1) A group of five friends are choosing what to cook for dinner. The options are kangaroo vindaloo (k), pad thai (p), meatloaf (m), and grilled salmon (g).
Two have the following preference ranking: k>p>m>g
Two have the following preference ranking: g>m>k>p
One has the following preference ranking: p>g>m>k
They use a round-robin tournament to make their decision. What happens? First check to see if if Sen’s value-restriction condition is met, and then play out the round-robin tournament–do they manage to come up with a coherent group preference, and if so, what is it?
Now try it with a very minor change. See the preference rankings below and note that just one person has changed their preferences—and they have only reversed the order of two options.
Two have the following preference ranking: k>p>m>g
Two have the following preference ranking: g>m>k>p
One has the following preference ranking: g>p>m>k
What happens here? Again, first check to see if Sen’s value-restriction condition is satisfied and then play out the round-robin tournament–do they manage to come up with a coherent group preference, and if so what is it?
2) The Senate Finance Committee, in debating a health care reform bill, contains some members who demanded a full government-run insurance scheme (a “public option�), others who were strongly opposed and who favor no change, and a third group who favored a compromise health care cooperative (a “co-op�). Both those in favor of a full public option and those opposed agree that the co-op is the second-best compromise option.
Let’s use c for the co-op option, p for the public option, and n for the no-change option. Let’s use 1 for those who favor the public option, 2 for those who favor no change, and 3 and 4 for those who favor the co-op option. Is the Single-Peaked Condition satisfied? That is, can the options be represented on a line such that each of the utility functions has a maximum at some point on the line and slopes away from that maximum on either side?
3) A seven-member committee is deciding the total amount to be spent on year-end bonuses. The preferences of each member over alternative total amounts are single-peaked and symmetric about each member’s ideal points and are arrayed as follows: Bobby: 0; Amy: 1500; Cathy: 6000; Frank: 7500; Geri and Emma: 10,000; and David: 12,000. The decision will be made by the method of majority rule. What will the decision be? Explain your answer.