Download Arrow`s Theorem - Columbia University

Instutional Analysis
Lecture 5: Legislative
Organization
How a bill becomes a law
Arrow’s Theorem
 Completeness - (Transitivity) If x beats y and y beats z in
the social ordering then x beats z.
 Universal Domain - the rules must apply to all possible
combinations of individual preference orderings.
 Pareto Optimality- x beats y whenever everyone
individually prefers x to y.
 Independence of irrelevant alternatives - The collective
preferences between two alternatives never depend on
individual preferences regarding other alternatives.
 Non-Dictatorship. No individual is so powerful that, for
every pair of alternatives x and y, x beats y socially even if
she prefers x to y while everyone else prefers y to x.
CUPID
Implications
Arrow showed that these 5 criteria are
incompatible with any system of
preference aggregation.
Example: Simple Majority Rule
Simple Voting Paradox
Majority Rule Violates Transitivity
Suppose we have a preference ordering as follows:
A
x
y
z
B
y
z
x
C
z
x
y
Problem:
Collective preferences are cyclic and every feasible
alternative is unstable, such that: x>y, y>z; and z>x.
No unique stable outcome:
x
z
y
Significance
We can't predict what will be the outcome
under a majority rule setting.
There is no true social welfare maximizing
outcome
Then majority government cannot be
modeled as maximizing anything, if any
outcome is possible no matter how
unrepresentative.
When Does Stability Occur?
Black’s Theorem:
One-dimensional issue space
Ideal (bliss) points: the one point on the line
they prefer to all others
The only stable point on the line is the median
x1 x2
x3
xm
x5
x6
x7
Xm is the Condorcet winner because it is the
point that beats all others.
Median Voter Result
In general, for any distribution of voters
with single peakedness there will be a
median voter.
If we assume that
1) Individuals have single peaked preference in a
one-dimensional space and
2) A proposal can be freely amended
Then the outcome is always the ideal point
of the median voter.
Plott Conditions
In more than one dimension, however, no
single equilibrium point exists.
Unless… the Plott Conditions are met:
Each individual’s ideal point can be paired
with another that is exactly on the opposite
side of the Median XM.
x5
x7
x1
x6
xM
x3
x2
Chaos Theory
Divide the Dollar Game
Distributive politics has no unique outcome
Devil Agenda Setter…
things are getting worse:
If no Condorcet winner exists, then majority rule
voting can lead to anywhere in the policy space.
x2
x1
x3
Strategic Voting
Assume individuals vote sophisticatedly:
A bill is proposed along with an amendment
• a = amended bill
• b = bill
• q = status quo
Preference are:
Sequence:
1
a
b
q
2
b
q
a
3
q
a
b
An amendment is voted on against the bill
Then which ever wins is pitted against the status quo.
Strategic Voting (continued)
In sincere or myopic voting:
2
1, 3
b
b
a
q
1
2,3
a
q
In sophisticated voting:
1,2
3
b
1,2
b
a
3
q
a
q
Structure Induced Equilibrium
Question: Why do we observe so much
stability?
Governmental institutions are stable
Policy change in incremental
Answer: Institutions Create Stability
If a policy space is divided among legislators
And these legislators have agenda control and
special rights
Then outcomes will not be chaotic
Procedures induce Institutional Equilibrium
Committee System & Stability
issue 2
x = outcome
committe 2
committe 1
Issue 1
Committees are given monopoly power over a
jurisdiction.
Policy may be unrepresentative of median
chamber, but it is stable.