10 The Manipulability of Voting Systems

Skills Check

Question 10.1

1. A "unilateral change in ballot" refers to the fact that

only one candidate's position is being altered.
no communication is taking place.
only one voter is changing his or her ballot.

Question 10.2

2. The quote "My scheme is intended only for honest men!" is from _________________________.

Borda

Question 10.3

3. If a voter has sincere preferences of $A$ over $B$ over $C$ over $D$ , then

she will prefer a tie between $A$ and $D$ to a tie between $B$ and $C$ .
she will prefer a tie between $B$ and $C$ to a tie between $A$ and $D$ .
it's not at all clear which tie— $A D$ or $B C$ .—she will prefer.

Question 10.4

4. A ballot that misrepresents a voter's true preference is referred to as _________________________.

an insincere (or disingenuous) ballot

Question 10.5

5. A ballot that does not represent a voter's true preference is often called

an insincere ballot.
a disingenuous ballot.
either (a) or (b).

Question 10.6

6. Suppose Voter 1 ranks $A$ over $B$ over $C$ over $D$ and Voter 2 ranks $B$ over $C$ over $A$ over $D$ . Assume the Borda count is being used, so that $B$ wins. If Voter 1 knows that Voter 2 will submit his or her true preferences, then Voter 1 can secure a win for $A$ by submitting the following ballot: _________________.

A over D over C over B

Question 10.7

7. In presenting an example of a voting system's susceptibility to manipulation, we present two elections (Election 1 and Election 2). We assume that

all ballots in Election 1 are sincere.
all ballots in Election 2 are sincere.
both (a) and (b).

Question 10.8

8. Nonmanipulability and monotonicity are equivalent if the number of candidates is ___________________.

two

Question 10.9

8. The two-candidate voting system in which the winner is the alternative (or alternatives) with the fewest first-place votes

is manipulable.
treats all voters and candidates equally.
both (a) and (b).

two

Question 10.10

10. An example of a two-candidate voting system that is not monotone is ________________________________.

10.

one in which the winner has the fewest first-place votes

Question 10.11

11. Suppose that two elections show that a voting system is manipulable. Then

neither election results in a tie.
the winners are the same in both elections.
every voter has changed his or her ballot.

11.

Question 10.12

12. In the two-candidate case, nonmanipulable is equivalent to ________________________________.

12.

monotonicity

Question 10.13

13. Condorcet's method

can be manipulated but always produces a winner.
is nonmanipulable but sometimes produces no winner.
sometimes results in a tie, so manipulability is hard to assess.

13.

Question 10.14

14. May's theorem for manipulability says that, with an odd number of voters, among all voting systems for two candidates that never result in a tie, majority rule is the only one that is nonmanipulable and _______________.

14.

treats both candidates equally and all voters equally

Question 10.15

15. With the Borda count, two ballots "cancel each other out" if

they are identical.
each is arrived at by turning the other one upside down.
other voters also hold these same ballots.

15.

Question 10.16

16. The Borda count is nonmanipulable in the special case in which _________________________.

16.

there are only three candidates

Question 10.17

17. A 6-voter example of manipulation with the Borda count can be modified to yield a 10-voter example by

adding 4 ballots that are identical to each other.
adding 4 ballots that are identical to Voter 1's ballot.
adding 2 pairs of ballots, with the ballots in each pair canceling each other out.

17.

Question 10.18

18. With any voting system that satisfies the Pareto condition, an $n$ -voter example of manipulation with $k$ candidates can be modified to yield an $n$ -voter example with $k + j$ candidates by ________________.

18.

placing the additional $j$ candidates at the bottom of each ballot (in any order whatsoever)

Question 10.19

19. Of the Hare system and the plurality runoff method,

only the Hare system is manipulable.
only plurality runoff is manipulable.
both are manipulable.

19.

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Question 10.20

20. Sequential pairwise voting is susceptible to a kind of manipulation called ____________________.

20.

agenda manipulation

Question 10.21

21. Plurality voting

cannot be manipulated by a single voter.
can be manipulated by a single voter.
is subject to agenda manipulation.

21.

Question 10.22

22. Plurality voting is susceptible to a kind of manipulation called __________________________.

22.

group manipulation

Question 10.23

23. Group manipulability was discussed in connection with

Condorcet's method.
the Borda count.
sequential pairwise voting.
plurality voting.

23.

Question 10.24

24. One strategy weakly dominates another strategy if it yields an outcome that is ______________________.

24.

at least as good as, and sometimes better than, the other

Question 10.25

25. Agenda manipulation was discussed in connection with

Condorcet's method.
the Borda count.
sequential pairwise voting.
plurality voting.

25.

Question 10.26

26. The deep result in this chapter that is related to Arrow's impossibility theorem is called the ____________.

26.

Gibbard–Satterthwaite theorem

Question 10.27

27. The Gibbard-Satterthwaite theorem says that with three or more candidates and any number of voters, there is no voting system that

is not a dictatorship.
is nonmanipulable and is not a dictatorship.
satisfies the Pareto condition, is nonmanipulable, and is not a dictatorship.
always yields a unique winner, satisfies the Pareto condition, is nonmanipulable, and is not a dictatorship.

27.

Question 10.28

28. The weak version of the Gibbard-Satterthwaite theorem asserts that if we have a voting system that agrees with Condorcet's method whenever there is a Condorcet winner and that also produces a unique winner when confronted by the ballots in the Condorcet voting paradox, then the system is _________________.

28.

manipulable

Question 10.29

29. The voters′ preferences in the chair's paradox are

precisely the Condorcet voting paradox ballots.
all the same.
dictated by the chair.

29.

Question 10.30

30. The chair's paradox is paradoxical because ________________________.

30.

the chair has the most power, but fares the worst