff, who will advance to the second round? A majority of 48 total votes is required to win. Begin by finding 50 percent of 48, which is calculated as follows: 0.50 ( 48 ) = 24 . A majority is 25 or more. No candidate has a majority, so there is no winner based on the first round. Abou and Dali advance to the second round. Steps to Determine Winner by Plurality or Majority Election with Runoff To determine the winner by plurality or when a majority election with runoff occurs, we take these three steps: Step 1: If a majority is required to win the election, determine the number of votes needed to achieve a majority. This is the least whole number greater than 50 percent of the total votes. If a majority is not required, move to Step 2. Step 2: Count the number of votes for each candidate in the current round of voting. If a single candidate has enough votes to win a plurality, or a majority as appropriate, then you are done! Otherwise, eliminate a predetermined number of candidates based on the rules of the election. Elimination conditions may vary. For example, the rules may state that the candidate(s) with the fewest votes will be eliminated (as in the Hare method), or that only the candidates meeting a certain threshold will move on (as in a two-round system). Once the appropriate candidates are eliminated, move on to Step 3. Step 3: Hold a runoff election. If the runoff is simulated using a list of voter’s preferences, renumber the preferences to reflect the remaining number of options in such a way that the original order of preference is retained. Then repeat Step 2. Note: The second and third steps may be repeated as many times as necessary for voting procedures that allow multiple runoffs. Family Dinner Night The five members of the Chionilis family—Annette, Rene, Seema, Titus, and Galen—have decided to get takeout for dinner. They are trying to decide on a restaurant. The options are Rainbow China, Dough Boys Pizza, Taco City, or Caribbean Flavor. They will use majority election with runoffs where the restaurant with the fewest votes is eliminated in each round. The preferences of each family member are listed by first initial in the table below. An entry of 1 represents the person’s first choice; 2, their second; and so on. For example, Annette’s second choice is Dough Boys Pizza. Options A R S T G Rainbow China 1 3 3 1 3 Dough Boys Pizza 2 2 1 2 1 Taco City 3 4 2 4 2 Caribbean Flavor 4 1 4 3 4 Use the information in the table to answer the following questions. Which common type of runoff voting method is this? List the results of each round of voting based on this information and determine which restaurant was ultimately chosen. the Hare Method Step 1: Determine the number of votes necessary to have a majority. There are five family members, so 50 percent of 5 is 0.50 ( 5 ) = 2.5 . A majority is three or more votes. Step 2: Count the number of votes for each restaurant in the first election. In a list of voter preferences, the 1s represent the top choice of each voter, which corresponds to their vote in the first round. Results of Round 1: Rainbow China — 2 votes Dough Boys — 2 votes Taco City — 0 votes Caribbean Flavor — 1 vote No restaurant received a majority. Eliminate Taco City, which has the fewest first place votes: Options A R S T G Rainbow China 1 3 3 1 3 Dough Boys Pizza 2 2 1 2 1 Caribbean Flavor 4 1 4 3 4 Step 3: Hold a runoff election. In other words, hold a second round. Since we have a list of the voters’ preferences with the eliminated option removed, we will renumber the preferences as first, second, and third so that we keep the original order of preference. The result is that we will count the second-place vote of any voter whose first choice was eliminated. Options A R S T G Rainbow China 1 3 2 1 2 Dough Boys Pizza 2 2 1 2 1 Caribbean Flavor 3 1 3 3 3 Step 4: Repeat the process from Step 2. Count the number votes for each restaurant in the first-round election. Since we are using a list of preferences, we need to count the number of 1s received by each restaurant. Results of Round 2: Rainbow China — 2 votes Dough Boys — 2 votes Caribbean Flavor — 1 vote No single restaurant has three votes. Eliminate Caribbean Flavor, which has the fewest first place votes. OPTIONS A R S T G Rainbow China 1 3 2 1 2 Dough Boys Pizza 2 2 1 2 1 Step 5: Repeat the process from Step 3. Hold another runoff election. This will be Round 3. Renumber the voters' preferences as first and second this time. Options A R S T G Rainbow China 1 2 2 1 2 Dough Boys Pizza 2 1 1 2 1 Step 6: Repeat the process from Step 2 one last time. Count the number of first place votes for each remaining restaurant. Results of Round 3: Rainbow China — 2 votes Dough Boys — 3 votes Determine whether any one choice has a majority. Yes! Dough Boys has three votes, so it is the winner! Ranked-Choice Voting In and Your Turn 11.4 , you were given a list that ordered each voter’s preferences. This ordering is called a preference ranking . A ballot in which a voter is required to give an ordering of their preferences is a ranked ballot , and any voting system in which a voter uses a ranked ballot is referred to as ranked voting . The vote for the Academy Awards uses a ranked ballot. The table below provides an example of a ranked ballot for the 2020 Academy Award nominees for Best Director. Candidate for Best Director Rank top choice as 1, next choice as 2, and so on. Martin Scorsese, The Irishman 1 2 3 4 5 Todd Phillips, Joker 1 2 3 4 5 Sam Mendes, 1917 1 2 3 4 5 Quentin Tarantino, Once Upon a Time in Hollywood 1 2 3 4 5 Bong Joon-ho, Parasite 1 2 3 4 5 As you decide on the voting methods that will be used in your new democracy, budget must be a consideration. You might consider a particular type of ranked voting called ranked-choice voting (RCV) , which simulates a series of runoff elections without the usual time and expense involved when voters must repeatedly return to the polls, like we did in . The method of ranked-choice voting (RCV), also called instant runoff voting (IRV) , is a version of the Hare Method, using preference ranking so that, if no single candidate receives a majority, the least popular selections can be eliminated and the results can be recounted, without the need for more elections. Ranked voting can be confused with ranked-choice voting, but ranked voting is a more general category which includes ranked-choice voting and several other voting methods. As we explore examples of ranked voting, we will summarize the voters’ preference rankings using a table in which the top row shows the number of ballots that ranked the options in the same order. Let’s practice interpreting the information in this type of table. Interpreting the Sample Preference Summary Refer to the table below containing voters’ preference rankings to answer the following questions. Number of Ballots 100 200 150 75 Option A 1 4 3 4 Option B 2 3 4 2 Option C 4 2 1 1 Option D 3 1 2 3 How many voters ranked the options in the following order: Option A in fourth place, Option B in second place, Option C in first place, and Option D in third place? How many ballots in total were collected? How many voters indicated that Option C was their first choice? The column farthest to the right displays this ordering. The top entry in this column is 75; so, there were 75 voters who ranked the options in this way. The sum of the top row gives the total number of ballots collected: 100 + 200 + 150 + 75 = 525 . So, there were 525 ballots collected. In the Option C row, there are two entries of 1 which indicate a first choice for that option. These occur in the last two columns. The sum of the top entries in these columns is 150 + 75 = 225 . So, 225 voters indicated Option C as their first choice. Now that we’ve covered how to read a summary of preference rankings, let’s practice using the ranked-choice method to determine the winner of an election. Recall that ranked-choice voting is still the Hare Method where the candidate with the very least number of votes is eliminated each round until a majority is attained. The difference here is that the voters have completed a ranked ballot, so they don't have to visit the polls multiple times. Here are the steps for ranked-choice voting. Steps to Determine Winner by Ranked-Choice Voting To determine the winner when ranked-choice voting occurs, we take these three steps: Step 1: Determine the number of votes needed to achieve a majority. This is the least whole number greater than 50 percent of the total votes. Step 2: Count the number of first place votes for each candidate. If a candidate has a majority, that candidate wins the election and we are done! Otherwise, eliminate the candidate(s) with the fewest votes and complete Step 3. Step 3: Reallocate the votes to the remaining candidates, and repeat Step 2. Be methodical to avoid arithmetic errors. Make sure that each time you count the number of first place votes they sum to the number of ballots. How Does Ranked-Choice Voting Work? Most Popular Color in Kindergarten Let’s review the kindergarten class color preferences again, and this time determine which color would be selected based on these results using the ranked-choice method. Number of Ballots 4 6 4 7 Red 2 6 2 5 Blue 1 2 1 4 Green 6 5 5 2 Yellow 5 4 6 3 Purple 4 3 4 1 Pink 3 1 3 6 Step 1: Determine whether any candidate received a majority. There were 21 ballots. Fifty percent of 21 is 0.50 ( 21 ) = 10.5 . A majority is 11. Step 2: Count the number of first place votes for each candidate. If a candidate has a majority, that candidate wins the election. Otherwise, eliminate the candidate(s) with the fewest votes. Red: 0 Blue: 4 + 4 = 8 Green: 0 Yellow: 0 Purple: 7 Pink: 6 Notice that 8 + 7 + 6 = 21 , which is the total number of ballots. Confirming this helps to catch any arithmetic or counting errors. No candidate has a majority with 11 or more votes. We must eliminate red, green, and yellow which had the fewest votes with 0 each. The remaining votes that must be counted for Round 2 are given in the table below. Number of Ballots 4 6 4 7 Blue 1 2 1 4 Purple 4 3 4 1 Pink 3 1 3 6 Step 3: Reallocate the votes to the remaining candidates. We can do this by numbering the choices as 1, 2, and 3 in such a way that the order of preference is retained as seen in the table below: Number of Ballots 4 6 4 7 Blue 1 2 1 2 Purple 3 3 3 1 Pink 2 1 2 3 Step 4: Repeat the process from Step 2. Count the number of first place votes for each candidate. If a candidate has a majority, that candidate wins the election. Otherwise, eliminate the candidate(s) with the fewest votes. Blue: 4 + 4 = 8 Purple: 7 Pink: 6 Confirm that 8 + 7 + 6 = 21 . Great! No candidate has 11 or more votes. We must eliminate pink which had the fewest votes with 6. The remaining votes that must be counted for Round 2 are shown in the table below. Number of Ballots 4 6 4 7 Blue 1 2 1 2 Purple 3 3 3 1 Step 5: Repeat the process from Step 3. Reallocate the votes to the remaining candidates. We can do this by numbering the choices as 1 and 2; Number of Ballots 4 6 4 7 Blue 1 1 1 2 Purple 2 2 2 1 Step 6: Repeat the process from Step 2 one last time. Count the number of first place votes for each candidate. If a candidate has a majority, that candidate wins the election. Otherwise, eliminate the candidate(s) with the fewest votes. Blue: 4 + 6 + 4 = 14 Purple: 7 Blue has a majority and wins the election! Determine Winner of Election by Ranked-Choice Method (aka Instant Runoff) Borda Count Voting Ranked-choice voting is one type of ranked voting that simulates multiple runoffs based on ranked ballots. Another type of ranked voting is the Borda count method , which uses ranked ballots that award candidates points corresponding to the number of candidates ranked lower on each ballot. To understand how this works, let’s review the favorite colors of our kindergarten class from the table below. Let’s focus on the votes represented by the first column of the preference summary. Number of Ballots 4 6 4 7 Red 2 6 2 5 Blue 1 2 1 4 Green 6 5 5 2 Yellow 5 4 6 3 Purple 4 3 4 1 Pink 3 1 3 6 Each student had six options. This first column tells us that four students ranked blue as their first choice, red as their second choice, pink as their third choice, purple as their fourth choice, yellow as their fifth choice, and green as their sixth choice. Blue was ranked higher than 6 - 1 = 5 other colors. For each of the four students who completed their ballot in this way, blue would receive five points. Since there were four ballots with this ordering, blue would receive 5 ( 4 ) = 20 points from the first column. To determine the total points for each candidate, we have to find the sum of the points they received in each column. To determine the winner of a contest using the Borda count method, we must compare total number of points earned by each candidate. The candidate with the most points is the winner. Each row of the preference summary corresponds to a single candidate. To find the number of points received by a particular candidate in the preference summary, or their Borda score , we will need to focus on the row in which that candidate appears. Before we practice determining the winner of a Borda count election, let’s examine how to find the Borda score for a single candidate. Most Popular Color in Kindergarten Revisited Let’s review the ballots from the kindergarten class again, as shown in the table below. This time, let’s determine the Borda score received by the color purple. Number of Ballots 4 6 4 7 Red 2 6 2 5 Blue 1 2 1 4 Green 6 5 5 2 Yellow 5 4 6 3 Purple 4 3 4 1 Pink 3 1 3 6 Step 1: Find the number of points received by the candidate in each column. Column 1: 4 × ( 6 - 4 ) = 4 × 2 = 8 Column 2: 6 × ( 6 - 3 ) = 6 × 3 = 18 Column 3: 4 × ( 6 - 4 ) = 4 × 2 = 8 Column 4: 7 × ( 6 - 1 ) = 7 × 5 = 35 Step 2: Find the sum of the points received in each column. This is the total number of points received by this candidate: 8 + 18 + 8 + 35 = 69 This process can also be combined into one step as shown here. 4 × ( 6 - 4 ) + 6 × ( 6 - 3 ) + 4 × ( 6 - 4 ) + 7 × ( 6 - 1 ) = 4 × 2 + 6 × 3 + 4 × 2 + 7 × 5 = 8 + 18 + 8 + 35 = 69 Purple received 69 points in this election. When calculating a Borda score in one step, be careful to use the correct order of operations. Perform the subtraction inside each pair of parentheses first, then perform each multiplication, and then perform each addition. Now let’s determine the winner of an election by comparing the Borda scores for each of the candidates. Determine the Winner by Two Ranked Voting Methods Use the table below, which displays a sample preference summary, to answer the questions that follow. Number of Ballots 95 90 110 115 Option A 4 4 1 1 Option B 2 2 2 2 Option C 3 1 3 4 Option D 1 3 4 3 Use the ranked-choice voting method to determine the winner of the election. Use the Borda count method to determine the winner of the election. Step 1: Determine the number of votes needed to achieve a majority. The number of ballots is 95 + 90 + 110 + 115 = 410 . Fifty percent of 410 is 0.50 ( 410 ) = 205 . So, 206 votes or more is a majority. Step 2: Count the number of first place votes for each candidate. Option A has 110 + 115 = 225 Option B has 0 Option C has 90 Option D has 95 Since Option A has a majority, Option A is the winner by the ranked-choice method. The Borda scores would be: Option A: 95 ( 4 - 4 ) + 90 ( 4 - 4 ) + 110 ( 4 - 1 ) + 115 ( 4 - 1 ) = 675 Option B: 95 ( 4 - 2 ) + 90 ( 4 - 2 ) + 110 ( 4 - 2 ) + 115 ( 4 - 2 ) = 820 Option C: 95 ( 4 - 3 ) + 90 ( 4 - 1 ) + 110 ( 4 - 3 ) + 115 ( 4 - 4 ) = 475 Option D: 95 ( 4 - 1 ) + 90 ( 4 - 3 ) + 110 ( 4 - 4 ) + 115 ( 4 - 3 ) = 490 Since Option B has a Borda score of 820 points, Option B is the winner by the Borda count method. The Borda count method may seem too complicated to even consider using for Imaginaria, but each voting method has its own pros and cons. The Borda count method, for example, favors compromise candidates over divisive candidates. A compromise candidate is not the first choice of most of the voters, but is more acceptable to the population as a whole than the other candidates. A divisive candidate is simultaneously the first choice of a large portion of the voters and the last choice of another large portion of the voters. In , Candidate A was ranked first by 225 voters, but was ranked last by 185 voters. No voters ranked Candidate A as second or third. It appears that, although Candidate A had the majority of first place votes, there was a significant minority who strongly disliked them. Candidate A was a divisive candidate. Candidate B, on the other hand, was the second choice of every voter, making Candidate B a good compromise. The Borda count method chose Candidate B, a compromise candidate, that was more acceptable to the population as a whole. This scenario is cited by both opponents and proponents of the Borda count method. Determine Winner of Election by Borda Count Method Pairwise Comparison and Condorcet Voting We have discussed two kinds of ranked voting methods so far: ranked-choice and Borda count. A third type of ranked voting is the pairwise comparison method , in which the candidates receive a point for each candidate they would beat in a one-on-one election and half a point for each candidate they would tie. If one candidate earns more points than the others, then that candidate wins. This method is one of several Condorcet voting methods , which are methods in which candidates are ranked and then compared pairwise to each other, a candidate having to beat all others in order to win. These methods vary in the way candidates are scored, and there is not always a clear winner. A candidate who wins each possible pairing is known as a Condorcet candidate . These terms are named after the Marquis de Condorcet, a French philosopher and mathematician who preferred the pairwise comparison method to the Hare method and made public arguments in its favor. Marquis de Condorcet Condorcet voting methods are named for the Marquis de Condorcet, a French philosopher and mathematician known for, among other accomplishments, writing “ Sur l'admission des femmes au droit de Cité ” (“On the Admission of Women to the Rights of Citizenship”), in 1789, the first published essay on the political rights of women. For more details visit this Web site. If you include a Condorcet voting method in the constitution of Imaginaria, the election supervisors may want to use a pairwise comparison matrix like the one in . It’s a tool used to list the number of wins associated with each pairing of two candidates. Each candidate will receive a point for each win and a half a point for each tie. Each pairing is listed twice, once for the number of wins of a candidate over a particular challenger and once for the number of wins of the challenger over that candidate. Pairwise Comparison Matrix for Three Candidates Steps to Determine a Winner by Pairwise Comparison Method Using a Matrix To determine the winner when the pairwise comparison method is used, we take these three steps: Step 1: On the matrix, indicate a losing matchup by crossing out a box, × , and tie match ups by drawing a slash through the box, \ . Step 2: Award each candidate 1 point for a win, half a point for a tie, and 0 points for a loss. Step 3: Identify the winner, which is the candidate with the most points. Determine Winner of Election by Using the Pairwise Comparison Method Before you decide on the pairwise comparison method for Imaginaria, review what’s involved in constructing a pairwise comparison matrix from a summary of ranked ballots. Then we can use the matrix to determine the winner of the election. Does the winner using the Borda method still win? Construct and Use a Pairwise Comparison Matrix Consider the summary of ranked ballots shown in the table below. Determine the winner of an election using the pairwise comparison method. Number of Ballots 95 90 110 115 Option A 4 4 1 1 Option B 2 2 2 2 Option C 3 1 3 4 Option D 1 3 4 3 Construct a pairwise comparison matrix for the sample summary of ranked ballots in the table above. Use the pairwise comparison method to determine a winner. Recall that in , Candidate A won by the ranked-ballot method, and Candidate B won by the Hare method. Did the same candidate win using the pairwise comparison method? Is the winner a Condorcet candidate? There are four candidates on the ballots. We will need a row and a column for each candidate in addition to the headings, so we will draw a five by five matrix. Pairwise Comparison Matrix for Four Candidates Step 1: Refer to to determine the values that belong in each cell. A over B: A is preferred to B in columns 3 and 4. So, A scores 110 + 115 = 225 points. A over C: A is preferred to C in columns 3 and 4. So, A scores 225 points again. A over D: Similarly, A scores 225 points. B over A: B is preferred to A in columns 1 and 2. So, B scores 95 + 90 = 185 points. Step 2: Continuing in this way, we complete the pairwise comparison matrix, as shown in . Pairwise Comparison Matrix for Four Candidates with Vote Counts Step 1: Losing pairings are crossed off with an × . In the event of a tie, we will draw a slash, \ . Step 2: Determine the number of points for each candidate by analyzing their row of wins. Each win is 1 point, each loss, × , is 0 points, and each tie, \ , is half a point. Construct an additional column for each candidate’s points. Pairwise Comparison Matrix for Four Candidates with Pairwise Winners and Points Column Added Step 3: The winner by the pairwise comparison method is Option A with 3 points. Option A was not the winner by the Hare method. The winner, Option A, is a Condorcet candidate because Option A won each pairwise comparison. Notice that the pairwise vote totals are not used to determine the points. Vote totals are only used to determine a win or a loss. Avoid the common error of adding the values in each row to get the points. Three Key Questions Before you decide if you want to use the pairwise comparison method for Imaginarian elections, let’s consider three questions that might affect your decision. Is there always a winner? If there is a winner, is the winner always a Condorcet candidate? If there is a Condorcet candidate, does that candidate always win? Let’s think about why these questions might be important to you if you chose the pairwise comparison method. First, if no candidate meets the criteria to win an election, you will need a backup plan such as a runoff election. Second, if the winner is not a Condorcet candidate, then there is at least one candidate who beat the winner in a pairwise matchup and the supporters of that candidate might question the validity of the election. Finally, if there is a Condorcet candidate who beat every other candidate in a pairwise matchup, it is reasonable to conclude that it would be unfair for anyone else to win. The rest of the examples in this section should illustrate these key concepts. Rock, Paper, Scissors by Pairwise Comparison Suppose that three people are playing the game Rock, Paper, Scissors. On the count of three, each person shows a hand signal for rock, paper, or scissors. Each hand signal beats another hand signal. The group keeps having a tie because Person A always picks rock, Person B always picks paper which beats rock, and Person C always picks scissors which beats paper, and is beaten by rock! This leads to a disagreement about which choice is best. They decide to use the pairwise comparison method determine the winner. Their preference rankings are given in the following table. Voters A B C Rock (R) 1 3 2 Paper (P) 2 1 3 Scissors (S) 3 2 1 Construct the comparison matrix: Rock, Paper, Scissors Pairwise Comparison Matrix There is a tie! There is no winner. illustrates the answer to the first key question. The pairwise comparison method does not always result in a winner. For example, much like the game of Rock, Paper, Scissors, it is possible for a cyclic pattern to emerge in which each candidate beats the next until the last candidate who beats the first. Rock, Paper, Scissors Cyclic Outcome Now, you have the answer to the second key question. The pairwise comparison matrix in YOUR TURN 11.10 is an example of a scenario where a winner is not a Condorcet candidate. The answer to the third question is not as clear. If there is a Condorcet candidate, does that candidate win? So far, we have not come across a contradictory example where the Condorcet candidate didn't win, but we cannot know with certainty that it is not possible by looking at examples. Instead, we will need to use some reasoning. Let’s review some particular cases of elections with a certain number of candidates, and then we will try to generalize the scenario to an election with n candidates. Does the Condorcet Candidate Win? Suppose there is an election with five candidates—A, B, C, D, and E—and that Candidate C is a Condorcet candidate. How many points did Candidate C win? What is the greatest number of points that any one of the other candidates could win? Is it possible for Candidate C to lose or tie? In any pairwise election with five candidates, each candidate must compete against four other candidates. It follows that the most points a single candidate can win is four points, which would occur if the candidate won every matchup. As a Condorcet candidate, Candidate C won all the pairwise matchups against Candidates A, B, D, and E, earning four points. The rest of the candidates lost to Candidate C. The most points a particular candidate could win if they won matchups with each of the other three candidates is three points. Since Candidate C has four points and the rest of the candidates have three points or less, Candidate C is the winner. Therefore, it is not possible for Candidate C to tie or lose. Let’s consider a general case where there are n candidates. One of the candidates is a Condorcet candidate. Since the Condorcet candidate wins all matchups, the Condorcet candidate wins n - 1 points. Since each of the other candidates lost to the Condorcet candidate, the most a single candidate could win is n - 2 . Since the Condorcet candidate won n - 1 points and each other candidate won n - 2 points or fewer, the Condorcet candidate is the winner. You have your answer to the third key question! If there is a Condorcet candidate, that candidate is always the winner. Approval Voting The last type of voting system you will consider for your budding democracy is an approval voting system . In this system, each voter may approve any number of candidates without rank or preference for one over another (among the approved candidates), and the candidate approved by the most voters wins. This voting system has aspects in common with plurality voting and Condorcet voting methods, but it has characteristics that distinguish it from both. An approval voting ballot lists the candidates and provides the option to approve or not approve each candidate. The term “approval voting” was not used until the 1970s Brams, Steven J.; Fishburn, Peter C. (2007), Approval Voting , Springer-Verlag, p. xv, ISBN 978-0-387-49895-9, although its use has been documented as early as the 13th century (Brams, Steven J. (April 1, 2006). The Normative Turn in Public Choice (PDF) (Speech). Presidential Address to Public Choice Society. New Orleans, Louisiana.) Approval voting has the appeal of being simpler than ranked voting methods. It also allows an individual voter to support more than one candidate equally. This has appeal for those who do not want a split vote among a few mainstream candidates to lead to the election of a fringe candidate. It also has appeal for those who want an underdog to have a chance of success because voters will not worry about wasting their vote on a candidate who is not believed likely to win. Rock, Paper, Scissors, Lizard, Spock Suppose that Person A/B/C were just about to give up on their game of Rock, Paper, Scissors when they were joined by Person D who reminded them that their updated version, Rock, Paper, Scissors, Lizard, Spock was a far superior game with the added rules that Lizard eats Paper, Paper disproves Spock, Spock vaporizes Rock, Rock crushes Lizard, Lizard poisons Spock, Spock smashes Scissors, and Scissors decapitates Lizard. Rock, Paper, Scissors, Lizard, Spock Dominance Person D encourages their friends to hold a new election. This time, for the sake of simplicity, the group decides to use approval voting to determine the best move in the game. The summary of approval ballots for Rock, Paper, Scissors, Lizard, Spock is given in the table below. Voters A B C D Rock Yes No No No Paper No Yes No No Scissors No No Yes No Lizard No No No Yes Spock Yes Yes Yes Yes Count the number of approval votes for each candidate by counting the number of “Yes” votes in each row of the table. Rock: 1 Paper: 1 Scissors: 1 Lizard: 1 Spock: 4 Spock is the winning candidate, approved by four voters! The Chionilis Family Is Hungry Again! The eight members of the Chionilis family—Annette, Rene, Seema, Titus, Galen, Elena, Max and Demitri—have another decision to make. Approval voting worked out nicely the last time. They are going to use it again, but this time, Annette, Rene, Seema, and Galen are feeling a little indecisive. They can't narrow their choice down to two. They will approve their three top choices, but the other family members will only approve two. These choices are reflected in the following table. Determine the restaurant that will be chosen. Options A R S T G E M D Rainbow China Yes Yes Yes Yes Yes No Yes Yes Dough Boys Pizza Yes Yes Yes Yes No Yes No Yes Taco City Yes No Yes No Yes Yes No No Caribbean Flavor No Yes No No Yes No Yes No Count the number of approval votes for each restaurant by counting the number of “Yes” votes in each row. Rainbow China: 7 Dough Boys Pizza: 6 Taco City: 4 Caribbean Flavor: 3 This time, Rainbow China won! Compare and Contrast Voting Methods to Identify Flaws Wow! We have covered a lot of options for the voting methods. Now, you need to decide which one is best for Imaginaria. Imaginarians might consider characteristics of certain voting systems desirable and others undesirable. In some cases, voters may consider these undesirable traits to be flaws in a voting system that are significant enough to motivate them to reject that system. If you are feeling a bit overwhelmed by this decision, maybe it would help to read about the experiences of others who have faced similar questions. Consider the 2000 U.S. presidential election in which Green Party candidate Ralph Nader and Reform Party candidate Pat Buchanan were on the ballet running against the mainstream candidates, Democrat Al Gore and Republican George W. Bush. The voting results for Florida are given in . Candidate Party Votes Percentage (G) George W. Bush Republican 2,912,790 48.85% (A) Al Gore Democrat 2,912,253 48.84% (R) Ralph Nader Green 97,488 1.63% (P) Pat Buchanan Reform 17,484 0.29% (H) Harry Brown Libertarian 16,415 0.28% (O) 7 Other Candidates Other 6,680 0.11% Total 5,963,110 Florida Results in the 2000 U.S. Presidential Election (source: https://www.fec.gov/resources/cms-content/documents/FederalElections2000_PresidentialGeneralElectionResultsbyState.pdf) In more than one state, Buchanan was able to split the Republican vote enough to allow Gore to win that state. Nader split the Democrat vote in Florida and New Hampshire by enough votes to prevent Gore from winning those states. Had Gore won either state, he would have had enough electoral votes to win the election. Instead, Bush won. This is an example of a flaw in the plurality system of voting: the spoiler. A spoiler is a less popular candidate who takes votes from a more popular candidate with similar positions, swinging the race to another candidate with vastly different views that they would not support. This encourages voters not to vote for the candidate that they perceive to be the best, but instead for the candidate they can live with who they perceive to have a better chance of winning. Some voters may prefer a method such as approval voting, which does not have this trait in common with plurality voting. The Spoiler Controversy Because the vote counts for George W. Bush and Al Gore differed by only 537 votes, many Democrats blamed Ralph Nader and the Green Party for their loss. Let’s consider how the election results might have differed if the approval voting method had been used. Use and the following assumptions to extrapolate the results of an approval method election: 100 percent of Pat Buchanan supporters would approve George W. Bush. 100 percent of Ralph Nader supporters would approve Al Gore. 72 percent of Libertarians would approve George W. Bush. 28 percent of Libertarians would approve Al Gore (as was roughly the known percentage at the time according to the Cato Institute). 50 percent of the supporters of other candidates would approve George Bush while 50 percent would approve Al Gore. Step 1: Create a summary of approval ballots based on the given assumptions. For the Libertarian candidate, 72 percent of 16,415 of the votes is 0.72 ( 16 , 415 ) = 11 , 819 and 28 percent is 0.28 ( 16 , 415 ) = 4 , 596 . For the other candidates, 50 percent of the votes is 0.50 ( 6 , 680 ) = 3 , 340 . Number of Votes 2,912,790 (G) 2,912,253 (A) 97,488 (R) 17,484 (B) 11,819 (72% H) 4,596 (28% H) 3,340 (50% O) 3,340 (50% O) (G) George W. Bush Yes No No Yes Yes No No Yes (A) Al Gore No Yes Yes No No Yes Yes No (R) Ralph Nader No No Yes No No No No No (P) Pat Buchanan No No No Yes No No No No (H) Harry Brown No No No No Yes Yes No No (O) 7 Other Candidates No No No No No No Yes Yes Step 2: Count the number of approval votes for each candidate. George W. Bush: 2 , 912 , 790 + 17 , 484 + 11 , 819 + 3340 = 2 , 945 , 433 Al Gore: 2 , 912 , 253 + 97 , 488 + 4 , 596 + 3 , 340 = 3 , 017 , 677 Ralph Nader: 97 , 488 Pat Buchanan: 17 , 484 Harry Brown: 11 , 819 + 4 , 596 = 16 , 415 Other Candidates: 6 , 680 In this scenario, Al Gore is the winner. The results in and Your Turn 11.14 highlight one of the characteristics of approval voting. Ralph Nader moved up from a distant third place finish to a close second place finish when Al Gore’s supporters approved him on their ballots. In this way, fringe candidates have a better chance of winning, which some voters consider a flaw but others consider a benefit. Another aspect of approval voting systems that is a concern to many voters is that candidates in approval elections might encourage their loyal supporters to approve them and only them to avoid giving support to any other candidate. If this occurred, the election in effect becomes a traditional plurality election. This is a flaw that cannot occur in an instant runoff system since all candidates are ranked. Three Habitable Planets In the future, humans have explored distant solar systems and found three habitable planets which could be colonized. Since it will take all available resources to colonize one planet, humans must agree on the planet. Planet A has the most comfortable climate and most plentiful resources, but it is the farthest from Earth making travel to the planet a challenge. Planet B is half the distance but will require more resources to make comfortable. Planet C is the least suitable of the three and terraforming will be required, but it is close enough to make travel between Earth and Planet C possible on a more regular basis. The table below provides the voter preferences for the colonization of each planet. Percentage of Voters 45% 15% 40% Planet A 1 3 3 Planet B 2 1 2 Planet C 3 2 1 If the entire population were able to vote, determine the winning planet using each of the methods listed below. Plurality Ranked-choice method Borda count The plurality method only considers the top choice of each voter. By this system, Planet A has 45 percent of the vote, Planet B has 15 percent of the vote, and Planet C has 40 percent of the vote. Planet A wins. Using either instant runoff or a two-round system, Planet B with only 15 percent of the vote will be eliminated in the first round. In Round 2, the 15 percent that voted for Planet B would vote for their second choice, Planet C. This leaves Planet A with 45 percent and Planet C with 55 percent. Planet C has a majority and wins the election. To find the Borda score for each candidate, imagine there are exactly 100 voters. Then the summary of ranked ballots looks like: Out of 100 Voters 45 15 40 Planet A 1 3 3 Planet B 2 1 2 Planet C 3 2 1 The Borda score for each candidate is as follows: Planet A: ( 3 - 1 ) ( 45 ) + ( 3 - 3 ) ( 15 ) + ( 3 - 3 ) ( 40 ) = 90 Planet B: ( 3 - 2 ) ( 45 ) + ( 3 - 1 ) ( 15 ) + ( 3 - 2 ) ( 40 ) = 115 Planet C: ( 3 - 3 ) ( 45 ) + ( 3 - 2 ) ( 15 ) + ( 3 - 1 ) ( 40 ) = 95 Planet B wins. The election in involves a scenario in which there are two extreme candidates, Planet A and Planet B, and a moderate candidate, Planet C. The supporters of the extreme candidates prefer the moderate candidate to the other extremist ones. This makes Planet C a compromise candidate. In this case, both the plurality method and ranked-choice voting resulted in the election of one of the extreme candidates, but the Borda count method elected the compromise candidate in this scenario. Depending on a person’s perspective, this may be perceived as a flaw in either ranked-choice and plurality systems, or the Borda count method. In Fairness in Voting Methods , we will analyze the fairness of each voting system in greater detail using objective measures of fairness. Voting Calculators It is possible to create Excel spreadsheets that complete the calculations necessary to determine the winner of an election by various voting methods. In some cases, this work has already been done and posted online. As you practice applying the various voting methods that could be used in Imaginaria, quick Internet search will lead to sites such as Ms. Hearn Math with free specialty calculators. These sites can be a great way to check your results! Check Your Understanding Key Terms majority plurality runoff election runoff voting system two-round system Hare Method preference ranking ranked ballot ranked-choice voting (RCV) instant runoff voting (IRV) Borda count method Borda score Compromise candidate divisive candidate pairwise comparison method Condorcet voting methods Condorcet candidate approval voting system approval voting ballot spoiler Key Concepts In plurality voting, the candidate with the most votes wins. When a voting method does not result in a winner, runoff voting can be used to do so. Ranked-choice voting, also known as instant runoff voting, is one type of ranked voting system. The Borda count method is a type of ranked voting system in which each candidate is given a Borda score based on the number of candidates ranked lower than them on each ballot. When pairwise comparison is used, the winner will be the Condorcet candidate if one exists. Approval voting allows voters to give equally weighted votes to multiple candidates. When a voter finds a characteristic of a particular voting method unappealing, they may consider that characteristic a flaw in the voting method and look for an alternative method that does not have that characteristic. Videos How Does Ranked-Choice Voting Work? Determine Winner of Election by Ranked-Choice Method (aka Instant Runoff) Determine Winner of Election by Borda Count Method Determine Winner of Election by Pairwise Comparison Method

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Fairness in Voting Methods Citizens strive to ensure their voting system is fair. (credit: “Governor Votes Early” by Maryland GovPics/Flickr, CC BY 2.0) Learning Objectives After completing this section, you should be able to: Compare and contrast fairness of voting using majority criterion. Compare and contrast fairness of voting using head-to-head criterion. Compare and contrast fairness of voting using monotonicity criterion. Compare and contrast fairness of voting using irrelevant alternatives criterion. Apply Arrow’s Impossibility Theorem when evaluating voting fairness. Now that we’ve covered a variety of voting methods and discussed their differences and similarities, you might be leaning toward one method over another. You will need to convince the other founders of Imaginaria that your preference will be the best for the country. Before your collaborators approve the inclusion of a voting method in the constitution, they will want to know that the voting method is a fair method. In this section, we will formally define the characteristics of a fair system. We will analyze each voting previously discussed to determine which characteristics of fairness they have, and which they do not. In order to guarantee one ideal, we must often sacrifice others. The Majority Criterion One of the most fundamental concepts in voting is the idea that most voters should be in favor of a candidate for a candidate to win, and that a candidate should not win without majority support. This concept is known as the majority criterion . With respect to the four main ranked voting methods we have discussed—plurality, ranked-choice, pairwise comparison, and the Borda count method—we will explore two important questions: Which of these voting systems satisfy the majority criterion and which do not? Is it always “fair” for a voting system to satisfy the majority criterion? Keep in mind that this criterion only applies when one of the candidates has a majority. So, the examples we will analyze will be based on scenarios in which a single candidate has more than 50 percent of the vote. Roommates Choose Fast Food It’s final exams week and seven college students are hungry. They must get food, but from which drive thru? Their preferences are listed in the table below. The majority have listed McDonald’s as their top choice. Let’s calculate what the results of the election will be using various voting methods. Voters A B C D E F G (M) McDonald’s 1 1 1 1 5 5 5 (B) Burger King 2 2 2 2 2 2 2 (T) Taco Bell 4 4 3 4 3 3 1 (O) Pollo Tropical 3 5 4 3 4 1 3 (I) Pizza Hut 5 3 5 5 1 4 4 Which restaurant is the winner using the plurality voting method? Which restaurant is the winner using the ranked-choice voting method? Does the majority criterion apply? If so, for which of voting method(s), if any, did the majority criterion fail? For plurality voting, we only need to count the first-place votes for each candidate. In this case, McDonald’s has four first place votes, which is a majority and wins the election automatically. For ranked-choice voting, McDonald’s also wins because it has a majority at the end of Round 1. The majority criterion does apply because one candidate had a majority of the first-place votes. The majority criterion did not fail by either method, because in each case the majority candidate won. From , it appears that the plurality and ranked-choice voting methods satisfy the majority criterion. In general, the majority candidate always wins in a plurality election because the candidate that has more than half of the votes has more votes than any other candidate. The same is true for ranked-choice voting; and there will never be a need for a second round when there is a majority candidate. Let’s examine how some of the other voting methods stand up to the majority criterion. Roommates Choose Fast Food Those seven college students are hungry again! Their preferences haven’t changed, as shown below. Let’s calculate if the results change when we use different voting methods. VOTERS A B C D E F G (M) McDonald’s 1 1 1 1 5 5 5 (B) Burger King 2 2 2 2 2 2 2 (T) Taco Bell 4 4 3 4 3 3 1 (O) Pollo Tropical 3 5 4 3 4 1 3 (I) Pizza Hut 5 3 5 5 1 4 4 Which restaurant is the winner using the pairwise comparison voting method? Which restaurant is the winner using the Borda count voting method? Does the majority criterion apply? If so, for which voting method(s), if any, did the majority criterion fail? For pairwise comparison, notice that McDonald’s is a Condorcet candidate because it wins every pairwise comparison. So, McDonald’s is the winner. For the Borda count, we must calculate the Borda score for each candidate: McDonald’s is 16, Burger King is 21, Taco Bell is 13, Pollo Tropical is 12, Pizza Hut is 8. The winner is Burger King! Yes, the majority criterion applies because McDonald’s has the majority of first place votes. The majority criterion only fails using the Borda method. demonstrates a concept that we also saw in Borda Count Voting —the Borda method frequently favors the compromise candidate over the divisive candidate. This can happen even when the divisive candidate has a majority, as it did in this example. Although a majority of the voters were in favor of McDonald’s, a significant minority was strongly opposed to McDonald’s, ranking it last. Since the Borda score includes all rankings, this strong opposition has an impact on the outcome of the election. Pairwise comparison will always satisfy the majority criterion because the candidate with the majority of first-place votes wins each pairwise matchup. While it is possible for the majority candidate to win by the Borda count method, it is not guaranteed. So, the Borda method fails the majority criterion. A summary of each voting method as it relates to the majority criterion is found in the following table. Voting Method Majority Criterion Plurality Satisfies Ranked-choice Satisfies Pairwise comparison Satisfies Borda count Violates If you prefer the Borda method, you might argue that its failure to satisfy the majority criterion is actually one of its strengths. As we saw in and , the majority have the power to vote for their own benefit at the expense of the minority. While four students were very enthusiastic about McDonald’s, three students were strongly opposed to McDonald’s. It is reasonable to say that the better option would be Burger King, the compromise candidate, which everyone ranked highly and no one strongly opposed. The Borda method is designed to favor a candidate that is acceptable to the population as a whole. In this way, the Borda method avoids a downfall of strict majority rule known as the tyranny of the majority , which occurs when a minority of a population is treated unfairly because their situation is different from the situation of the majority. The people of Imaginaria should know that the power of the majority to vote their will has serious implications for other groups. For example, according to the UCLA School of Law Williams Institute, the LGBTQ+ community in the United States makes up approximately 4.5 percent of the population. When elections occur that include issues that affect the LGBTQ+ community, members of the LGBTQ+ community depend on the 95.5 percent of the population who do not identify as LGBTQ+ to consider their perspectives when voting on issues such as same-sex marriage, the use of public restrooms by transgender people, and adoption by same-sex couples. Robert Dahl In his book, Democracy and Its Critics , Robert Dahl wrote, “If a majority is not entitled to do so, then it is thereby deprived of its rights; but if a majority is entitled to do so, then it can deprive the minority of its rights.” Dahl was a renowned political theorist, but he is also considered to be a mathematician since his work utilizes ideas from an area of mathematics known as Game Theory (Mathematics Genealogy Project, NDSU Department of Mathematics with the American Mathematical Society). Three Branches of Government Concerns about the consequences of majority rule are not new. In 1788, John Adams warned of the consequences of majority rule and he argued for three branches of government as a way to temper them. In the early 1800s, a young French aristocrat named Alexis de Tocqueville toured the United States and wrote Democracy in America , which focused on the impact of democracy on political and civil societies. He observed, even then, the dominance of the white majority over the Indigenous people and enslaved people, which was perpetuated by majority rule. Separation of Powers and Checks and Balances Head-to-Head Criterion Another fairness criterion you must consider as you select a voting method for Imaginaria is the Condorcet criterion , also known as the head-to-head criterion. An election method satisfies the Condorcet criterion provided that the Condorcet candidate wins the election whenever a Condorcet candidate exists. A Condorcet method is any voting method that satisfies the Condorcet criterion. Recall from Three Key Questions that not every election has a Condorcet candidate; the Condorcet criterion will not apply to every election. Also recall that a Condorcet candidate cannot lose an election by pairwise comparison. So, the pairwise comparison voting method is said to satisfy the Condorcet criterion. Spending Tax Refund A survey asked a random sample of 100 people in the United States to rank their priorities for spending their tax refund. The options were (V) go on vacation, (S) put into savings, (D) pay off debt, or (T) other. The pairwise comparison matrix for the results is in . Determine whether the Condorcet criterion applies. Pairwise Comparison Matrix for Tax Refund Spending The Condorcet criterion only applies when there is a Condorcet candidate. “Pay off debt” (D) is a Condorcet candidate because D wins every matchup. Yes, the Condorcet criterion applies to this election. Spending Tax Refund Let’s return to the survey about tax refund spending from . We know that the Condorcet criterion applies because Option D, “Pay off debt,” is a Condorcet candidate, which wins every pairwise match up. Use the information in the ballot summary from the table below to find the winner and determine whether the Condorcet criterion is satisfied in this election when each of the following voting methods are used. Votes 33 32 31 4 On a vacation (V) 1 3 3 2 Put into savings (S) 3 1 2 1 Pay off debt (D) 2 2 1 4 Other (T) 4 4 4 3 Plurality Ranked-choice voting Borda count V wins 33 first place votes; S, 36; D, 31; and T, 0. So candidate S, “Put into savings,” has a plurality and wins. Since the Condorcet candidate D didn’t win, the Condorcet criterion is violated. Use the steps outlined in Ranked-Choice Voting for determining the winner of an election by ranked-choice voting, the application of the Hare method in which instant runoffs are used. Step 1: The number of votes needed to achieve a majority is 51. Step 2: As illustrated in part 1, no candidate has a majority of first-place votes; so the candidate with the fewest votes, T, must be eliminated. Step 3: Reallocate votes to the remaining candidates for the second round: Votes 33 32 31 4 On a vacation (V) 1 3 3 2 Put into savings (S) 3 1 2 1 Pay off debt (D) 2 2 1 3 Step 4: Repeat the process from Step 2. Count the first-place votes for each candidate: V has 33 votes, S has 36 votes, D has 31. votes. Eliminate candidate D, “Pay off debt,” for the third round. Step 5: Repeat the process from Step 3. Reallocate votes to the remaining candidates for the third round: Votes 33 32 31 4 On a vacation (V) 1 2 2 2 Put into savings (S) 2 1 1 1 Step 6: Repeat the process from Step 2 one last time. Count the first-place votes for each candidate: V has 33, S has 67. Candidate S, “Put into savings,” has a majority and wins. Since the Condorcet candidate D candidate D, “Pay off debt,” didn’t win, the Condorcet criterion is violated. Calculate the Borda score for each candidate. V: 33 ( 4 - 1 ) + 32 ( 4 - 3 ) + 31 ( 4 - 3 ) + 4 ( 4 - 2 ) = 170 S: 33 ( 4 - 3 ) + 32 ( 4 - 1 ) + 31 ( 4 - 2 ) + 4 ( 4 - 1 ) = 203 D: 33 ( 4 - 2 ) + 32 ( 4 - 2 ) + 31 ( 4 - 1 ) + 4 ( 4 - 4 ) = 223 T: 33 ( 4 - 4 ) + 32 ( 4 - 4 ) + 31 ( 4 - 4 ) + 4 ( 4 - 3 ) = 4 Candidate D, “Pay off debt,” has the highest Borda score and wins. Since D was the Condorcet candidate, this election satisfies the Condorcet criterion. As we have seen, the plurality method, ranked-choice voting, and the Borda count method each fail the Condorcet criterion in some circumstances. Of the four main ranked voting methods we have discussed, only the pairwise comparison method satisfies the Condorcet criterion every time. A summary of each voting method as it relates to the Condorcet criterion is found in the following table. Voting Method Condorcet Criterion Plurality Violates Ranked-choice Violates Pairwise comparison Satisfies Borda count Violates Monotonicity Criterion The citizens of Imaginaria might be surprised to learn that it is possible for a voter to cause a candidate to lose by ranking that candidate higher on their ballot. Is that fair? Most voters would say, “Absolutely not!!” This is an example of a violation of the fairness criterion called the monotonicity criterion , which is satisfied when no candidate is harmed by up-ranking nor helped by down-ranking, provided all other votes remain the same. Consider a scenario in which voters are permitted a first round that is not binding, and then they may change their vote before the second round. Such a first round can be called a “straw poll.” Now, let’s suppose that a particular candidate won the straw poll. After that, several voters are convinced to increase their support, or up-rank , that winning candidate and no voters decrease that support. It is reasonable to expect that the winner of the first round will also win the second. Similarly, if some of the voters decide to decrease their support, or down-rank , a losing candidate, it is reasonable to expect that candidate will still lose in the second round. You might be wondering why it’s called the monotonicity criterion. In mathematics, the term monotonicity refers to the quality of always increasing or always decreasing. For example, a person’s age is monotonic because it always increases, whereas a person’s weight is not monotonic because it can increase or decrease. If the only changes to the votes for a particular candidate after a straw poll are in one direction, this change is considered monotonic. If you are going to make an informed decision about which voting method to use in Imaginaria, you need to know which of the four main ranked voting methods we have discussed—plurality, ranked-choice, pairwise comparison, and the Borda count method—satisfy the monotonicity criterion. Favorite Dog Breed by Plurality The local animal shelter is having a vote-by-donation charity event. For a $10 donation, an individual can complete a ranked ballot indicating their favorite large dog breed: standard poodle, golden retriever, Labrador retriever, or bulldog. Use the summary of ballots below to answer each question. Votes 42 53 61 24 (S) Standard Poodle 1 3 2 1 (G) Golden Retriever 3 1 4 4 (L) Labrador Retriever 4 2 1 2 (B) Bulldog 2 4 3 3 Determine the winner of the election by plurality. Suppose that the 53 voters in the second column increased their ranking of the winner by 1. Determine the winner by plurality with the new rankings. Does this election violate the monotonicity criterion? Do you think the result of part 3 is also true for plurality voting and the monotonicity criterion in general? Why or why not? The number of votes for each candidate are: S 66, G 53, L 61, and B 0. The winner is the standard poodle. If the 53 voters in the second column rank S as 2 and L as 3, then the number of votes for each candidate are: S with 66, G with 53, L with 61, and B with 0. The winner is still the standard poodle. This election does not violate the monotonicity criterion because the winner was not hurt by up-ranking. In general, increasing the ranking for a winner of a plurality election will either leave them with the same or more first place votes while leaving the other candidates with the same or fewer first place votes. So a plurality election will never violate the monotonicity criterion. Monotonicity Criterion in a Pairwise Comparison Election Earlier, we discovered that the summary of ranked ballots shown in the table below results in the pairwise comparison matrix in . Use this information to answer the questions. Number of Ballots 95 90 110 115 Option A 4 4 1 1 Option B 2 2 2 2 Option C 3 1 3 4 Option D 1 3 4 3 Analyzed Pairwise Comparison Matrix for Sample Summary of Ranked Ballots Determine the winner of the election by the pairwise comparison method. Suppose that the 95 voters in the first column increased their ranking of the winner by 1. Determine the winner by the pairwise comparison method with the new rankings. Does this election violate the monotonicity criterion? Do you think the result of part 3 is true for the pairwise comparison method and the monotonicity criterion in general? Why or why not? By the pairwise comparison method, Option A wins with three points. If the 95 voters in the first column of increased their ranking of the winner by 1, then C would fall into fourth place and A would move up to third place on those ballots. This would only affect the matchup between A and C, and the result would be that A would gain 95 votes while C would lose 95 votes. This means A would have 320 votes and C would have 90. Since A already was ahead of C, this just puts A further ahead and causes no change to the election results. Since the winner A is not hurt by an up-rank, and the loser C is not helped by a down-rank, this election is fair by the monotonicity criterion. Yes, the monotonicity criterion would be satisfied by the pairwise comparison method, because an up-rank of the winner can never decrease the number of pairwise wins. Similarly, a down-rank can never increase the number of pairwise wins. The last few examples illustrate that the plurality method, pairwise comparison voting, and the Borda count method each satisfy the monotonicity criterion. Of the four main ranked voting methods we have discussed, only the ranked-choice method violates the monotonicity criterion. A summary of each voting method as it relates to the Condorcet criterion is found in the table below. Voting Method Monotonicity Criterion Plurality Satisfies Ranked-choice Violates Pairwise comparison Satisfies Borda count Satisfies Irrelevant Alternatives Criterion We have covered a lot about voting fairness, but there is one more fairness criterion that you and the other Imaginarians should know. Consider this well-known anecdote that is sometimes attributed to the American philosopher Sidney Morgenbesser: A man is told by his waiter that the dessert options this evening are blueberry pie or apple pie. The man orders the apple pie. The waiter returns and tells him that there is also a third option, cherry pie. The man says, “In that case, I would like the blueberry pie.” ( Gaming the Vote: Why Elections Aren’t Fair (and What We Can Do About It) , William Pound stone, p. 50, ISBN 0-8090-4893-0) This story illustrates the concept of the Irrelevant Alternatives Criterion, also known as the Independence of Irrelevant Alternatives Criterion (IIA ), which means that the introduction or removal of a third candidate should not change or reverse the rankings of the original two candidates relative to one another. In particular, if a losing candidate is removed from the race or if a new candidate is added, the winner of the race should not change. Apple, Blueberry, or Cherry? Suppose that 30 students in a class are going to vote on whether to have apple, blueberry, or cherry pie. Use the summary of ranked ballots in below to answer each question. Number of Ballots 14 12 4 (A) Apple Pie 1 3 3 (B) Blueberry Pie 2 1 2 (C) Cherry Pie 3 2 1 Determine the winner of the election by plurality. Which candidate would win a plurality election if cherry pie were removed from the ballot? Does this election violate the IIA? The number of first place votes for each candidate is: A with 14, B with 12, and C with 4. Apple pie has the most first-place votes and wins the election. If cherry pie is removed from the ballot, then the four voters in the third column now rank blueberry pie as their first choice. So the four votes for C now belong to B. This means that blueberry pie has 16 votes compared to the 14 votes for apple pie. Blueberry pie now wins the plurality election. Yes, the election violates the IIA because the removal of a losing candidate from the ballot changed the winner of the election. Best Fourth Wall Breaking Stare on The Office The NBC sitcom The Office ran for nine years and has been one of the most popular streamed television shows of all time. One of the trademarks of the show was that characters would often break the fourth wall to communicate with the audience just by staring directly into the camera. In fact, there is a website dedicated to " The Office stares" where you can watch over 700 of these stares! Suppose that 36 fans were asked which character had the best " The Office stare." Use the ballot summary in below to answer each question. Number of Ballots 9 11 7 6 3 (J) Jim Halpert (John Krasinski) 1 2 4 2 4 (P) Pam Beesly-Halpert (Jenna Fischer) 4 1 2 4 3 (D) Dwight Schrute (Rainn Wilson) 2 3 3 1 2 (M) Michael Scott (Steve Carell) 3 4 1 3 1 Determine the winner of the election by the pairwise comparison method. Determine the winner of the election by the pairwise comparison method if Michael Scott is removed from the ballot. Does this election violate IIA? Construct and analyze a pairwise comparison matrix: Pairwise Comparison Matrix for Jim, Pam, Dwight, and Michael Jim Halpert wins with 2 points. If Michael Scott is removed, the summary of ranked ballots becomes: Number of Ballots 9 11 7 6 3 (J) Jim Halpert (John Krasinski) 1 2 3 2 3 (P) Pam Beesly-Halpert (Jenna Fischer) 3 1 1 3 2 (D) Dwight Schrute (Rainn Wilson) 2 3 2 1 1 Construct and analyze a pairwise comparison matrix: Pairwise Comparison Matrix for Jim, Pam, and Dwight Pam wins with 1 1 2 points. Yes, this violates the IIA, because the winning candidate was hurt by the elimination of a losing candidate. We have seen that all four of the main voting systems we are working with fail the Irrelevant Alternatives Criterion (IIA). A summary of each voting method as it relates to the IIA criterion is found in the table below. Voting Method Irrelevant Alternatives Criterion Plurality Violates Ranked-choice Violates Pairwise comparison Violates Borda count Violates Electronic Voting: Does Your Vote Count? In order for an election to be fair, voting must be accessible to everyone and every vote must be counted. When hundreds of thousands to millions of votes must be collected and counted in a short period of time, deciding it can be challenging to be on counting procedures that are accurate and secure. Electronic voting machines or even Internet voting can speed up the process, but how reliable are these methods? This has been a subject for debate for years. In a press release on August 3, 2007, California Secretary of State Debra Bowen explained the results of an extensive review of electronic voting systems in her state. She said that transparency and auditability were key. She went on to say, “I think voters and counties are the victims of a federal certification process that hasn’t done an adequate job of ensuring that the systems made available to them are secure, accurate, reliable and accessible. Congress enacted the Help America Vote Act, which pushed many counties into buying electronic systems that—as we’ve seen for some time and we saw again in the independent UC review—were not properly reviewed or tested to ensure that they protected the integrity of the vote.” Secretary Bowden subsequently ordered that voting machines must have tighter security to be used in California. (DB07:042, Secretary of State Debra Bowen Moves to Strengthen Voter Confidence in Election Security Following Top-to-Bottom Review of Voting Systems, https://sos.ca.gov/elections.) In some instances, the use of electronic voting in parliamentary elections has been discontinued completely for security reasons. For example, according to the National Democratic Institute, the Netherlands returned to all paper ballots and hand counting in 2006. Will you use voting machines or Internet voting in Imaginaria? So far, every one of the voting methods we have analyzed has failed one or more of the fairness criteria in one election or another. Voting Method Majority Criterion Condorcet Criterion (Head-to-Head Criterion) Monotonicity Criterion (Independence of) Irrelevant Alternatives Criterion Plurality Satisfies Violates Satisfies Violates Ranked-choice Satisfies Violates Satisfies Violates Pairwise comparison Satisfies Satisfies Violates Violates Borda count Violates Violates Satisfies Violates You might be wondering if there is a voting system you could recommend for Imaginaria that satisfies all the fairness criteria. If there is one, it remains to be discovered and it is not a voting system that is based solely on preference rankings. In 1972, Harvard Professor of Economics Kenneth J. Arrow received the Nobel Prize in Economics for proving Arrow’s Impossibility Theorem , which states that any voting system, either existing or yet to be created, in which the only information available is the preference rankings of the candidates, will fail to satisfy at least one of the following fairness criteria: the majority criterion, the Condorcet criterion, the monotonicity criterion, and the independence of irrelevant alternatives criterion. This theorem only applies to a specific category of voting systems—those for which the preference ranking is the only information collected. There are other types of voting systems to which the Impossibility Theorem does not apply. For example, there is a class of voting systems called Cardinal voting systems that allow for rating the candidates in some way. “Rating” is different from “ranking” because a voter can give different candidates the same rating. Consider the five-star rating systems used by various industries, or the thumbs up/thumbs down rating system used on YouTube. Could there be a Cardinal voting system that does not violate any of the fairness criteria we have discussed? It’s possible, but more research must be done in order to prove it. Kenneth J. Arrow In 1972, Kenneth J. Arrow, a Harvard Professor of Economics, received the Nobel Prize in Economics jointly with Sir John Hicks, another world renowned economist, for their contributions to economic theory. In particular, Professor Arrow proved mathematically that no ranked voting system meets all four of the fairness criteria discussed in this section. The statement of this fact is known as Arrow’s Impossibility Theorem. Visit this site for more details on Professor Arrow Check Your Understanding Key Terms majority criterion tyranny of the majority Condorcet criterion Condorcet method monotonicity criterion up-rank down-rank independence of irrelevant alternatives criterion (IIA) Arrow’s Impossibility Theorem cardinal voting system Key Concepts There are several common measures of voting fairness, including the majority criterion, the head-to head criterion, the monotonicity criterion, and the irrelevant alternatives criterion. According to Arrow’s Impossibility Theorem, each voting method in which the only information is the order of preference of the voters will violate one of the fairness criteria. Videos Separation of Powers and Checks and Balances

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Standard Divisors, Standard Quotas, and the Apportionment Problem Every person at a party gets their fair slice of the cake. (credit: “apple spice cake” by Mark Bonica/Flickr, CC BY 2.0) Learning Objectives After completing this section, you should be able to: Analyze the apportionment problem and applications to representation. Evaluate applications of standard divisors. Evaluate applications of standard quotas. The Apportionment Problem In the new democracy of Imaginaria, there are four states: Fictionville, Pretendstead, Illusionham, and Mythbury. Each state will have representatives in the Imaginarian Legislature. You might now have an agreement on which voting method your citizens will use to elect representatives. However, before that process can even begin, you must decided on how many representatives each state will receive. This decision will present its own challenges. When sharing your birthday cake, it’s only fair that everyone gets the same portion size, right? You were portioning the cake by dividing it up equally and giving everyone a slice. A great thing about cake is that you can slice it any way you want, but how do you apportion , or divide and distribute, items that can't be sliced? Suppose that you have a box of 16 Ring Pops™, gem-shaped lollipops on a plastic ring. You are going to share the box with four other kids. Dividing the 16 Ring Pops™ among the group of five leads to a problem; after each person in the group gets three Ring Pops™, there is still one left! Who gets the last one? The apportionment problem is how to fairly divide, or apportion, available resources that must be distributed to the recipients in whole, not fractional, parts. The apportionment problem applies to many aspects of life, including the representatives in the Imaginarian legislature. The table below provides a short list of examples of resources that must be apportioned in whole parts, and the recipients of those resources. Resource Recipients Covid-19 Vaccines Nations around the world Airport Terminals Airlines Faculty Positions at a University Departments Public Schools Communities U.S. House of Representatives Seats States Parliamentary Seats Political Parties Fair division of a resource is not necessarily equal division of the resource like when distributing cake slices. When distributing airport terminals amongst airlines, there are many factors to consider such as the size of the airline, the number and types of aircraft they have, and the demand for the service. In most cases, fairness is defined as being proportional ; two quantities are proportional if they have the same relative size. In the case of the Covid-19 vaccine, the expectation would be that countries with larger populations get more doses of the vaccine. In the Imaginarian legislature, the expectation may be that the states with larger populations will receive the larger number of representatives. This concept is referred to as a part-to-part ratio . Suppose that a supermarket has a special on pies, two for $5. The first customer purchases four pies for $10, and the second customer purchases eight pies for $20. The dollar to pie ratio for the first customer is 10 dollars 4 pies = 2.5 dollars per pie and the dollar to pie ratio for the second customer is 20 dollars 5 pies = 2.5 dollars per pie . So, the dollar to pie ratio is constant. Although the customers do not spend the same amount of money, the amount each spent was proportional to the number of pies purchased. Now suppose that the supermarket changed the special to $5 for the first pie, and $2 for each additional pie. In that case, four pies would cost $ 5 + 3 ( $ 2 ) = $ 11 , while 8 pies would cost $ 5 + 7 ( $ 2 ) = $ 19 . The dollar to pie ratios would be 11 dollars 4 pies = 2.75 dollars per pie and 19 dollars 8 pies = 2.375 dollars per pie , respectively. This special does not result in a constant part to part ratio. The dollars spent are not proportional to the number of pies purchased. What Is a Ratio? What Are the Different Types of Ratios? Ratio of Faculty to Students at a College The following table provides a comparison of the number of faculty members in each department at a particular college to the student head count in that department and the number of class sections in that department in the Spring semester. Use this information to answer the questions. Department Mathematics English History Science (S) Student Head Count 4800 2376 1536 2880 (C) Class Sections 120 108 48 96 (T) Total Faculty 30 27 12 24 (F) Full-Time Faculty 10 9 4 8 (P) Part-Time Faculty 20 18 8 16 Determine the ratios for each department: S to C, C to T, S to T, F to P What are the units of the ratios that you found? Which of these pairs, if any, has a constant part to part ratio? State the ratio. Does it appear that the total number of faculty positions were allocated to each department based on student head count, the number of class sections, or neither? Justify your answer. Divide the first quantity by the second, for each department, as shown in the table below. Answers are provided in last column of the table. Department Mathematics English History Science Units of Ratios Found S to C 4800 120 = 40 2376 108 = 22 1536 48 = 32 2880 96 = 30 Students per class section C to T 120 30 = 4 108 27 = 4 48 12 = 4 96 24 = 4 Class sections per faculty member S to T 4800 30 = 160 2376 27 = 88 1536 12 = 128 2880 24 = 120 Students per faculty member F to P 10 20 = 1 2 9 18 = 1 2 4 8 = 1 2 8 16 = 1 2 Full-time faculty member per part-time faculty member The ratio of class sections to faculty members is a constant ratio of four. The ratio of full-time faculty to part-time faculty is a constant ratio of 1 2 . It appears that the faculty positions were allocated based on the number of class sections because there is a constant ratio of four class sections per faculty member. There are some useful relationships between quantities that are proportional to each other. When there is a constant ratio between two quantities, the one quantity can be found by multiplying the other by that ratio. Remember the supermarket special on pies, 2 pies for $5? The ratio of dollars to pies is 5 dollars 2 pies = 2.5 dollars per pie and the ratio of pies to dollars is 2 pies 5 dollars = 0.4 pies per dollar . These two values are reciprocals of each other, 1 2.5 = 0.4 and 1 0.4 = 2.5 . This means that multiplying by one has the same effect as dividing by the other. This also means that knowing either constant ratio allows us to calculate the price given the number of pies. To find the cost of 20 pies, multiply by the ratio of dollars to pies or divide by the ratio of pies to dollars. 20 pies × 2.50 dollars per pie = 50 dollars 20 pies ÷ 0.4 pies per dollar = 50 dollars These patterns are true in general. Let A be a particular item and B another such that there is a constant ratio of A to B ratio of B 's to A 's = 1 ratio of A 's to B 's and ratio of A 's to B 's = 1 ratio of B 's to A 's units of A = ( units of B ) × ( ratio of A 's to B 's ) = units of B ratio of B 's to A 's units of B = ( units of A ) × ( ratio of B 's to A 's ) = units of A ratio of A 's to B 's Ratio of Faculty to Students at a College Refer to the information given in . If there are 32 class sections each semester in the Fine Art department, and the same ratio is used to determine the number of faculty members, how many faculty members would you expect to see in the Fine Art department? If the Health Sciences department has 6 full-time faculty members, how many part-time faculty members are in the department? Multiply the number of class sections by the ratio of faculty members to class sections to find the number of faculty. Since there are 4 classess per faculty member, the ratio of faculty members to classs sections is 1 4 . This means that the number of faculty members for 32 classes should be 32 × 1 4 = 8 faculty members. Multiply the number of full-time faculty by the ratio of part-time to full-time to find the number of part-time. Since ratio of full-time faculty to part-time faculty at the college is 1 2 or 1 full-time per 2 part time, the ratio of part-time to full-time is 2 1 = 2 part-time to 1 full-time; so the number of part-time faculty in a department with 6 full-time faculty should be 6 . 4 = 24 part-time faculty. The apportionment application that will be important to the founders of Imaginaria occurs in representative democracies in which elected persons represent a group. The United Kingdom, France, and India each have a parliament, and the United States has a Congress, just as Imaginaria will have a legislature! The citizens of a country must decide what portion of the representatives each group, such as a state or province or even a political party, will have. A larger portion of representatives means greater influence over policy. Ratio of U.S. Representatives to State Population contains a list of the five U.S. states with the greatest number of representatives in the U.S. House of Representatives, along with the population of that state in 2021. Use the information in the table to answer the questions. State Representative Seats State Population (CA) California 53 39,613,000 (TX) Texas 36 29,730,300 (NY) New York 27 19,300,000 (FL) Florida 27 21,944,600 (PA) Pennsylvania 18 12,804,100 The First through Fifth Ranked States by Number of Representatives (sources: https://www.census.gov/popclock/ [state population], https://www.britannica.com/topic/United-States-House-of-Representatives-Seats-by-State-1787120 [representative seats]) What is the ratio of State Population to Representative Seats for each state to the nearest hundred thousand? What is the ratio of Representative Seats to State Population for each state rounded to seven decimal places? What is the ratio of Representative Seats to State Population for each state rounded to six decimal places? Does there appear to be a constant ratio? Justify your answer. CA 700,000; TX 800,000; NY 700,000; FL 800,000; PA 700,000 CA 0.0000013; TX 0.0000012; NY 0.0000014; FL 0.0000012; PA 0.0000014 CA 0.000001; TX 0.000001; NY 0.000001; FL 0.000001; PA 0.000001 The ratio of State Population to Representative Seats seems to be either 700,000 or 800,000. There does appear to be a constant ratio of about 0.000001 of Representative Seats to State Population if we round off to the sixth decimal place. Math Antics – Rounding You might be wondering why the ratio doesn't appear to be quite the same depending on the rounding of the values. We will see that the key to this variation lies in the fractions. Just like the five children sharing 16 Ring Pops™, there are going to be leftovers and there are many methods for deciding what to do with those leftovers. The Standard Divisor There are two houses of congress in the United States: the Senate and the House of Representatives. Each state has two senators, but the number of representatives depends on the population of the state. The number of representative seats in the U.S. House of Representatives is currently set by law to be 435. In order to distribute the seats fairly to each state, the ratio of the population of the U.S. to the number of representative seats must be calculated. The ratio of the total population to the house size is called the standard divisor , and it is the number of members of the total population represented by one seat. Although apportionment applies to many other scenarios, such as the pencil distribution during the SAT, the terminology of apportionment is based on the House of Representatives scenario. Thus, several government-related terms take on a more general meaning. The states are the recipients of the apportioned resource, the seats are the units of the resource being apportioned, the house size is the total number of seats to be apportioned, the state population is the measurement of the state's size, and the total population is the sum of the state populations. Standard Divisor = Total Population House Size The Standard Divisor of the U.S. House of Representatives 2021 As of this writing, the Census.gov website U.S. Population clock showed a population of 332,693,997. There are 435 seats in the U.S. House of Representatives. Find the standard divisor rounded to the nearest tenth. Dividing 330,147,881 people by 435 seats, there are 758,960.6 people per representative. Whether the standard divisor is less than, equal to, or greater than 1 depends on the ratio of the population to the number of seats. The standard divisor will be equal to 1 if the total population is equal to the number of seats. This would mean that each member of the population is allocated their own personal seat. The standard divisor will be a number between 0 and 1 when the total population is less than the number of seats. This means that each member of the population is allocated more than one seat. The standard divisor will be a number greater than 1when the total population is greater than the number of seats. This means that a certain number of members of the population will share 1 seat. If the population is five children and the house consists of five pieces of candy, the standard divisor is 5 children 5 candies = 1 child per candy meaning each child gets one candy. If the population is five children and ten pieces of candy, the standard divisor is 5 children 10 candies = 0.5 child per candy meaning that each child gets more than one candy. If the population is five children and four pieces of candy, the standard divisor is 5 children 4 candies = 1.25 child per candy meaning that each child gets less than one candy. If the seats in the Imaginarian legislature are distributed to the states based on population, then the house size will be less than the population and we should expect the standard divisor to be a number greater than 1. School Resource Officers in Brevard County, Florida The public schools in a certain county have been allotted 349 school resource officers to be distributed among 327 public schools attended by approximately 271,500 students. Identify the states, seats, house size, state population, and total population in this apportionment scenario. Describe the ratio the standard divisor represents in this scenario and calculate the standard divisor to the nearest tenth. The states are the schools in that county. The seats are the school resource officers. The house size is the number of school resource officers, which is 349. The state population is the number of students in a particular school, which was not given. The total population consists of the sum of the school populations, which is 271,500. The standard divisor is the ratio of the total population to the house size, which is the number of students served by each resource officer. Divide 271 , 500 students ÷ 349 officers = 777.9 students per officer. The Standard Quota Once the standard divisor for the Imaginarian legislature is calculated, the next task is to determine the number of seats that each state should receive, which is referred to as the state’s standard quota . Unless all the states have the same population, each state will receive a different number of seats because the quantities will be proportionate to the state populations. To determine those amounts, we will use an idea we learned earlier. Recall that, when the number of units of item A is proportionate to the number of units of item B , we have: units of A = units of B ratio of B 's to A 's In this case, we are trying to calculate the number of seats a state should be apportioned, the state’s standard quota. So A So A would refer to seats allocated to a particular state, while B would refer to the state population. This means that the ratio of B to A is the ratio of the total population to house size, which is the standard divisor. So in apportionment terms, we have the following formula. State's Standard Quota = State Population Standard Divisor seats The Standard Quota of the U.S. House of Representatives 2021 Example 27 outlined that the Census.gov website U.S. Population clock showed a population of 330,147,881, there are 435 seats in the U.S. House of Representatives, and the standard divisor was 758,960.6 people per representative. The state of California has a population of approximately 39,613,000. Use these values to determine the standard quota for California to two decimal places. California's standard Quota = California's population Standard Divisor = 39 , 613 , 000 758 , 960.6 = 52.19 representatives Apportionment of Laptops in a Science Department The science department of a high school has received a grant for 34 laptops. They plan to apportion them among their six classrooms based on each classroom’s student capacity. Use the values in the table below to find the standard quota for each classroom. Room Students A 30 B 25 C 28 D 32 E 24 F 27 Step 1: Identify the state population, total population, and the house size. The states are the classrooms, and the state populations are listed in the table. The total population is the sum of the state populations, which is 166. The house size is the number of seats, or laptops, to be allocated, which is 34. Step 2: Calculate the standard divisor by dividing the total population by the house size. Standard Divisor = Total of Room Capacities Number of Laptops = 166 34 ≈ 4.88 students per laptop. Step 3: Calculate the standard quota by dividing the state population by the standard divisors , as shown in the table below. Room's Standard Quota = Room Capacity Standard Divisor Room Room Capacity Room’s Standard Quota A 30 30 ÷ 4.88 ≈ 6.15 laptops B 25 25 ÷ 4.88 ≈ 5.12 laptops C 28 28 ÷ 4.88 ≈ 5.74 laptops D 32 32 ÷ 4.88 ≈ 6.56 laptops E 24 24 ÷ 4.88 ≈ 4.92 laptops F 27 27 ÷ 4.88 ≈ 5.53 laptops Step 4: Find the sum of the standard quotas. 6.15 + 5.12 + 5.74 + 6.56 + 4.91 + 5.53 = 34.01 . This is only slightly off from the number of laptops—34—which can be caused by rounding off in previous steps. This is a good indication that the calculations were correct. If you find that the value of the sum of the standard quotas is significantly different from the house size (number of seats), it is possible that the standard divisor was calculated using too few decimal places. Calculate the standard divisor and standard quotas again but round off to a greater number of decimal places. Check Your Understanding Key Terms apportion apportionment problem proportional part-to-part ratio representative democracies standard divisor states seats house size state population total population standard quota Key Concepts The apportionment problem is how to fairly divide and distribute available resources to recipients in whole, not fractional, parts. To distribute the seats in the U.S. House of Representatives fairly to each state, calculations are based on state population, total population, and house size, or the total number of seats to be apportioned. The standard divisor is the ratio of the total population to the house size, and the standard quota is the number of seats that each state should receive. Formulas Let A be a particular item and B another such that there is a constant ratio of A to B . ratio of B 's to A 's = 1 ratio of A 's to B 's and ratio of A 's to B 's = 1 ratio of B 's to A 's units of A = ( units of B ) × ( ratio of A 's to B 's ) = units of B ratio of B 's to A 's units of B = ( units of A ) × ( ratio of B 's to A 's ) = units of A ratio of A 's to B 's Standard Divisor = Total Population House Size State's Standard Quota = State Population Standard Divisor seats Videos What Is a Ratio? What Are the Different Types of Ratios? Math Antics – Rounding

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