35. Because approval voting is very different from the social choice pro- cedures we studied earlier,...

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35. Because approval voting is very different from the social choice pro- cedures we studied earlier, not all of the desirable properties we used to evaluate the procedures even apply. We can modify them somewhat, however, to analyze approval voting more closely. For example, we could define a notion similar to the Pareto condition as follows: If every voter approves of alternative a, then a is a social choice. (a) Prove that with this modification, approval voting does satisfy the Pareto condition. A slightly different modification of the original Pareto condition is the following: If every voter prefers alternative a to alternative b, then b is not a social choice. This version of Pareto is more similar to the original than the one mentioned above, but with the presence of the word "prefer," we are implicitly assuming that a voter has a ranked preference list of alter- natives (which may include ties), but has the power under approval voting to draw the cutoff line between approval and disapproval at any point in the list. (b) Prove that with this modification, approval voting does not satisfy the Pareto condition. (c) Write down a modified version of the monotonicity condition that makes sense for approval voting. Does approval voting satisfy this modified monotonicity condition? 35. Because approval voting is very different from the social choice pro- cedures we studied earlier, not all of the desirable properties we used to evaluate the procedures even apply. We can modify them somewhat, however, to analyze approval voting more closely. For example, we could define a notion similar to the Pareto condition as follows: If every voter approves of alternative a, then a is a social choice. (a) Prove that with this modification, approval voting does satisfy the Pareto condition. A slightly different modification of the original Pareto condition is the following: If every voter prefers alternative a to alternative b, then b is not a social choice. This version of Pareto is more similar to the original than the one mentioned above, but with the presence of the word "prefer," we are implicitly assuming that a voter has a ranked preference list of alter- natives (which may include ties), but has the power under approval voting to draw the cutoff line between approval and disapproval at any point in the list. (b) Prove that with this modification, approval voting does not satisfy the Pareto condition. (c) Write down a modified version of the monotonicity condition that makes sense for approval voting. Does approval voting satisfy this modified monotonicity condition?