Public Choice
Dr. Martin A. Leroch
Institute of SocioEconomics
5. Representative Democracy
Summer Term 2010
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1 | Introduction
▹ Last week we introduced the median voter theorem as predictive tool
for social (voting) outcomes
▹ In general, models which “geometrically” position parties / voters in
issue spaces are called spacial (voting) models
▹ We also began to relax some of the assumptions
↝ Not all voters vote
↝ Two dimensions
▹ This week we will relax the assumption of 2 parties only
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2 | Multiparty Models
▹ Similar results to the median voter theorem are also available with more
than two parties
▹ One remarkable result: The voting rule may influence the number of
“effective” parties
▹ Effective number of parties:
v
based on number of votes: ENV = (∑np=1 ( vp )2 )−1
s p 2 −1
n
based on number of seats: ENS = (∑p=1 ( s ) )
▹ (You may recognize the similarity to the Herfindahl-Index)
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2 | Multiparty Models
Duverger’s Law
▹ If voters vote strategically,
▹ majority rule will lead to two leading parties
▹ Why? Voters will focus on candidates they regard as crucial, i.e. the top
two
▹ Of these two, they will pass their vote for the party closest to their own
position
▹ Similarly, in multi-party systems where two or more candidates are
elected, focus will be on the marginal seat
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2 | Multiparty Models
▹ Median line
↝ A median line is a line separating the N voters into two groups
with
↝ NL ≤ N2 to the left and on the line
↝ and NR ≤ N2 to the right and on the line
▹ The yolk as “solution concept”
↝ The yolk is the smallest circle touching all median lines
↝ It has been argued that the yolk is a “reasonable” space in which
outcomes may occur
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2 | Multiparty Models
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2 | Multiparty Models
▹ With more than two parties, no unique result is obtained even in one
dimension if parties have no aim but winning elections
▹ (Unique results may be obtained by probabilistic models)
▹ Things may change, however, if parties want to implement specific
platforms
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2 | Multiparty Models
▹ Consider for instance the case that parties occupy the median position
of their supporters
▹ (One) Justification: If cyclical majorities occur anyway, one could
likewise be interested in specific policies
▹ Another justification: Honesty (a matter of valence) could be important
▹ Therefore, we will assume that parties want to win elections and
determine the social outcome (which should be close to their platform)
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2 | Multiparty Models
▹ In order to implement social outcomes, it will probably be necessary to
join coalitions
▹ Different forms of coalitions are considered:
↝ Minimal winning coalitions
Removing any member will turn winning coalition to a loosing
one
↝ Minimum winning coalitions
Smallest minimal winning coalition
↝ Minimal-connected-winning coalitions
Consider only minimal winning coalitions with “neighboring
parties”
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2 | Multiparty Models
The heart as “solution concept”
▹ It is the area “encased” by the median lines
▹ It is used as “reasonable” space for which coalitions may form
▹ Note the similarity to the yolk and the uncovered set: We are looking
for the space where outcomes will lie
▹ Note the difference: We are focussing on coalitions, not single ideal
points
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2 | Multiparty Models
▹ The (weak) claim is that one or more partners of the winning coalition
will be in the heart
▹ If all median lines intersect, this intersection will be the core:
No party can successfully form a coalition (if necessary) without the
party at the core
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2 | Multiparty Models
▹ Besides answering the question which coalition may form, it may be
interesting to see which party takes which ministry
▹ Put differently, we are now considering the formation of cabinets
▹ The idea is that each minister will implement exactly its ideal point as
policy
▹ Note the correlation to the logrolling example given earlier
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2 | Multiparty Models
Procedure:
▹ Identify the median positions in each dimension
▹ The intersection of the “median lines” is the most probable allocation of
ministries
▹ Note that even the coalitions formed could improve over this solution
(but are by assumption “forbidden” to do so)
▹ If another lattice point lies within the winset (as compared to the status
quo), this will be a likely candidate to win
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