Math 110-02, Fall 2008 Assignment 9: Mathematics of Voting

This assignment is due on Wednesday, November 5.
Reminder: There is a test coming up on Friday, November 7.

The first part of the assignment is simply to determine the winners in some elections, using a variety of voting methods. The candidates in each elections are Red, Blue, and Green. Some of these questions are very easy; some of them take more work. A calculator might be useful for some of the exercises.

Exercise 1: Suppose that there are 12 voters. An election is held using the plurality method. There are 3 votes for Red, 4 votes for Blue, and 5 votes for Green. Who wins? Why?

Exercise 2: Now, suppose that voters indicate their preference using a ranked ballot. When the ballots are tabulated, the results are as follows:

 Number ofBallots FirstChoice SecondChoice ThirdChoice 12 Red Blue Green 5 Blue Green Red 10 Green Blue Red 1 Green Red Blue

In this table, the first row (for example) means that 12 voters ranked Red first, Blue second, and Green third.

(a) In a simple plurality election, who would win this election? Why?

(b) Suppose that the election is run using instant runoff voting. Who wins the election in that case? Show your work.

(c) Who wins the election, if anyone, under Condorcet voting? Show your work.

(d) Who wins the election under the Borda Count method? Show your work.

(e) Who do you think really deserves to win this election, if the goal is to best reflect the overall "will of the voters"? Explain.

Exercise 3: For this exercise, suppose that a range ballot is used, where each voter rates each candidate on a scale of 1 to 5, where 1 is the worst rating and 5 is the best. Here are the results when the ballots are tabulated:

 Number ofBallots Ranking forRED Ranking forGREEN Ranking forBLUE 6 ballots 3 5 2 3 ballots 5 4 1 5 ballots 5 3 1 5 ballots 1 4 5 7 ballots 1 3 5 3 ballots 2 1 5

In this table, the first row, for example, says that 6 voters assigned a rating of 3 out of 5 to Red, 5 out of 5 to Green, and 2 out of 5 to Blue.

(a) In range voting, the winner is the candidate with the highest average rating. Which candidate wins the election, using range voting? Show your work.

(b) Let's assume that if a voter were asked to rank the candidates (first choice, second choice, third choice), then the voter's ranking would reflect their ratings of the candidates. For example, a voter who rated Red 3 out 5, Green 5 out of 5, and Blue 2 out of 5 would have Green as first choice, Red as second choice, and Blue as third choice. Make a table, similar to the one for Exercise 2, that shows the complete set of rankings for the voters in this election.

(c) Given the table of rankings in part (b), one of the candidates would win a plurality election with an actual majority. Which one? Why?

(d) Based on the table of rankings in part (b), which candidate, if any, would be the winner under the Condorcet voting method? Why?

(e) Based on the table of rankings in part (b), which candidate would be the winner under the Borda Count voting method? Show your work.

Majority Rule?

Exercise 4: Since we tend to believe in "majority rule," it seems like a candidate who is the first choice of a majority of the voters should be declared the winner. But is that really a good rule? Exercise 3 showed that some voting systems can violate this rule; that is, it is possible that there is a candidate who is the first choice of a majority, but a different candidate is declared to be the winner. Is that a reason for rejecting those voting systems? Or is it possible that a majority choice does not always represent the overall "will of the voters" most accurately?

(a) Write a paragraph making the case that a candidate that is the first choice of a majority of the voters should always be declared the winner.

(b) Write a paragraph making the case that that sometimes a candidate that is the first choice of a majority of the voters should not be declared the winner.

(c) Which position do you agree with? What would you say to someone who takes the opposite position?

Parliamentary Procedure

We have been talking about elections where a winner is chosen from a group of options based on ballots filled in by the voters. But there are other ways to make a choice among alternatives. One is to have a sequence of elections, where each election matches up two options. The winning option from one election goes on to the next election, where it will be matched against another option. Although this might be a strange way to run an election among candidates for a political office, it's done all the time in another type of social choice situation: parliamentary debate.

When a proposed bill is debated by a legislative body, members of the body can propose amendments to the bill. An amendment is a proposed modification to the bill that is under consideration. The legislators vote on the amendment. If it passes, the bill is modified by the amendment, so that it is actually a different bill that is then under consideration. We can consider a vote on the amendment to be an "election" between two options: the unmodified bill without the amendment and the bill as modified by the amendment. One of these options is selected, and that option goes on to the next "election": another proposed amendment or a final vote on the bill.

Let's consider the case where there is only one proposed amendment. Then there are really only three possible final outcomes: (1) The amendment is rejected, and the bill without the amendment is then passed. (2) The amendment is passed, and the bill with the amendment is then passed. (3) The bill does not pass in the end; this option represents the status quo where no change to current law is made (and it doesn't really matter whether or not the amendment passed in this case, because the end result is the same).

We assume that the preferences of a legislator can be expressed as a ranking of these three alternatives. As an example, suppose there are three groups of legislators. Legislators in Group A prefer the status quo, but if the bill is passed, they would prefer the unamended bill to the amended bill. Legislators in Group B would prefer to pass the bill without the amendment, but they don't like the amendment at all, and prefer not passing the bill to passing the bill with the amendment. Legislators in Group C like the amendment and would most prefer to pass the bill without the amendment, but they prefer the unamended bill to the option of not passing the bill at all. Let's suppose that there are 6 voters in Group A, 5 voters in Group B, and 8 voters in Group C. We can collect the rankings of all the voters in the following table. This is similar to the ranking table in Exercise 2, even though this information would not actually be collected from the legislators on a single ballot:

 Number ofVoters FirstChoice SecondChoice ThirdChoice Group A:6 Voters Don't passthe bill Pass the billwithout amendment Pass the billwith amendment Group B:5 Voters Pass the billwithout amendment Don't passthe bill Pass the billwith amendment Group C:8 Voters Pass the billwith amendment Pass the billwithout amendment Don't passthe bill

Exercise 5: Given this table, consider what could happen when the bill is debated and someone proposes the amendment. First, there will be a vote on whether or not to amend the bill. Then, there will be a vote on the bill itself, either with or without the amendment based on the result of the first vote.

(a) Suppose that everyone votes honestly. First comes the vote on whether to amend the bill. Groupts A and B, who don't like the amendment, vote against the amendment while Group C votes for it. What happens to the amendment, and does the bill pass in the end? Explain.

(b) Now suppose instead that the voters in Group A vote for the amendment, while all the other votes are the same. Does the amendment pass? Does the bill pass in the end? Explain.

(c) Discuss the result in part (b). Why would Group A vote for an amendment that they actually oppose? Does the final result reflect the true will of the voters? Explain how strategic voting is illustrated by this example.

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