Exam 9: Social Choice: the Impossible Dream

Arrow's Impossibility Theorem states that any voting system can give undesirable outcomes. Explain what this means.

Use the following information to answer the Questions Twenty-nine voters rank three candidates according to the following preference table. $\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\text { Number of voters }$ 3 9 8 9 First choice Second choice Third choice -Which candidate wins using sequential pairwise voting with agenda A, B, C?

Answer the following Questions using the preference table. $\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\text { Number of voters }$ 6 3 4 2 First choice A C C B Second choice B B D A Third choice C D A C Fourth choice D A B D -Which candidate will be chosen if plurality voting is used?

Use the following information to answer the Questions Suppose that a nine-member committee needs to elect one of the four alternatives-A, B, C, or D. Their preference schedule is shown below. $\quad$$\quad$$\quad$$\quad$$\quad$$\text { Number of members }$ 4 3 2 First choice Second choice Third choice Fourth choice -Which alternative wins using the Hare system?

Twenty-nine voters must choose from among three alternatives: A, B, and C. The voters' preference schedules are shown below. C wins in a Borda count. Is the outcome different in a rank system that assigns five points for first choice, two points for second, and one point for third? $\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\text { Number of voters}$ 12 8 6 3 First choice Second choice Third choice

Use the following information to answer Questions Fifty voters who elect one of the five candidates -A, B, C, D, or E-have the preference schedule shown below. $\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\text { Number of voters }$ 20 14 10 6 First choice Second choice Third choice Fourth choice Fifth choice -Which candidate will be elected using the plurality-runoff voting method?

A group of 12 students have to decide among three types of pizza: sausage (S), mushroom (M), and beef (B). Their preference rankings are shown below. What is the Borda score of the beef topping? $\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\text { Number of students }$ 3 3 2 2 2 First choice Second choice Third choice

Use the following information to answer Questions Fifty voters who elect one of the five candidates -A, B, C, D, or E-have the preference schedule shown below. $\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\text { Number of voters }$ 20 14 10 6 First choice Second choice Third choice Fourth choice Fifth choice -Which candidate will be elected using plurality voting?

Use the following information to answer the Questions: Seventeen board members vote on four candidates, A, B, C, or D, for a new position on their board. Their preference schedules are shown below. $\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\text { Number of members }$ 7 6 4 First choice Second choice Third choice Fourth choice -Using the Hare system according to the preference schedules shown, what happens if candidate A rejects the offer before the ranking?

Use the following information to answer the Questions Twenty-nine voters rank three candidates according to the following preference table. $\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\text { Number of voters }$ 3 9 8 9 First choice Second choice Third choice -Which candidate wins using Hare's system?

Use the following information to answer the Questions Your fly fishing club is attempting to decide where to go for the next day trip. The choices are Alder River (A), Boulder Pond (B), or Collie Creek (C). The preference rankings of the members are shown below. $\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\text { Number of members }$ 6 12 15 11 First choice Second choice Third choice -Which location will be chosen if plurality-runoff voting is used?

Sequential pairwise voting satisfies the Condorcet criterion.

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. $\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\text { Number of voters}$ 10 8 4 First choice Second choice Third choice -Which choice wins using sequential pairwise voting with the agenda Z, B, M?

In how many ways can a voter rank five candidates, without allowing ties?

Use the following preference table to answer the Questions $\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\text { Number of voters }$ 1 1 1 First choice A C B Second choice B A D Third choice D B C Fourth choice C D A -Which candidate wins using sequential pairwise voting with agenda B, D, C, A?

Use the following information to answer the Questions Twenty-nine voters rank three candidates according to the following preference table. $\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\text { Number of voters }$ 3 9 8 9 First choice Second choice Third choice -Which candidate wins using majority rule?

Use the following preference table to answer the Questions: $\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\text { Number of voters }$ 3 3 2 3 First choice A C A D Second choice B D D C Third choice C B C B Fourth choice D A B A -Which candidate wins using sequential pairwise voting with agenda C, A, B, D?

Use the following information to answer the Questions Twenty-nine voters rank three candidates according to the following preference table. $\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\text { Number of voters }$ 3 9 8 9 First choice Second choice Third choice -Which candidate wins using Condorcet's method?

Use the following information to answer Questions Fifty voters who elect one of the five candidates -A, B, C, D, or E-have the preference schedule shown below. $\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\text { Number of voters }$ 20 14 10 6 First choice Second choice Third choice Fourth choice Fifth choice -Which candidate will be elected using sequential pairwise voting with the agenda A, B, C, D, E?

Explain the Condorcet Winner Criterion.