Public Choice
Dr. Martin A. Leroch
Institute of SocioEconomics
5. Representative Democracy
Summer Term 2010
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1 | Introduction
▹ Last week we began to analyze how collective decisions work
▹ In the beginning, social welfare as normative measure was shown to be
problematic
▹ Unanimous consent was then taken as first criterion - with different
results, depending on the procedure
▹ Building on a cost calculation, majority vote was introduced as
alternative
▹ Today, we will push focus to the behavior of representatives
↝ Before doing so, see why majority vote entails redistribution
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1 | Introduction
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1 | Introduction
Especially in deciding distributional matters via majority vote, cycling
may occur
▹ Think of the division of 100,- EUR
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2 | The Median Voter Theorem
The Median Voter Theorem
In a (simple) majority election, the most preferred alternative of the median
voter will win if:
▹ two parties / candidates compete over
▹ a single policy-dimension over which
▹ individuals’ preferences are single-peaked
▹ and voters vote for the candidate committing to the alternative closest
to their preference
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2 | The Median Voter Theorem
Note the following assumptions of the Median Voter Theorem
▹ Representatives are only interested in maximizing votes (i.e. winning
the election)
▹ All voters vote
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2 | The Median Voter Theorem
Voters may abstain from voting for instance due to
1
2
Indifference (i.e. parties platforms are “too close”)
Alienation (i.e. parties platforms are “too far away” from one’s ideal
point)
In the first case, the median voter theorem still applies, in the second it
does if voters’ distribution is symmetric and unimodal
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2 | The Median Voter Theorem
Relaxing the assumption of a single dimension:
▹ What we are looking for are either dominant points (i.e. policy positions
which cannot loose to any other point)...
▹ or at least points which are not dominated
▹ According to the Median Voter Theorem, this point was the median
position if one dimension is at stake
▹ How about two dimensions?
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2 | The Median Voter Theorem
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2 | The Median Voter Theorem
In two dimensions, there may be no single dominant point. However,
there can be a dominant point which is a “median point” in a more
general sense:
▹ Any line drawn through this dominant point separates the rest of the
voters into two equal sets
If no such point exists, choosing a platform from outside the
Pareto-set may seem implausible
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2 | The Median Voter Theorem
The Uncovered Set
▹ An alternative x is uncovered if
↝ no other alternative y beats all alternatives that x beats in a
pairwise comparison and
↝ y does not defeat x in a pairwise comparison
▹ The uncovered set is the set of all alternatives which are uncovered (also
called Fishburn winners)
▹ The uncovered set lies within the Pareto set
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2 | The Median Voter Theorem
An Example:
xPy
xPw
yPz
yPw
zPx
wPz
▹ The uncovered set is (x, y, z)
▹ w is covered by y
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2 | The Median Voter Theorem
Two remarks:
1
2
The spatial model may be generalized to certain rational orderings,
though restrictions in this case are quite severe
The criterium Pareto-optimality as it was used so far requires
information on intensities of tastes:
↝ Technically, Pareto-optimality requires that the marginal rates of
substitution (MRS) are equal among people
↝ The MRS is a measure for the relative importance of one good as
compared to others
↝ This relative importance is (usually) subject to variation when the
relative amounts consumed vary
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3 | Logrolling
Logrolling occurs when two (or more) players agree to voting in favor
of the respective other
▹ You can say it is a form of “quid pro quo”
▹ It is a form of “sophisticated voting”
An example:
A
B
C
X
-2
5
-2
Y
-2
-2
5
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3 | Logrolling
▹ Logrolling reveals different intensities of taste to a certain degree
▹ Note that in the above example logrolling is welfare enhancing (if we
assume an additive welfare function)
▹ However, negative externalities are still present
▹ It is not a Pareto-improvement (even according to the
Kaldor-Hicks-criterion)
▹ In case that all three members could win “something”, we are caught
back in a cycle
▹ An agenda setter can, in the case of cyclical majorities, determine the
outcome
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3 | Logrolling
THE core result of the previous topics was a form of instability:
cyclical majorities
However, the (political) world we live in tends to be somewhat stable.
Where does this stability come from?
1
An agenda setter may exist
2
Rules may be such that defeated alternatives are excluded
3
Alternatives may be ordered in a single dimension (an preferences
single-peaked)
4
Voting on issues may be sequential if multiple dimensions exist
5
Logrolling may occur (once)
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