Chapter 1 Review. 1. Write a summary of what you think are the important points of this chapter. 2. Consider the following set of preferences.

Transcription

1 hapter 1 Review 1. Write a summary of what you think are the important points of this chapter. 2. onsider the following set of preferences. E E E E a. etermine a winner using a orda count. b. etermine a plurality winner. c. etermine a runoff winner. d. etermine a sequential runoff winner. e. etermine a ondorcet winner. f. Suppose that this election is conducted by an approval model and all voters approve of the first two choices on their preference schedules. etermine an approval winner. 3. omplete the following table for the recurrence relation n = 2 n 1 + n. n n (3) + 2 =

2 hapter 1 Review In this chapter, you encountered several paradoxes involving groupranking models. a. One of the most surprising paradoxes occurs when a winning choice becomes a loser when the choice s standing actually improves. In which group-ranking model(s) can this occur? b. iscuss at least one other paradox that occurs with groupranking models. 5. In the 1912 presidential election, polls showed that the preferences of voters were as follows. Wilson Roosevelt Taft Roosevelt Taft Wilson Taft Roosevelt Wilson 45% 30% 25% a. Who won the election? Was he a majority winner? b. How did the majority of voters feel about the winner? c. How could one of the groups of voters have changed the results of the election by voting insincerely? d. iscuss who might have won the election if a different model had been used. 6. Your class is ranking soft drinks and someone suggests that the names of the soft drinks be placed in a hat and the group ranking be determined by drawing them from the hat. Which of rrow s conditions does this method violate? 7. fter their final round of skating in the 1995 World Figure Skating hampionship, hen Lu of hina, Nicole obek of the United States, and Surya onaly of France were in first, second, and third place, respectively, with little chance of any remaining skater passing them. However, when merican Michelle Kwan skated, she did well enough to move into fourth place. ut something else quite surprising happened. Kwan s scores reversed the positions of obek and onaly. Which of rrow s conditions did the scoring system violate in this case? Explain.

3 50 hapter 1 Election Theory: Modeling the Voting Process 8. State rrow s theorem. In other words, what did rrow prove? 9. an the point system used to do a orda count affect the ranking (for example, a system instead of a system)? onstruct an example to support your answer. 10. The 1992 presidential election was unusual because of a strong third-party candidate. In that election ill linton received 43% of the popular vote, George ush 38%, and Ross Perot 19%. Steven rams and Samuel Merrill III used polling results to estimate the percentage of those voting for one candidate who also approved of another. pproximately 15% of linton voters approved of ush and approximately 30% approved of Perot. pproximately 20% of ush voters approved of linton and approximately 20% approved of Perot. pproximately 35% of Perot voters approved of linton and approximately 30% approved of ush. a. Estimate the percentage of approval votes each candidate would have received if approval voting had been used in the election. b. Find the total of the three percentages you gave as answers in part a. Explain why the total is not 100%. 11. hoose an election model from those you have studied in this chapter that you think best to use to determine a winner for the following preferences. Explain why you think your choice of method is best

4 hapter 1 Review In 2012, Washington Nationals outfielder ryce Harper won the National League Jackie Robinson Rookie of the Year ward. (There are two winners each year: one in the National League and one in the merican League.) The results of the National League Voting are shown in the following table. Player First-Place Votes Second-Place Votes Third-Place Votes Total ryce Harper Wade Miley Todd Frazier Wilin Rosario Norichika oki Yonder lonso 1 1 Matt arpenter 1 1 Jordan Pachecho 1 1 What type of voting model is used to select rookie of the year? 13. There are conditions other than rrow s that some experts consider important to a fair group-ranking model. For example, onald Saari thinks that a good method should not retain the same winner if the voters reverse their preferences. Mathematician of Note a. To see how this reversal effect works, find a plurality winner for the following set of voter preferences. Then reverse the order of the rankings on each schedule and find a plurality winner. onald Saari (1940 ) Professor Saari is istinguished Professor of Mathematics and Economics at University of alifornia Irvine

5 52 hapter 1 Election Theory: Modeling the Voting Process b. o any other models that you studied in this chapter demonstrate a reversal effect in the set of preferences in part a? That is, do any other models leave the winner unchanged when preferences are reversed? Explain. c. Examine the following set of preferences for reversal effects onsider a situation in which voters,,, and have 4, 3, 3, and 2 votes, respectively, and 7 votes are needed to pass an issue. a. List all winning coalitions and their vote totals. b. Find a power index for each voter. c. o the power indices reflect the distribution of votes? Explain. d. Suppose the number of votes necessary to pass an issue increases from 7 to 8. How does this change the voters power indices? 15. county planning commission has five members. Each member s vote is weighted to reflect the population of the community the member represents. Member has 1 vote, has 1 vote, has 2 votes, has 5 votes, and E has 6 votes. simple majority of the vote total is required to pass an issue. o any of the members seem to have considerably more or less power than intended? Explain.

6 hapter 1 Review 53 ibliography my, ouglas J Real hoices/new Voices: New York: olumbia University Press. my, ouglas J ehind the allot ox: itizen s Guide to Voting Systems. Westport T: Praeger. alinski, MIchel and Rida Laraki Majority Judgment: Measuring, Ranking, and Electing. ambridge/new York: MIT Press. rams, Steven J pproval Voting on ills and Propositions. The Good Society 5(2): rams, Steven J Paradoxes in Politics. New York: Free Press. rams, Steven J., and Peter Fishburn pproval Voting. New York: Springer. unch, ryan Mathematical Fallacies and Paradoxes. New York: over. OMP For ll Practical Purposes: Introduction to ontemporary Mathematics. 9th ed. New York: W. H. Freeman. ole, K The Universe and the Teacup: The Mathematics of Truth and eauty. New York: Mariner. avis, Morton Mathematically Speaking. New York: Harcourt race Jovanovich. Falletta, Nicholas The Paradoxican. New York: Wiley. Gardner, Martin From ounting Votes to Making Votes ount: The Mathematics of Elections. Scientific merican 251(4): 16. Hoffman, Paul rchimedes Revenge. New York: W. W. Norton & ompany. Lijphart, rend Patterns of emocracy: Government Forms and Performance in Thirty-Six ountries. New Haven: Yale University Press. Lucas, William F Fair Voting: Weighted Votes for Unequal onstituencies. Lexington, M: OMP, Inc. Lum, Lewis, and avid. Kurtz Voting Made Easy: Mathematical Theory of Election Procedures. Greensboro, N: Guilford Press. Niemi, Richard G., and William H. Riker The hoice of Voting Systems. Scientific merican 234(6): 21.

7 54 hapter 1 Election Theory: Modeling the Voting Process Pakhomov, Valery emocracy and Mathematics. Quantum January-February: 4-9. Rush, Mark E., and Richard L. Engstrom Fair and Effective Representation: ebating Electoral Reform and Minority Rights. Lanham, M: Rowman & Littlefield. Saari, onald haotic Elections!: Mathematician Looks t Voting. Providence, RI: merican Mathematical Society. Saari, onald ecisions and Elections: Explaining the Unexpected. New York: ambridge University Press. Saari, onald isposing ictators, emystifying Voting Paradoxes: Social hoice nalysis. New York: ambridge University Press. Szpiro, George G Numbers Rule: The Vexing Mathematics of emocracy. Princeton, NJ: Princeton University Press. Taylor, lan Mathematics and Politics: Strategy, Voting, Power and Proof. New York: Springer. Taylor, lan Social hoice and the Mathematics of Manipulation. New York: ambridge University Press.