Exam 10: The Manipulability of Voting Systems

A vote using the Borda count method is:

Consider the following preference table for four voters: First choice B D D C Second choice C B A A Third choice A C B B Fourth choice D A C D If a Borda count is used, can the fourth voter manipulate the outcome so that C wins?

Use the following information to answer the Questions: Twenty-nine voters must choose from among three alternatives: A, B, and C. The voters' preference schedules are shown below. $\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\text { Number of voters }$ 12 8 6 3 First choice Second choice Third choice -Using the agenda A, B, C, and sequential pairwise voting, alternative B wins. Is there an agenda that produces C as a winner?

Use the following information to answer the Questions A group of 22 young people must decide whether to go to the beach (B), the mountain (M), or the zoo (Z) on a field trip. Their preference rankings are summarized in the table below, and the decision will be made using a Borda count. $\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\text { Number of voters }$ 10 8 4 First choice Second choice Third choice Z Z -Can the four voters in the last column change the results of the vote to their favor by changing their preference rankings?

Consider the following preference table for three voters to answer Questions First choice Second choice Third choice Fourth choice -If a Borda count is used, can the voter on the left manipulate the outcome to his or her benefit?

A vote among exactly three candidates using the Borda count method is:

If a voting system has three or more alternatives, satisfies the Pareto condition, always produces a unique winner, and is not a dictatorship, what conclusion follows from the GS theorem?

Use the following information to answer the Questions Twenty-nine voters must choose from among three alternatives-A, B, and C-using the Borda count method. The voters' preference schedules are shown below. $\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\text { Number of voters }$ 12 8 6 3 First choice Second choice Third choice -Suppose the group of voters anticipates that the group of six plans to insincerely rank candidate B above candidate A. How can the remaining voters respond in their own rankings?

Consider the following preference table for two voters: First choice C B Second choice B A Third choice D C Fourth choice A D If a Borda count is used, can the voter on the left manipulate the outcome to his or her benefit?

Use the following preference table to answer the Questions . $\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\text { Number of voters }$ 4 6 8 4 First choice Second choice Third choice Fourth choice -Who wins using plurality? Could the four voters who most prefer candidate B vote insincerely to change the outcome in a way that would benefit them?

Use the following information to answer the Questions A 17-member committee must elect one of four candidates: R, S, T, or W. See the preference schedule below. $\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\text { Number of }$ $\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\text { metnbers }$ 6 4 3 4 First choice Second choice Third choice Fourth choice -R wins using the plurality method. Could those members who most prefer T vote strategically in some way to change the outcome in a way that will benefit them?

Use the following preference table to answer the Questions . $\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\text { Number of voters }$ 4 6 8 4 First choice Second choice Third choice Fourth choice -Who wins using plurality? Could the four voters who most prefer candidate D vote insincerely to change the outcome in a way that would benefit them?

The GS theorem states the nonexistence of a voting system with three or more candidates which satisfies a list of certain properties. Which of the following is not on the list of those properties?

According to the Chair's Paradox, what surprising situation can befall the tie-breaking chair of a committee?

Consider the following preference table for three voters to answer Questions First choice Second choice Third choice Fourth choice -If a Borda count is used, can the voter on the right manipulate the outcome to his or her benefit?

Consider the following preference table for four voters to answer the Questions First choice Second choice Third choice Fourth choice Fifth choice -If a Borda count is used, can the first voter manipulate the outcome so that B wins?

When there are only two or three candidates, nonmanipulability and monotonicity are exactly the same thing.

Use the following information to answer the Questions Consider an 11-member committee that must choose one of three alternatives: X, Y, or Z. Their schedule of preferences is shown below. $\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\text { Number of voters }$ 5 4 2 First choice Second choice Third choice -If the Hare system is used, alternative X wins. Could the voters who most prefer Y vote insincerely in some way to change the outcome in a way that would benefit them?

Use the following information to answer the Questions: Consider an 11-member committee that must choose one of three alternatives -X, Y, or Z- using the Hare system. Their schedule of preferences is shown below. $\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\text { Number of voters }$ 5 4 2 First choice Second choice Third choice -Who wins? Is it possible for the group of two voters to change the outcome in a way that would benefit them?

Determine whether plurality voting method is manipulable.