Public Economies

7/31/2019 Public Economies

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Public Economics

Paper 7

Lecture 3

Multi-dimensional policy spaces and the

Probabilistic Voting Model


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Outline

Critique of the median voter model

The probabilistic voting model


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The Median Voters Theorem

The two parties will propose the same platform

and that platform is equal to the

most-preferred policy of the median voter.

A Condorcet Winner

A policy that beats all others in

a pair-wise comparison.


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When does a Condorcet winner

exist? The policy space has one dimension.

Voters have single peaked preference

over policy along that one dimension.

Each voter has a most-preferred policy and the

further the policy is away from this ideal point

the worse.

g

)(gWi

(size of government)Ideal point

Single peaked.

Not single peaked.


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More policy dimensions

A Condorcet winner will not exist in

general.

Policy instability and cycles.

Chaos


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The logic of chaos

Two public goods, g and h.

Each voter has an ideal point ),(**

ii hg

g

h

*

ig

*

ih

Ideal point

Indifference

curve


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g*

1g

*

1

*

2hh

h

*

2g

*

3h

*

3g

A

Win set 12

1

3

2

B


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The Chaos Theorem(McKelvey, Journal of Economic Theory, 1976)

When there are at least two issues and at least three

voters every platform can be defeated.

Political instability => incumbents are always defeated.

The median voter model has no predictive power

in the realistic case where the policy space is complex

Reinstated domain assumption =>

collective rationality has to give


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Why so much stability?

Institutions prevent instability

Legislative bargaining (next week)

Uncertainty about voter preferences

restores predictability

Probabilistic voting (now)


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Probabilistic voting


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The Idea Voters care about two things:

Policy platforms

Fixed characteristics of the parties that are unrelated totheir platform endorsements; they have a bias towardsone of the parties:

Ideology

Charisma or competency of leaders

Voters tend to vote for party that offers the bestplatform but only if the difference is big enough to overturntheir bias.


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The Idea Let ibe the ideological bias if voters i

Each voter knows i

The parties dont know i but they know thedistribution of the biases.

The parties therefore perceive that there only is

a probability that voter i will support itsplatformeven it is better for that voter thanthe one proposed by the other party.


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Deterministic vs. probabilistic voting

Voters vote for sure for the platform

that comes closest to their ideal point.

Let

iiWWiff1 BAAip

be the probability that voter i votes for party A.Aip

i

B

i

A

WWiff0 Ai

pi

B

i

A2

1 WWiff Aip

g

h

*

ig

*

ih

Party A

Party B


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Deterministic vs. probabilistic voting

Voters dont vote for sure for the platform

that comes closest to their ideal point.

i

BAAip ii

WWiff1

i

Aip i

B

i

A WWiff0

i

Aip i

B

i

A2

1 WWiff

g

h

*

ig

*

ih

Party A

Party B?


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Assumptions

Two parties, A and B

Platforms are xA and xB (many dimensions, e.g.

x={g,h,t}) Different ideologies (or other fixed characteristics).

Two types of voters, J=1,2

WJ(x) is strictly concave in x (single-peaked).

xJ* is the most-preferred policy of a voter of type J

nJ voters of type J with n1 n2.


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Assumptions

The ideological bias of voter i in favour of

party B is i

iBAAi xxp )(W)(Wiff1

ii

The distribution of the biases is type

specific

i is distributed uniformly between

[lJ

,hJ

] for J=1,2.


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17

Distribution of ideological biases

i

0

Type 1

l1 h1

1/(h1+ |l1|)

Type 2

l2 h2

1/(h2+ |l2|)

2211 lhlh

0)()( 21 ii EE

)(0)(21 ii EE

Swing versus committed voters:

Equally popular parties:

Type 1 (2) is faithful to party B (A):

i

BAAi xxp )(W)(Wiff1ii


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The timing

Politiciansannounce platforms

Citizens vote Elected politicianimplements his platform

{xA, xB}

Citizens learn their

ideological bias

i

BAAi xxp )(W)(Wiff1ii

Simple majority

Proportional representation

First-past-the-post in multiple districts


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Objectives of the parties

Maximize the probability of winning a

majority in the election:

)2

1Prob(

21

2211

nn

nn AA

JA (JB) is the share of voters of type J that

votes for party A (B)

Common popularity shock in favour of

party A: .


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The political Equilibrium

1. Express the vote shares as a function of theplatforms.

2. Find the platform that maximise the probabilityof winning for each party

3. Find the Political (Nash) equilibrium by solvingthe resulting first order conditions

A pair of platforms that maximise eachparties chance of winning given the platform

choice of the other party and the voting rule

followed by voters.


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The vote shares Step 1: Express share of votes for party A (and B) amongst

type 1 voters (A

1 and B

1) as a function of xA

and xB

Fix two platform proposals xA and xB Find the voter of type 1 who is indifferent between the two parties

when these platforms are proposed

W1(xA

) = W1(xB

) + s1

s1

= W1

(xA)W

1

(xB)

This voter, indexed by s, is the swing voter of type 1.

Who is the swing voters depends on xA and xB

All voters in group 1 with i1< s1 vote for party A

All voters in group 1 with i1s1 vote for party B

Step 2: Calculate the vote share for party A and party B

respectively:

Step 3: Repeat step 1 and 2 for voters of type 2 to get A

2

and B

2


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22

Shares of votes for party A and B among type 1 voters

Voting rule: vote for A ifW1(xA) > W1(x

B) + i

i

0l1

1/(h1+ |l1|)

s1(xA, xB)

h1

11

11

1

hl

ls

A

11

111)'()'(

hl

xWxWlBA

11

111

11)'()'(

1

hl

xWxWhAB

AB


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The platform choice

Party A proposes the platform xA that maximisesits probability of winning given the platform

proposed by party B xB

2

1nnProbmax

21

2211

nn

AAxA

2211nnmax AAxA

22

222

2

11

111

1)()()()(

maxhl

xWxWln

hl

xWxWln

BABA

xA

for given xB


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22

222

2

11

111

1)()()()(

maxhl

xWxWln

hl

xWxWln

BABA

xA

for given xB

First order condition:

02

2

22

1

1

11

A

A

A

A

x

W

Wn

x

W

Wn

0

112

22

2

1

11

1

AA x

W

hlnx

W

hln

Similarly for party B


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011

2

22

21

11

1

AA x

W

hln

x

W

hlnParty A:

Party B: 011

2

22

2

1

11

1

BB x

W

hln

x

W

hln

Equilibrium exists: Even if the policy space is bigand complex.

Policy convergence: The two parties propose the

same platform xA

=xB

=x*

Policy compromise: The equilibrium policy is a

compromise between what type 1 and type 2

voters want.


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The Representation Theorem

The equilibrium platform maximizes

2

1

)(maxJ

J

Jx

xWq

A representative democracy with probabilistic voting

behaves as ifthe government is maximizing a weighed

social welfare function, where the weights represent

the voters responsiveness to marginal platform changes

and group sizes.

JJ

J

Jhl

nq


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Result overview

Groups whose vote decision is mostly

influenced by policy as opposed to ideology

get a bigger weight.Policy caters to the swing voters as opposed to

the median voter.

Large groups of voters get bigger weight.


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What is Next?

Probabilistic voting model and electoral

systems

Legislative Bargaining


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g*

1g

*

1

*

2hh

h

*

2g

*

3h

*

3g

Pareto set


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g*

1g

*

1

*

2hh

h

*

2g

*

3h

*

3g

A

Win set 32

Win set 31

Win set 12

1

3

2

Documents

Public Economies