Exam 10: The Manipulability of Voting Systems

Consider the following preference table for three voters to answer the Questions First choice Second choice Third choice Fourth choice Fifth choice -Is there an agenda for which C would win a sequential pairwise election?

Consider the following preference table for three voters to answer the Questions First choice Second choice Third choice Fourth choice Fifth choice -Is there an agenda for which D would win a sequential pairwise election?

Consider the following preference table for two voters: First choice C A Second choice D D Third choice B B Fourth choice A C If a Borda count is used, can the voter on the left manipulate the outcome to his or her benefit with no ties for the winner?

Use the following preference table for an election to be decided by sequential pairwise voting to answer the Questions . $\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\text { Number of voters }$ 1 1 1 First choice Second choice Third choice Fourth choice Fifth choice -Is there an agenda for which candidate E wins? If so, give an example.

Consider the following preference table for three voters to answer the Questions First choice A D B Second choice D C A Third choice B B D Fourth choice C E C Fifth choice E A E -Is there an agenda for which B would win a sequential pairwise election?

Use the following preference table for an election to be decided by sequential pairwise voting to answer the Questions . $\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\text { Number of voters }$ 1 1 1 First choice Second choice Third choice Fourth choice Fifth choice -Is there an agenda for which candidate A wins? If so, give an example.

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 -Using plurality, A wins. Could the four voters who most prefer D (and prefer A least) vote insincerely to change the outcome in a way that would benefit them?

Use the following information to answer the Questions: An 11-member committee must choose one of the four applicants -K, L, M, and N-for membership on the committee. $\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\text { Number of }$ $\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\text { metnbers }$ 6 2 3 First choice Second choice Third choice Fourth choice -The committee members have preferences among the applicants as given in the table. If the committee uses pairwise sequential voting with the agenda K, L, M, N, applicant K wins. Is it possible that another agenda will yield a different winner?

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 -In a plurality runoff election, candidate S wins. What would happen if the four voters who prefer T insincerely voted for S instead? Is this in their best interests?

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 -Who wins Borda count? Can the group of six voters change their preference list to produce an outcome they like better?

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 -C wins in a Borda count. Can the 12 voters in the first column change their preference list to produce an outcome they like better?

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 eight voters in the middle column change the results of the vote to their favor by changing their preference rankings?

A vote using Condorcet's method is:

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 five voters to change the outcome in a way that would benefit them?

After considering all possible one-on-one contests, Copeland's rule chooses as the winner:

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 six voters who most prefer candidate C 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 S vote strategically in some way to change the outcome in a way that will benefit them?

Consider the following preference table for three voters to answer the Questions First choice A D B Second choice D C A Third choice B B D Fourth choice C E C Fifth choice E A E -Is there an agenda for which D would win a sequential pairwise election?

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 -Using plurality, A wins. Could the six voters who most prefer C (and prefer A least) vote insincerely to change the outcome in a way that would benefit them?

Which of the following is NOT a property of the Condorcet method of voting?