Game Theory (ECO4400)
Book: Games of Strategy, by Dixit & Skeath
Chapter 15: Strategy and voting
Because there are many types of election procedures used to determine a winner in an
election, there can be many different election outcomes.
The procedure used for a specific election might be open to manipulation: voters might
be able to alter the outcome by misrepresenting their preferences (strategic voting).
1. Voting rules and procedures
This is just a simple list (far from complete).
a. Binary methods
Binary methods require voters to choose between only 2 alternatives at a time.
If there are only 2 candidates, the majority rule can be used to determine the winner.
If there are more than 2 alternatives, pairwise voting can be used (a repetition of binary
votes). Pairwise procedures are multistage.
One pairwise procedure (where each alternative is put against each of the others) is called
the Condorcet method.
Then, a candidate beating all the others in a series of one-to-one contests is termed a
Another pairwise procedure is the amendment procedure (US Congress).
b. Plurative methods
Plurative methods allow for more than 2 alternatives simultaneously.
One group of plurative voting methods is called positional methods, where points are
assigned to each alternative according to the position the voter gives them on his ballot.
The plurality rule is a special-case positional method where the voter gives a vote only
to his most preferred alternative.
That alternative is assigned a single point and the alternative with the most votes wins
(not necessarily with 51% of the votes, though). Another positional method is the
antiplurality method, where you choose your least preferred alternative and it gets -1
One of the most well-known positional methods is the Borda count. It requires voters to
rank-order all the alternatives, and they get points according to the number of
In a 3-alternative election, for example, the most preferred alternative gets 3 points, the
second one 2 points and the least preferred alternative gets 1 point. Or you can give more
points to the top alternative… or to the top 2 alternatives… or whatever…
Another alternative: approval voting, where each voter says which alternatives he
“approves” (the position in the ballot is not distinguished in this case). The alternative
with most approvals wins.
c. Mixed methods
Some multistage procedures combine plurative and binary voting in mixed methods.
The majority runoff procedure is a two-stage method used to decrease a large group of
possibilities to a binary decision.
In a first stage, voters indicate their most preferred alternative. In the second stage (if
nobody got a majority in the first stage) the two most preferred alternatives are the only
choices to vote for.
Another mixed procedure consists of voting in successive rounds (eliminating the worst
performer and voting again and again). Example: Olympic Games site.
2. Voting paradoxes
Even when people vote according to their true preferences, specific conditions on voters’
preferences and voting procedures can give rise to curious outcomes.
a. The Condorcet paradox
It arises when no Condorcet winner emerges from a Condorcet method of voting (each
alternative against all the others, one at a time).
Example with three alternatives (G=generous; A=average; D=decrease) about welfare
benefits and three voters (L=left; C=center; R=right).
See fig 15.1 for a description of their preferences. In pairwise voting: G wins over A, A
wins over D and D wins over G. The group’s preferences are cyclical and no winner
This is an example of an intransitive ordering of preferences: even when the voters have
transitive preferences, the group does not.
That is the paradox: even if all individual preference orderings are transitive, there is no
guarantee that the social preference ordering induced by the Condorcet’s voting
procedure will also be transitive.
Even more generally, there is no guarantee that the social ordering induced by any formal
group voting process will be transitive just because individual preferences are.
b. The agenda paradox
This example considers the ordering of alternatives in a binary voting procedure. Suppose
preferences are the ones from fig 15.1 and one of the councilors decides which two
alternatives go first, such that the winner then faces the remaining alternative.
The chair can get any final outcome he wants. For example, if L is the chair he can make
G win by setting A vs D in the first round, where A will win.
In the second stage, however, A will lose against G. The determinant of the outcome is
just the ordering of the agenda.
Then, the real game is who is chosen to be the chair of the council.
But this assumed that in the first stage everybody voted according to their preferences.
However, C could vote strategically in the first stage: vote for D instead of A such that D
wins the first round and also beats G in the second round.
And C prefers D versus G (which would have arisen if she voted truthfully). But
everybody realizes that strategic voting is possible… rollback analysis applies here…
more to come.
c. The reversal paradox
Positional methods can also lead to paradoxical results. The Borda count, for example,
can yield to the reversal paradox when the set of alternatives changes.
This paradox arises in an election with at least 4 alternatives when one of them is
removed from consideration after votes were submitted.
Suppose there are 4 alternatives (pitchers) and 7 voters, who rank the alternatives
according to fig 15.2.
The top ranked candidate gets 4 points… etc. For example: Seaver gets 3 points from 2
voters (ordering 1), 2 points from 3 voters (ordering 2) and 4 points from the other 2
voters (ordering 3), for a total of 20 points.
Koufax gets 19, Carlton 18 and Roberts 13. Seaver wins, followed by K and C and R.
What if R is removed due to a mistake? Recalculation is necessary.
Three points to the top ranked and so on… Now, C gets 15, K gets 14 and S gets only 13
(2 points from 2 people, 1 point from 3 people and 3 points from 2 people).
The winner turns to loser!
d. Change the voting method, change the outcome.
Consider now 100 voters who can be grouped as in fig 15.3. Depending on the method
used, any alternative could win the election.
With simple plurality rule, A wins with only 40% of the votes (B gets 25% and C gets
35%). The Borda count, however, gives 180 points to A (40 #1 position (i.e. 3 points) and
60 #3 position (i.e. 1 point), 225 points to B (75 #2 position (i.e. 2 points) and 25 #1
position (i.e. 3 points) and 195 points to C.
Then B wins (C second and A last).
On the other hand, C can win with an instant runoff system.
3. General ideas pertaining to paradoxes
Is there a voting system that satisfies certain regularity conditions and most accurately
captures the preferences of the voters?
According to Arrow’s impossibility theorem, the answer is no. He said that no
aggregation method could simultaneously satisfy all these 6 principles:
1. The group ranking must rank all alternatives
2. It must be transitive
3. It should satisfy positive responsiveness (given A and B, if voters
unanimously prefer A, the aggregate ranking should also place A
4. The ranking must not be imposed by external considerations
independent of individual preferences.
5. It must not be dictatorial
6. It should be independent of irrelevant alternatives (IIA).
All the systems considered violate at least one of these principles. No research has been
successful in reducing these requirements.
Some form of compromise is considered necessary when choosing an aggregation
a. Black’s condition
Black imposed a restriction (to avoid paradoxes) on the preference orderings of
individual voters: single-peaked preferences.
This requires that the preferences should be able to be ordered along some dimension
(example: expenditure level for the former example).
See fig 15.4. (see my interpretation as well). Black shows that with single-peaked
preferences, the pairwise (majority) voting procedure must produce a transitive social
The Condorcet paradox is prevented and pairwise voting satisfies Arrow’s transitivity
Complicated. Just take a look.
c. Intensity ranking
The mere ranking of alternatives is not enough: the intensity of the preference should also
The requirement #6 (IIA) is replaced by another (weaker) one. With this change, the
Borda count is the only method satisfying the revised Arrow’s theorem.
4. Strategic manipulation of votes
Generally, voters can vote for candidates, issues or policies that are not actually their
most-preferred outcomes among the alternatives presented in an early round if such
behavior can alter the final election results in their favor.
a. Plurality rule
In presidential elections there are usually two major candidates in contention.
Ruled by the plurality rule, there is room for a third candidate to divert votes away from
the leading candidate, threatening him to lose if the race was very close. These are called
The 1992 (Perot helped Clinton winning?) and 2000 (Nader helped Bush winning?)
presidential elections have apparently showed some strategic voting.
b. Pairwise voting
Back to the example where Left is the chair, and he sets A vs D in the first round and the
winner vs G in the second round.
With truthful voting, A beats D and then G beats A. But what is actually the optimal way
to vote in this set up? Use rollback.
With this outcome, Center observes his least preferred outcome (G) winning. However, if
C votes for D in the first round (even when not preferred), D beats A and then D beats G
in the second round.
At least, D is the final winner (2nd in C’s ranking) and not G. But will the other two
voters vote truthfully? No.
Then, both rounds should be analyzed using rollback to find the NE. See fig. 15.5.
Suppose the order is still set by L, the chair, putting A vs D in the first round and the
winner against G in the second. Then, see the fig with the two possible second round
options: A vs G or D vs G.
There is a dominant strategy for each voter at A vs G (G for Left, A for Center and G for
Right) and G wins this election. At D vs G, however, D wins (D is dominant for Center
and Right while G is dominant for Left). Notice that everybody votes according to their
preferences (it is the last round).
Rolling back to the first round, fig 15.6 shows the final outcome depending on the first
round votes, which determine who faces G in the second round.
In the first round, Left votes A and Right votes D (dominant strategy and truthful voting).
However, Center votes D even though she prefers A (misrepresentation of preferences).
Final outcome (NE): D (not G as the chair originally thought when setting the agenda).
Then, the chair should anticipate this and set a different agenda!
In fact, setting D vs G in the first round and the winner against A, the NE is actually G,
the chair’s most preferred outcome.
At that NE, Right misrepresents preferences voting for G over D in the first round to
prevent A from winning at the end (verify yourself).
c. Strategic voting with incomplete information
Now suppose that the preferences of the other voters are only imperfectly known by a
particular councilor. How to vote?
Suppose we are back to the original agenda set by Left, where A and D meet first and the
winner faces G in the second round.
Assume that each councilor knows his preferences but only knows the probabilities pL, pC
and pR that the other voters will have preferences like the original Left, Center and Right
The imperfect information affects only the (former) misrepresentation of preferences by
Center in the first round.
We need to assume payoffs for Center preferences: assume U(A)=1, U(D)=u with 0<u<1
Suppose now Center needs to vote for A and D in the first round, supposing the other two
councilors vote truthfully (even if they also are Center type).
Only case in which Center affects the decision is if the other two voters split votes (1 for
A and 1 for D).
In that case, the D vote came from a Right type voter (see fig 15.1), while the A vote
could come from a Left type or a (truthful) Center type (see fig 15.1 again).
If the A vote came from a L type, there is one voter of each type. If she votes truthfully,
the final outcome is G (payoff 0).
If she votes strategically, final outcome is A (payoff u). But if the A vote came from a C
type, there is no L in the council.
Voting truthfully for A makes A the final outcome (beats D and G) for a payoff of 1. If
voting strategically for D, the final outcome is D for payoff u.
Now, center needs to compare the expected payoff of voting truthfully (for A) or not
(vote for D):
Voting truthfully for A: prob that the other vote came from a L type (pL/(pL+pC)) times
0 + prob the other A vote came from a C type (pC/(pL+pC)) times 1.
Voting strategically for D: D is the final outcome for sure (payoff u).
Then, compare pC/(pL+pC) vs u.
5. General ideas pertaining to manipulation
6. The median voter theorem
Strategic analysis can be applied to candidates as well. When there are just two
candidates, when voters are distributed in a “reasonable” way and have single-peaked
preferences, the median voter theorem says that the candidates will position at the same
place as the median voter (the “middle” voter).
The full game: first the candidates position themselves and then voters choose. With only
2 candidates, there is no misrepresentation of preferences in the second stage so the first
stage defines the NE.
a. Discrete political spectrum
Assume a population of 90 million voters with preferences described in fig 15.7 (10
million prefer far-left, 20 million prefer far-right…etc).
You vote for the candidate closer to your preference. If you prefer L, and the two
candidates propose FL and C, you are indifferent between them (50/50).
The table in fig 15.8 shows how many votes you get depending on your strategy and the
rival’s (where to locate).
Solve the game: FL is dominated by L and FR is dominated by R (for both candidates).
Then, R is dominated by C for both. Finally, C is dominated by L for both.
The NE: both choose L.
Features of the NE:
Both candidates locate at the same position (principle of minimum
differentiation). It explains why cars are so similar or why there is never a
single gas station at an important intersection (my counterexample: Univ and
Both candidates locate at the position of the median voter (L here, since that is
where voter # 45 million is located). Half voters to the left and half voters to the
The location of the median voter is not necessarily in the “center” (here, the
distribution is skewed to the left).
Usual way of stating the theorem: the position of the median voter is the equilibrium
location position of the candidates in a 2 candidate election.
b. Continuous political spectrum
A continuous distribution assumes that there are an infinite number of political positions.
Also, assume that voters are distributed according to a distribution function (see fig 15.9).
These are now symmetric, so the median voter is actually at 0.5 (unlike the discrete
example). A payoff table is not possible here, so use calculus.
Suppose Dolores locates at x. If you locate to the left of x, you get all the voters to your
left plus half of the voters between you and Dolores.
If you locate to the right of x, you get all the voters to your right plus half the voters
between you and her. If you locate at x, you get half of the population.
So what is the BR? It actually depends where x is located with respect to the median
Do it sequentially and see that the NE is when both locate exactly at the median voter’s
Formal proof: using integration.
SUGGESTED EXERCISES: 1, 2, 5a and 5b.
Also valid for ice-cream vendors on a beach, or feature choices by electronics firms.