Median Voter
Hisahiro Naito
Graduate School of Humanities and Social
Sciences
1
MECHANISMS FOR AGGREGATING
INDIVIDUAL PREFERENCES
 This section discusses how voting can serve to
aggregate individual preferences into a social
decision.
 For now, we focus on direct democracy, whereby
voters directly cast ballots in favor of or in
opposition to particular public projects.
2
1
Majority Voting: When It Works
 A common mechanism
h i used
d to aggregate iindividual
di id l
votes into a social decision is majority voting, in
which individual policy options are put to a vote,
and the option that receives the majority of votes is
chosen.
3
Majority Voting: When It Works
 Majority voting does not always provide a consistent means
of aggregating preferences.
 To be consistent, an aggregation mechanism must satisfy
three goals:



Dominance: If one choice is preferred by all voters, then the
aggregation mechanism much be such that this choice is made
by society.
Transitivity: Choices must satisfy this mathematical property.
Independence of Irrelevant Alternatives: The introduction of a third
choice does not change the ranking of the first two choices.
4
2
Majority Voting: When It Works
 It turns out that with these three conditions,
majority voting can only produce a consistent
aggregation of individual preferences if preferences
are restricted to take a certain form.
 This is best illustrated with some examples.
 Table 1 shows a case when majority voting works.
5
A town is deciding on education taxes (and
Consider
pair-wise
High
Since
Medium
High
vs
Mvs
Medium:
has
vsLow:
Low:
beaten
Parents
Parents
Parents
both
spending). There are 3 possibilities:
high,
voting:
High
vs
Low,
High
Table 1
H
and
vote
and
Young
L,
for
M
H,
is
vote
Elderly
the
for
overall
M,
&
medium, and low spending. There are also 3
Finally,
Theirthe
preferences
“young couples”
are for
vs do
Medium,
and
Medium
winner
Young
Elderly
Young
in
vote
vote
vote
this
for
for
for
case.
M.
L.
L.
groups,
represented
in equal proportions.
Majority
voting
delivers
a consistent
outcome
the
preferences
the
The preferences
of While
parents
are
medium
not haveof
spending,
kids
and do
then
notlow,
want
then
to M
Low.
L vs
wins
wins
2-1.
2-1.
elderly
are exactlypay
opposite.
for high spending, then
medium
high taxes
high.
right now.
Types
of voters
spending, then low spending.
Preference
Parents
Elders
Young
rankings
Couples
First
H
L
M
Second
M
M
L
Third
L
H
H
3
Majority Voting: When It Doesn’t Work
 Table 2 shows a different scenario, where majority
voting does not work.
work
7
Table 2
Majority voting
AThus,
towntheir
is again
deciding
on
education
taxes
Private
Assuming
parents,
that
ordering
first
doesn’t
and
is low,
happen,
foremost,
then high,
however,
want
then
low
High
There
High
Medium
This
vsis
vs
Hmmm
Low:
no
violates
Medium:
vsclear
Low:
Only
…the
winner.
“public
Only
Only
pair-wise
voting:
(andsospending).
The Consider
elderly
have
been
taxes
private
they
parents
can medium.
want
afford
high
to sent
quality
their
public
kids to
Young
private
transitivity
parents”
L
is
preferred
Marrieds
parents
vote
assumption
for
for
vote
to
H,
L,
H.vsfor
L
so
High
vs
Low,
High
replaced with “private
parents.” The other 2
private
education.
schools.
and
H
M,
is
leads
M
so
preferred
wins
wins
H
wins
to
2-1.
2-1.
cycling.
to
2-1.
M.
Medium,
and
Medium
vs
groupsaare
the same as before.
doesn’t deliver
consistent
M isoutcome
preferred
Low. to L.
Types of voters
Preference
rankings
Parents
Private
Parents
Young
Couples
First
H
L
M
Second
M
H
L
Third
L
M
H
4
Majority Voting: When It Doesn’t Work
 This set of outcomes is problematic because there is no clear
winner. These results violate the principle of transitivity
resulting in cycling–when majority voting does not deliver a
consistent aggregation of individual preferences.
 Note that the failure to get a consistent winner from majority
voting does not reflect a failure on the part of individuals–
each group has a sensible set of preferences.
 The problem is aggregation–we are unable to use voting to
aggregate these individual preferences into a consistent social
outcome.
9
Majority Voting: When It Doesn’t Work
 This creates the problem of the agenda setter, the
person who decides the sequencing of the votes.
votes
 In the second situation, he can affect the outcome.


For low spending to win, for example, first set up a
vote between H and M. H wins. Then a vote
between L and H means L will win.
Any outcome can win with appropriate ordering.
ordering
10
5
Arrow’s Impossibility Theorem
 In fact, there is no voting system that will produce a
consistent outcome here.
here
 Arrow’s Impossibility Theorem states that there
is no social decision (voting) rule that converts
individual preferences into a consistent aggregate
function without either restricting preferences or
imposing a dictatorship.
dictatorship
11
Restricting Preferences to Solve the Impossibility
Theorem
 One way to solve this problem is to restrict preferences
to “single-peaked”
single peaked preferences.


A “peak” in preferences is a point that is preferred to
all its immediate neighbors. Utility falls in any
direction away from this point.
Multi-peaked preferences means that utility may first
rise, then fall, then rise again.
 If preferences
f
are single
i l peaked,
k d majority
j it voting
ti will
ill
yield a consistent outcome.
 We can visualize our earlier examples. See Figure 2.
2
12
6
(a)
(b)
Utility
Utility
Elders
Young
marrieds
Private
parents
Parents
Young
marrieds
Public
parents
M
H
Private
Their utility
parents
goesare
in
indifferent
either direction
in the
second
from case.
M.
And young marrieds
The elderly
are single peaked
are single
at “M”.
peaked at “L”.
L
Figure 2
Parents are
single peaked
at “H”.
H
M
School
spending
L
School
spending
Voting rules
Restricting Preferences to Solve the Impossibility
Theorem
 The failure of these preferences for the “private
parents in this second case is what leads to the
parents”
inability of majority voting to consistently aggregate
preferences.
 Fortunately, single-peakedness is a reasonable
assumption in most cases.
14
7
Median Voter Theory
 When preferences are single-peaked, then majority
voting will deliver a consistent aggregation of
preferences of the individual voters.
 Even stronger, the median voter theorem states
that majority voting will yield the outcome preferred
by the median voter if preferences are single peaked.

The median voter is the voter whose tastes are in the
middle of the set of voters, so an equal number of
other voters prefer more and prefer less of the public
good.
15
Empirical studies of Median Voter
Theorem
 Several people tried to examine whether the
median voter theorem holds or not in Japan
Japan.
 The question is which subject we should
examine.
 National lower house election(shugi-in)?
 National upper house election(sangi-in)?
 Or other election?
 Many researchers realized that it is not a
good idea to use national election because
the national election is more like indirect
democracy
16
8
Examining the median voter
theorem
 Thus, many studies examined the prefecture
governor election
election.
 This is because the prefecture governor
election is the direct democracy.
 Income data is available, especially median
income
 Also,
Al
when
h the
th governor changes,
h
the
th
expenditure pattern changes dramatically
within the prefecture.
17
Empirical examination of the
median voter theorem
 Furthermore, the prefecture residence tax
has a large share in the tax revenue in each
prefecture.
 Takahashi and Miyamoto examined
empirically examined whether the median
voter theorem hold in the Japanese
prefecture data
data.
 They use the Japanese panel data of the
expenditure and income
 They found the followings
18
9
Their findings
 In the general prefecture expenditure, the
median voter theorem does not hold
hold.
 That is the median’s preference does not
affect the general prefecture expenditure.
 However, they found that the expenditure on
construction and public project are affected
by the median’s
median s preference
preference.
19
Finding
 Their finding makes sense.
 As
A for
f the
th generall prefecture
f t
expenditure
dit
is
i
like the expenditure for public school teacher
and expenditure for police service and health
care.
 Those are determined by the national rule.
There is not so much choice for the
prefecture.
 But, the expenditure for construction and
public project, the prefecture has much
choices.
20
10
Findings
 For robustness checks, they also examine
whether the mean income instead of the
median income affect the prefecture
expenditure.
 They found that the mean income does not
have so much explanatory power.
 This supports the median voter theorem
theorem.
21
11