# ch16_solutions_solved edit.doc

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16 Apr 2012
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Solutions to Chapter 16 Exercises
SOLVED EXERCISES
S1. Under truthful voting, A should match Geology and Sociology in the first vote, with the winner
(Geology) to face Philosophy in the second round. Under strategic voting, A should match Philosophy
and Sociology in the first round. In order to prevent a Geology outcome, B casts a strategic vote for
Philosophy; Philosophy goes on to beat Geology.
S2. Allocate points in the following way: 5 for a first-place vote, 2 for a second-place vote, and 1 for
a third-place vote. A then receives 5 + 5 + 1 + 1 = 12 points, B gets 2 + 2 + 5 + 2 = 11, and C gets 1 + 1 +
2 + 5 = 9.
S3. (a) Under a plurality voting system (with truthful voting), Proposal 1 gets 20 votes and
Proposals 2 and 3 get 15 votes each.
(b) Under a 3-2-1 Borda count with truthful voting, Proposal 1 gets 60 + 15 + 30 = 105
points, Proposal 2 gets 40 + 45 + 15 = 100 points, and Proposal 3 gets 20 + 30 + 45 = 95 points.
(c) For both the second and third type of voter to gain, Proposal 3 must win rather than
Proposal 1. This can be achieved if types 2 and/or 3 change their votes in such a way that Proposal 1 gets
fewer points and/or Proposal 3 gets more. One example would be if types 2 and 3 vote as though their
rankings were Proposal 3, Proposal 2, Proposal 1. With this voting pattern, Proposal 1 gets 60 + 15 + 15 =
90 points, Proposal 2 gets 40 + 30 + 30 = 100, and Proposal 3 gets 20 + 45 + 45 = 110.
Note two things. First, other voting schemes will also result in Proposal 3’s winning. For example,
if only the type 2s vote with a 3-2-1 ranking, Proposal 3 still wins. Second, a single member of types 2 or
3 can’t alter the outcome by changing only her vote; coordinated action is required.
S4. (a) The preferences create a standard intransitive ordering, so none of the three alternatives
would win a majority vote with truthful voting.
Solutions to Chapter 16 Exercises 1 of 4
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