Exam 10: The Manipulability of Voting Systems

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 uses the Borda count method. The committee members have preferences among the applicants as given in the table. Who wins the election? Can the group of two voters favorably affect the results through insincere voting?

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 uses the Borda count method. The committee members have preferences among the applicants as given in the table. Suppose the group of six suspects that the group of two intends to insincerely exchange M and L in their rankings. Can the group of six counteract to protect K as the winner?

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 left manipulate the outcome to his or her benefit?

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

The Chair's paradox is the fact that the chair, who can break ties, can:

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 D would win a sequential pairwise election?

What is agenda manipulation?

When the voter with tie-breaking power can end up with his or her least-preferred candidate as the election winner, this phenomena is called:

A vote using plurality can be manipulated by:

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. Can the three voters who least prefer K vote strategically in some way to change the outcome to one they find more favorable? Why or why not?

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

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 W vote strategically in some way to change the outcome in a way that will benefit them?

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 A withdrew from the race before the ranking?

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 uses the Borda count method. The committee members have preferences among the applicants as given in the table. Who wins the election? Can the group of three voters favorably affect the results through insincere voting?

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 -Is it possible to manipulate the results of a sequential pairwise election?

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 fourth voter manipulate the outcome so that C wins?

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 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 D wins?

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 C withdrew from the race before the ranking?

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 B would win a sequential pairwise election?