Exam 10: The Manipulability of Voting Systems

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 W insincerely voted for T instead? Is this in their best interests?

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 B 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 C would win a sequential pairwise election?

Which of the following method can be used to break ties, which can extend Condorcet's method?

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 -Is there an agenda for which Z would win under sequential pairwise voting?

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 -The committee suspects that the group of five plans to insincerely reorder their preferences as Y, Z, X. How can the group of four respond? $\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\text { Number of voters }$ 5 4 2 First choice Second choice Third choice

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 Z vote insincerely in some way to change the outcome in a way that would benefit them?

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

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?

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 A would win a sequential pairwise election?

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

Determine whether the Borda count method is manipulable. Are there situations in which the Borda count method is known to be never manipulable?

Use this preference table to answer the following Questions $\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\text { Number of voters }$ 1 1 1 First choice A C B Second choice B B C Third choice C A A Fourth choice D D D -How could the voter on the left rank the candidates to produce a tie using Borda count?

Use this preference table to answer the following Questions $\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\text { Number of voters }$ 1 1 1 First choice A B B Second choice B A A Third choice C C D Fourth choice D D C -Which candidate wins using Borda count?

Use this preference table to answer the following Questions $\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\text { Number of voters }$ 1 1 1 First choice A B B Second choice B A A Third choice C C D Fourth choice D D C -How can the voter on the left change his or her ballot to give A the victory using Borda count?

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

There are 21 delegates to a political party's convention at which four people-A, B, C, and D-have been nominated as the party's candidate for governor. The delegates' preference schedule is shown below. If the party uses a Borda count, candidate B would be elected. Can the four voters who most prefer candidate C vote strategically in some way to change this outcome to one they would find more favorable? Why or why not? $\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\text { Number of }$ $\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\text { delegates }$ 8 9 4 First choice A B C Second choice B A B Third choice C D A Fourth choice D C D

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 second voter manipulate the outcome so that D wins?

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 B vote insincerely to change the outcome in a way that would benefit them?

Use this preference table to answer the following Questions $\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\text { Number of voters }$ 1 1 1 First choice A C B Second choice B B C Third choice C A A Fourth choice D D D -Which candidate wins using Borda count?