Lesson 1.2. Group-Ranking Models

Transcription

1 Lesson 1.2 Group-Ranking Models If the soft drink data for your class are typical, you know that the problem of establishing a group ranking is not without controversy. Even among professionals, there isn t agreement on the best way to do so. This lesson examines several common models for determining a group ranking from a set of voter preferences. s you examine these models, consider whether any of them are similar to the ones devised by members of your class in Lesson 1.1. onsider the preferences of Exercise 9 of the previous lesson, which are shown again in Figure 1.1. Total number of voters: = 2. Figure 1.1. Preferences of 2 voters. Many voting situations, such as elections in which there is only one office to fill, require the selection of a single winner. lthough most such elections in the United States do not use a preferential ballot, they could. For example, in the set of preferences shown in Figure 1.1, choice is ranked first on eight schedules, more often than any other choice.

2 hapter 1 Election Theory: Modeling the Voting Process If wins on this basis, is called the plurality winner. plurality winner is based on first-place rankings only. The winner is the choice that receives the most votes. Note, however, that is first only on about 30.% of the schedules. If is first on over half the schedules, is a majority winner. Mathematician of Note Jean-harles de orda ( ) French cavalry officer, naval captain, mathematician, and scientist, he preferred methods that assign points to the rankings of individuals because he was dissatisfied with plurality methods. orda Models id anyone in your class determine the soft drink ranking by assigning points to the first, second, third, and fourth choice of each individual s preference and obtaining a point total? If so, these groups used a type of orda count. The most common way that orda methods are applied to a ranking of n choices is by assigning n points to a firstplace ranking, n 1 to a second-place ranking, n 2 to a third-place ranking,... and 1 point to a last-place ranking. The group ranking is established by totaling each choice s points. In the example of Figure 1.1, is ranked first by people and fourth by the remaining 1, so s point total is (4) + 1(1) = 0. Similar calculations give totals of 3, 9, and for,, and, respectively, as summarized below. : (4) + (1) + (1) + (1) = 0 Technology Note : (3) + (4) + (3) + (3) = 3 : (2) + (3) + (4) + (2) = 9 : (1) + (2) + (2) + (4) = In this case, the plurality winner does not fare well under a orda system. orda counts can be done quickly on a graphing calculator: after the first calculation is typed and entered, it is replayed and edited.

3 Lesson 1.2 Group-Ranking Models 9 Runoff Models Some elections in the United States and other countries require a majority winner. If there is no majority winner, a runoff election between the top two candidates is held. Runoff elections are expensive because of the cost of holding another election and time-consuming for voters because they require a second trip to the polls. However, if voters use a preferential ballot, both disadvantages can be avoided. To conduct a runoff, determine the number of firsts for each choice. In the example of Figure 1.1, is first eight times, is first five times, is first six times, and is first seven times. High ost of Runoff Elections Has Some in lbany Thinking It's Time to End Them New York aily News Oct. 4, 2009 labama will spend $3 million to conduct a runoff election Tuesday that s expected to attract only one-fifth of the state s registered voters. Turnout for the primary June 4 was 34 percent. The runoff is for races where one candidate didn t get at least one vote over 0 percent in the primary. Eliminate all choices except the two with more first-place votes than the others: hoices and are eliminated; and are retained. Now consider each preference schedule on which the eliminated choices are ranked first. hoice is first on the second schedule. Of the two remaining choices, and, is ranked higher than, so these votes are transferred to. Similarly, the votes from the third schedule are transferred to. The totals are now for and + + = 1 for, and so is the runoff winner (see Figure 1.2). The runoff method eliminates all choices except the two with the most firsts: : : : : (Eliminate &.) The five votes for are transferred to, and the six votes for are transferred to. : : + + = 1. Figure 1.2. runoff.

4 10 hapter 1 Election Theory: Modeling the Voting Process Sequential Runoff Models Some elections are conducted by a runoff variant that eliminates only one choice at a time. However, if there are several choices and voters must vote again when one is eliminated, sequential runoff methods compound the disadvantages of runoff methods. s with the runoff method, if voters use a preferential ballot, they need vote only once. In the example of Figure 1.1, is eliminated first because it is ranked first the fewest times. The first-place votes for are transferred to. The point totals are now for, + = 11 for, and for. There are three choices remaining. Now s total is the smallest, so is eliminated next. The votes are transferred to the remaining choice that is ranked highest by these. Thus, is given an additional votes and so defeats by 1 to (see Figure 1.3). The sequential runoff method eliminates only one choice at a time: : : : : (Eliminate.) The five votes for are transferred to : : : + = 11 : (Eliminate.) The seven votes for are transferred to : : : 11 + = 1. Figure 1.3. sequential runoff. When a preferential ballot is used and computers process the data, sequential runoff results are obtained almost instantaneously. Thus, the term instant runoff is often applied to sequential runoffs.

5 Lesson 1.2 Group-Ranking Models 11 Exercises For Exercises 1 4, apply the methods as they are described in this lesson. 1. Find a plurality winner for the soft drink voting in your class. Is it also a majority winner? Explain. 2. Find a orda winner for the soft drink voting in your class. 3. Find a runoff winner for the soft drink voting in your class. 4. Find a sequential runoff winner for the soft drink voting in your class.. The International Olympic ommittee uses sequential runoff voting to choose Olympic sites. Since the committee is relatively small, separate rounds of voting are used rather than a single round with preferential ballots. There were three cities competing to host the 2014 winter games. This table summarizes the voting. Sochi 34 1 Pyeongchang 3 4 Salzburg 2 a. Write a short summary of the voting, including the order in which the cities were eliminated. b. Five cities competed to host the 201 summer games. Read the news article about the voting and construct a similar table. Rio to Stage 201 Olympic Games Oct. 2, 2009 razil will become the first South merican country to host the Olympics after the city of Rio de Janeiro was chosen to stage the 201 Games. hicago's early exit was a surprise, after bookmakers made them favorites. hicago received only 1 of the 94 votes available in the first round poll of IO delegates. Madrid came out top with 2, followed by Rio with 2. In the second round, however, Rio almost secured the absolute majority needed to win outright, with 4 of the 9 votes cast. Madrid came a distant second with 29, while Tokyo was eliminated after receiving 20. The final ballot saw Rio win by a comprehensive margin of votes to 32.

6 12 hapter 1 Election Theory: Modeling the Voting Process. For the example of Figure 1.1, find the percentage of voters that rank each choice first and last. a. Enter the results in a table like this: hoice First Last b. On the basis of these percentages only, which choice do you think is most objectionable to voters? Least objectionable? Explain your answers. c. Which choice do you think most deserves to be ranked first for the group? Explain your reasoning. d. Give at least one argument against your choice.. The 2010 race for governor of Minnesota had three strong candidates. The following are the results from the general election. Mark ayton 919,232 Tom Emmer 910,42 Tom Horner 21,4 Others 2,40 a. What percentage of the vote did the winner receive? Is the winner a majority winner? b. What is the smallest percentage that a plurality winner can receive in a race with three candidates? Explain.. etermine plurality, orda, runoff, and sequential runoff winners for the following set of preferences. pply the methods as they are described in this lesson

7 Lesson 1.2 Group-Ranking Models orda models produce a complete group ranking, but the other modelss examined in this lesson determine a winner only. However, each of them can be extended to produce a complete group ranking. a. escribe how a plurality model could be extended to determine a second, third, and so forth. pply your extension to the example in Figure 1.1 and list the second, third, and fourth that it produces. b. escribe how runoff models could be extended to determine a second, third, and so forth. pply your extension to the example in Figure 1.1 and list the second, third, and fourth that it produces. c. escribe how a sequential runoff model could be extended to determine a second, third, and so forth. pply your extension to the example in Figure 1.1 and list the second, third, and fourth that it produces. 10. Each year the Heisman Trophy recognizes one of the country s outstanding college football players. In 2012, Texas & M freshman quarterback Johnny Manziel received the award. The results of the voting follow. Each voter selects a player to rank first, another to rank second, and another to rank third. 1st 2nd 3rd Points 1. Johnny Manziel , Manti Te o ,0 3. ollin Klein Marqise Lee raxton Miller Jadeveon lowney Jordan Lynch Tavon ustin Kenjon arner Jarvis Jones a. How many points are awarded for a first-place vote? For a second-place vote? For a third-place vote? b. oes the ranking produced by this model differ if a plurality model is used? Explain.

8 14 hapter 1 Election Theory: Modeling the Voting Process 11. In most merican runoff elections, voters do not use a preferential ballot and therefore must return to the polls to vote in the runoff. Voters in some countries, such as Ireland, use a preferential ballot and therefore go to the polls only once. Examine the vote totals in the two runoffs shown below. What do the totals tell you about the merits of preferential ballots in runoff elections? President of Ireland: 2011 Results General Election Runoff Michael Higgins 01,101 1,00,104 Seán Gallagher 04,94 2,114 Martin McGuinness 243,030 Gay Mitchell 113,321 avid Norris 109,49 ana Scallon 1,220 Mary avis 4, U.S. House Louisiana istrict 3: 2012 Results General Election Runoff harles oustany 139,123,20 Jeff Landry 93,2 3,4 Ron Richard,00 ryan arrilleaux,90 Jim Stark 3, Michael Higgins ecomes Irish President The Guardian Oct. 29, 2011 The Irish Labour party's Michael Higgins has been confirmed as the ninth president of the republic after winning a landslide victory in the most fractious campaign in the country's history. The poet and campaigner gained a.% share of the vote, putting him comfortably ahead of his rivals ragons' en star Seán Gallagher, who came second, and former IR commander Martin McGuinness, who ended in third place. The result capped a two-day count of ballots to determine who would succeed Mary Mcleese as Ireland's head of state.

9 Lesson 1.2 Group-Ranking Models In sequential runoffs, the number of choices on a given round is 1 less than the number of choices on the previous round. Let n represent the number of choices after n rounds and write this as a recurrence relation. 13. procedure for solving a problem is called an algorithm. This lesson discusses various algorithms for obtaining a group decision from individual preferences. lgorithms are often written in numbered steps in order to make them easy to apply. The following is an algorithmic description of a runoff model as discussed in this lesson. 1. For each choice, determine the number of preference schedules on which the choice is ranked first. 2. Eliminate all choices except the two that are ranked first most often. 3. For each preference schedule, transfer the vote total to the remaining choice that ranks highest on that schedule. 4. etermine the vote total for the preference schedules on which each of the remaining choices is ranked first.. The winner is the choice ranked first on the most schedules. a. Write an algorithmic description of the sequential runoff model discussed in this lesson. b. Write an algorithmic description of the orda model discussed in this lesson. 14. The number of first-, second-, third-, and fourth-place votes for each choice in an election can be described in a table, or matrix, as shown below. The preferences: The matrix: 1st nd rd th

10 1 hapter 1 Election Theory: Modeling the Voting Process The number of points that a choice receives for first, second, third, and fourth place can be written in a matrix, as shown below. 1st 2nd 3rd 4th Points new matrix that gives orda point totals for each choice can be computed by writing this matrix alongside the first, as shown below [ ] Entries for a new matrix can be computed in this way: Multiply each entry of the first matrix by the corresponding entry in the first column of the second matrix and find the sum of these products. 4(20) + 3(0) + 2(2) + 1(12) = 142 This number is the first entry in a new matrix that gives orda point totals for choices,,, and : Point totals: [142 ] a. alculate the remaining entries of the new matrix. b. If you have this new matrix but not the voter preference schedules, by which methods is it possible to determine the winner? Explain. omputer/alculator Explorations 1. Enter the soft drink preferences of your class members into the election machine computer program that accompanies this book. ompare the results given by the computer to your answers to the first four exercises of this lesson. Resolve any discrepancies.

11 Lesson 1.2 Group-Ranking Models 1 Projects 1. Write a short report on the history of any of the models discussed in this lesson. Look into the lives of people who were influential in developing the model. iscuss factors that led them to propose the model. 1. Find at least two examples of group-ranking models that are currently used somewhere in the world but not discussed in this lesson. escribe how the group ranking is determined. ompare each new model with those described in this lesson. What are some advantages and disadvantages of each new model? 1. Select one or more countries that are not discussed in this lesson and report on the models they use to conduct elections. ayern Munich Stays est Team in P Global Poll ssociated Press pril 30, 2013 fter dominating the first legs of their hampions League semifinals, it's no surprise to see ayern Munich and orussia ortmund leading the way in the latest ssociated Press global football poll. ayern pipped ortmund to be voted the world's top team for the fifth straight week, with its 4-0 hammering of arcelona impressing the P's panel of 1 journalists. P Global Football Rankings for the week ending pril 29. ased on 1 voters, using 10 points for first, nine for second, one for bottom place. Previous rankings in parentheses. 1. ayern Munich (1), orussia ortmund (), Juventus (2), 1. helsea,. Liverpool, 0. Real Madrid (4), 3. Paris Saint-Germain (), Real Sociedad 10. arcelona (), 24