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CHAPTER 15
Strategy and Voting
Teaching Suggestions
Most students will be familiar with at least a few of the vari ous aggregation methods
used to determine winners in elections. You may want to poll the class and come up with a list
of different methods that they know about or have used. Even a simple discussion of this type
will lay the groundwork for coverage of the many methods available. This is another
opportunity to let the political science students in your class use their field knowledge and lead
the discussion, thereby engaging their interest. There are additional examples beyond those
found in the text; some of these are explained in the symposium on “Economics of Voting” in
the Journal of Economic Perspectives, vol. 9, no. 1 (winter 1995).
The best way to get students to appreciate that different voting methods can lead to
very different final outcomes is to have them work through an example. The John Allen Paulos
example found in Exercise S5 of this chapter is an excellent one; you can change the characters
to fit your own state, local, or university situation to make the idea more interesting to
students. Another classic example of how the choice of procedure can affect the outcome is
based on a Roman legal case, extracted from the letters of Pliny the Younger (see Robin
Farquharson, Theory of Voting [New Haven, Conn.: Yale University Press, 1969], pp. 6–7, or
Peter Ordeshook, Game Theory and Political Theory, pp. 54–55). Pliny, as a member of the
Roman senate, had a distinct preference for the particular aggregation rule that would lead to
his most-preferred outcome.
Many other examples of voting paradoxes can be found in the literature. Those who
want to emphasize political science applications may want to refer to William Riker’s
Liberalism Against Populism, some of the papers of Peter Fishburn (including “Paradoxes of
Voting,” American Political Science Review, vol. 68 [1974]), and the Farquharson and
Ordeshook books mentioned above.
Games of Strategy, Fourth Edition Copyright © 2015 W. W. Norton & Company
For a more-detailed discussion of the Condorcet paradox, you can pick up on the idea
mentioned briefly in the text that in the City Council example, the paradox arose “because the
Council completely disagreed not only about which alternative was best but also about which
was worst.” This argument can be spelled out it more detail; doing so will help to clarify the
conditions under which an agenda setter will be able to determine the outcome.
In a three-by-three case (in which votes are cast “truthfully”), it is clear that a
Condorcet paradox can exist only when all three members rank a different alternative best. If
two members agreed on their favorite, that alternative would beat both others. As the text
implies, however, disagreement about the best alternative is not sufficient for the Condorcet
paradox to exist. In addition, the three voters must also disagree about which alternative is
worst.
Suppose that this isn’t true, so that the three voters (who each prefer a different
alternative) agree that one alternative, say A, is not the worst. In this case, alternative A is
ranked first by one voter, and second by the other two. Consider the voting outcome when
alternative A is matched against (for example) alternative B: A gets two votes, one from the
person who likes A best and one from the person who likes C best but prefers A over B.
Similarly, A gets two votes when matched against C, the second coming from the person who
likes B best but ranks A over C. When all three voters agree that A is not the worst outcome,
therefore, a Condorcet paradox cannot exist. We conclude that in the 3 x 3 case, a Condorcet
paradox exists only when there is complete disagreement among the voters; they must disagree
not only about which alternative is best but also about which is worst.
It is only when disagreement is so complete that the outcome can be determined by the
preferences of the agenda setter. This is intuitive; if there is “enough” agreement among the
voters, the voters’ preferences fully determine the outcome. The power to set the agenda is
decisive only when voters’ preferences are quite diverse. One can note also that the above
description is, of course, just another way to describe the well-known
double-peaked-preferences condition. In addition, it may be worth pointing out to students that
Games of Strategy, Fourth Edition Copyright © 2015 W. W. Norton & Company
complete disagreement on rankings may be more likely if the three alternatives being voted on
are three completely different options—like spending money on a park, a library, or a clinic—
rather than a range of spending on a single program.
Another example that can be used in class comes from the story, mentioned briefly in
the text, concerning how opponents of a federal school-construction-funding bill were able to
defeat the bill by strategic voting. They voted in favor of an amendment, which they would
have opposed on its own merits, anticipating that the amended bill was more likely to fail than
the original bill was.
Bear in mind, however, that attempts to engage in such clever schemes sometimes fail.
The first federal law that banned discrimination on the basis of gender passed only as a result
of a failed strategic plan. The original bill banned racial discrimination (in certain contexts);
opponents supported adding the gender-discrimination language expecting it to sink the whole
bill. To their surprise, it passed.
For those of you with mathematically sophisticated students, you may want to spend
some time discussing the example of strategic voting with incomplete information in Section
4.C. If you carefully covered the material on incomplete information in Chapter 8, this is a
good opportunity to make the connection between the broad implications of information
asymmetries and the specific case of voting games. Exercise U7 considers the opposite case to
that described in the text (in which a possible center-type vote is assumed to vote truthfully). If
you want to do a more complete analysis of this example, you can refer to P. Ordeshook and T.
Palfrey, “Agendas, Strategic Voting, and Signaling with Incomplete Information,” American
Journal of Political Science, vol. 32, no. 2 (1988).
When covering the median voter theorem, you may find that students balk at the idea
that candidates all move toward the position of the median voter since this does not seem to be
observed in reality. In actual political campaigns, candidates will often take positions in order
to distinguish themselves from one another. It is worthwhile to explain to students that this
observation does not destroy the essential conclusions of the median voter theorem.
Games of Strategy, Fourth Edition Copyright © 2015 W. W. Norton & Company
As the text notes, part of the reason why candidates may distinguish themselves is that
the median voter theorem does not hold when the political spectrum has two or more dimen -
sions. There are also institutional (and other) features that explain some differentiation between
candidates. A brief list includes:
A candidate may need to win a primary election, by appealing to a left- or
right-leaning subset of the total population, before running in the general election
campaign.
A candidate may attempt to increase the turnout of voters by appealing to her base
of supporters.
Prior public statements by a candidate may tie her to a particular position.
A candidate’s own belief system may lead some candidates to accept a reduced
chance of winning an election in order to promote what she believes are correct stands.
In spite of these factors, however, the basic conclusion of the median voter theorem—a desire
of candidates to move toward the middle of the political spectrum—still holds, even if the
candidates do not meet at the exact midpoint.
For example, candidates who must take relatively extreme positions in order to win a
primary often begin to moderate their positions as they move into the general election cam-
paign. Consider also a candidate who has represented a congressional district that leans more
to the left or the right than the candidate’s whole state. If this person runs for a statewide
office, she may modify some previous statements in order to sound more mainstream.
Candidates may also alter or hide their own personal views.
In all of these cases, a candidate may not be able to move into the exact middle of the
voter distribution because doing so would require the candidate to reverse some of her pre vious
positions. The loss of credibility that might result could be very damaging to the candidate.
Still, it is worth emphasizing that a candidate’s incentive to moderate her positions is quite
strong. Many candidates will behave in such a manner, up to the point at which the resulting
loss of credibility becomes too costly.
Games of Strategy, Fourth Edition Copyright © 2015 W. W. Norton & Company
