Exam 10: The Manipulability of Voting Systems

Among all two-candidate voting systems that never result in a tie, majority rule is the only one that treats all voters equally, treats both candidates equally, and is nonmanipulable.

Use the following information to answer Questions . A group of 22 young people must decide whether to go to the beach (B), the mountains (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 -Who wins the vote? Can the 10 voters in the first column change the results of the vote to their favor by changing their preference rankings?

A person who has tie-breaking power:

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 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 in the middle manipulate the outcome to his or her benefit?

Which of the following is also known as an "insincere ballot"?

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 A would win a sequential pairwise election?

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 X would win under sequential pairwise voting?

Consider the following preference table for two voters: 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 two voters: First choice D B Second choice C A Third choice B 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?

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 first voter manipulate the outcome so that B wins?

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 A would win a sequential pairwise election?

Use the following information to answer Questions . A group of 22 young people must decide whether to go to the beach (B), the mountains (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 -Who wins the vote? Can the four voters in the last column change the results of the vote to their favor by changing their preference rankings?

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 D wins? If so, give an example.

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 C wins? If so, give an example.

Explain the difference between sincere and strategic voting.

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

A voter in a preference ballot decided using Borda count discovers that his neighbor is going to rank the candidates A over C over D over B. She wants to exactly cancel her neighbor's vote so that it will have no effect net effect on the Borda count result. How should she vote?

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 six voters in the third column change their preference list to produce an outcome they like better?

Use the following information to answer the Questions Thirty voters with the preference schedules below are to elect a union spokesman from among five candidates: A, B, C, D, and E. If the Borda count is used, candidate B would win. $\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\text { Number of voters }$ 12 10 8 First choice A B D Second choice D C B Third choice B A A Fourth choice C D E Fifth choice E E C -Would there be any difference in the result if candidate D withdrew from the race before the ranking?