9.1 An Introduction to Social Choice

9.2 Majority Rule and Condorcet’s Method

1. In a few sentences, explain why minority rule (the voting procedure for two alternatives that is described on page 408) satisfies conditions 1 and 2 on page 407, but not condition 3.

2. In a few sentences, explain why imposed rule (the voting procedure for two alternatives that is described on pages 407–408) satisfies conditions 1 and 3 on page 407, but not condition 2.

3. In a few sentences, explain why a dictatorship (the voting procedure for two alternatives that is described on page 407) satisfies conditions 2 and 3 on page 407, but not condition 1.

4. Find (or invent) a voting rule for two alternatives that satisfies condition

5. Construct a real-world example (perhaps involving yourself and two friends) where the individual preference lists for three alternatives are as in the voting paradox of Condorcet.

6. Condorcet’s voting paradox shows that with three voters (or three equal-size groups of voters) and the three alternatives , and , it is possible to have two-thirds prefer to , two-thirds prefer to , and two-thirds prefer to . Find four preference lists that show that with four voters and the four alternatives , and , it is possible to have three-fourths prefer to , three-fourths prefer to , three-fourths prefer to , and three-fourths prefer to .

7. Generalize the result in Exercise 6 from four alternatives to alternatives: .

8. The mathematics department is hiring a new faculty member and the five-person hiring committee has interviewed four candidates: Adam, Beth, Carol, and Dan. They have decided to use Condorcet’s method on their five ballots (reproduced in the following table). Who gets the offer?

9. Suppose that votes on the five mathematics department ballots described in Exercise 8 were distributed according to the table below. Who would get the offer now?

9.3 Other Voting Systems for Three or More Candidates

10. Plurality voting is illustrated by the 1980 U.S. Senate race in New York among Alfonse D’Amato (, a conservative), Elizabeth Holtzman (, a liberal), and Jacob Javits ( , also a liberal). Reasonable estimates (based largely on exit polls) suggest that voters ranked the candidates according to the following table:

434

11. Condorcet’s method can be used to create a new voting system that operates in a manner similar to the Hare system in that it involves repeatedly deleting candidates that are “least preferred.” But now we use Condorcet’s method to decide what least preferred means, and we do this in the following clever kind of way. We tip the ballots upside-down and we look for a Condorcet winner from these inverted ballots. Intuitively, a candidate that wins when all the ballots are reversed is “least preferred,” according to the original ballots. So this new system works by tipping the ballots upside-down and then repeatedly deleting a Condorcet winner, if there is one. Use this new system to find the winner for the following ballots:

12. Use the voting system introduced in the preceding problem to find the winner for the following ballots:

13. (Everyone wins.) Consider the following set of preference lists:

Note that the first list is held by three voters, not just one. Calculate the winner using

14. Consider the following set of preference lists:

Calculate the winner using

15. Consider the following set of preference lists:

Calculate the winner using

16. Consider the following set of preference lists:

435

Calculate the winner using

17. Consider the following set of preference lists:

Calculate the winner using

18. In a few sentences, explain why Condorcet’s rule satisfies

19. In a few sentences, explain why plurality voting satisfies

20. In a few sentences, explain why the Borda count satisfies

21. In a few sentences, explain why sequential pairwise voting satisfies

22. In a few sentences, explain why the Hare system satisfies the Pareto condition.

23. In a few sentences, explain why the plurality runoff method satisfies the Pareto condition.

24. Use the following ballots to show that the plurality runoff method does not satisfy the CWC:

25. Use the following ballots to show that the plurality runoff method does not satisfy monotonicity:

26. Consider the following two elections among Candidates , and :

27. Construct ballots for the alternatives , and to show that the Borda count does not satisfy the CWC.

436

28. Show that the nonmonotonicity of the Hare system can also be demonstrated by the following 17-voter, 4-alternative election. (In a number of recent books, this example is used to show the nonmonotonicity of the Hare system. The 13-voter, 3-alternative example given in the text was pointed out to us by Matt Gendron when he was an undergraduate at Union College.)

29. The following example illustrates how badly the Hare system can fail to satisfy monotonicity. Consider the following sequence of preference lists:

30. In a few sentences, explain why, with an odd number of voters,

31. Suppose there are three voters and three alternatives , and .

9.4 Insurmountable Difficulties: Arrow’s Impossibility Theorem

32. Complete the proof of the version of Arrow’s impossibility theorem from the text by showing that neither nor can be a winner in the situation described. (Your argument will be almost word for word the same as the proofs in the text.)

9.5 A Better Approach? Approval Voting

33. The 10 members of a board vote by approval voting on eight candidates for new positions on their board, as indicated in the following table. An indicates an approval vote. For example, Voter 1, in the first column, approves of Candidates , and , and disapproves of , and .

34. The 45 members of a school’s football team vote on three nominees, , and , by approval voting for the award of “most improved player,” as indicated in the following table. An X indicates an approval vote.

Chapter Review

35. In a sentence or two, explain why it’s impossible, with an odd number of voters, to have two distinct candidates win the same election using Condorcet’s method.

36. Consider the following set of preference lists:

Calculate the winner using

37. An interesting variant of the Hare system was proposed by psychologist Clyde Coombs. It operates exactly as the Hare system, but instead of deleting alternatives with the fewest first-place votes, it deletes those with the most last-place votes.

38. In a few sentences, explain why the plurality runoff method can never elect a candidate ranked last on a majority of ballots, assuming there are no ties for first or second place in the voting.

39. Produce ballots showing that plurality voting can, in fact, elect a candidate ranked last on a majority of the ballots.

40. A voting system is said to satisfy the majority criterion if a candidate ranked first by a majority of the voters is always among the winners. For each of the following, either give a sentence or two explaining why the answer is “yes,” or give a collection of ballots showing that the answer is “no.”

41. Every voting system can be used to create a new voting system in the manner that we did with Condorcet’s method in Exercise 11 on page 434. That is, works as follows: We tip the ballots upside- down and repeatedly delete the “winners” using the voting system . The last candidate (or group of candidates) to be eliminated is the winner. Describe in one sentence the voting system that yields the Hare system as .