Choosing in Groups - PowerPoint Presentation

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Choosing in Groups - PPT Presentation

Munger and Munger Slides for Chapter 6 Two Dimensions Elusive Equilibrium Outline of Chapter 6 Appropriations committee example Equilibrium Representing spatial preferences Nonseparability and complementarity ID: 244044

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Slide1

Choosing in GroupsMunger and Munger

Slides for Chapter 6Two Dimensions: Elusive EquilibriumSlide2

Outline of Chapter 6 Appropriations committee example

EquilibriumRepresenting spatial preferencesNonseparability and complementarityChaos and equilibria

Mathematical examples in multiple dimensions

Generalized Mean Voter Theorem (GMVT)

2Slide3

Appropriations committee examplePreferences over spending on two projects

Having two dimensions changes the problem:Salience: How much importance does each person place on the two issues?Separability:

Ideal point in each dimension depends on decision on other dimension

Conditional ideal points

Multidimensional Amendments

: Proposals can occur on multiple dimensions.

Table 6.1. Subcommittee ideal points on two projects

Member

Project 1 (millions of $)

Project 2 (millions of $)

Mr. A

150

120 (Median Project 2)

Mr. B

Ms. C

100

Ms. D

200

Mr. E

80 (Median Project 1)

150Slide4
Slide5

EquilibriumPolitical equilibrium is a status quo position that cannot be defeated

The equilibrium position defeats any other alternative that can be proposedRules or institutions may prevent feasible alternatives that would defeat the equilibrium from being proposedEquilibrium is rarely unique

5Slide6

Representing spatial preferences: utility functionsIf proposals concern only one issue at a time, Mean Voter Theorem applies

Outcome is median of each issueIndifference curves: Each point on an indifference curve is equally preferred Separable preferences with equal salience: Circular indifference curves

Separable preferences with unequal salience: Elliptical indifference curves

Nonseparable preferences: Indifference curves could take many shapes!

6Slide7
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Nonseparability: positive and negative complementarityNegative complementarity: if we choose more of A, I want less of B

E.g., Fixed budget: if we spend more on parks, I want to spend less on garbage collectionPositive complementarity: if we choose more of A, I want more of BE.g., every police officer should have a computer, so if we hire more officers, I want to buy more computers

Ideal point in each dimension is

contingent

on decision in the other dimension

10Slide11
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Voting on complex proposals: definitionsComplex proposals: changes on more than one dimension

Germaneness rule: one dimension at a timeWith germaneness rule and separable preferences, voting sequence does not matterPareto set: set of positions from which no one can be made better off without making someone else worse off

Condorcet winner: a position that beats or ties any other in pairwise contests

Example: median position with single-peaked preferences over one dimension

Win set (of

): positions that will get more votes than z in pairwise contests

Formally:

14Slide15
Slide16

ChaosGenerally the win set of the median proposal (in >1 dimensions) is not empty!

Chaos result: for sincere voting, sequence of accepted proposals can go anywhere!Outcome may not be in Pareto setMajority rule can lead to an outcome that

everyone agrees is bad!

16Slide17
Slide18

Conditions for equilibriumPlott

conditions: ordinal pairwise symmetrySufficient but not necessary for unique equilibriumFairly restrictive“Median in all directions”:

A line drawn through this point has at least half the voters on either side

Such a point is a Condorcet winner

This is a generalized version of the Median Voter Theorem

18Slide19
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Multiple dimensionsSimple Euclidean distance (SED):

SED implies separability and equal salienceWeighted Euclidean distance (WED):

WED does

not

imply separability or equal salience

21Slide22
Slide23

Multiple dimensions (2)A

i is the matrix of salience and interaction terms (shown below for two dimensions)Main diagonal elements are salience (weights)

We assume they are positive

Off-diagonal elements represent complementarity

We assume they are equal for simplicity (not required by theory)

Symmetry: If off-diagonal elements are 0 and main diagonal elements are 1

Then WED reduces to SEDIndifference: Two points are indifferent if their WED from the ideal point is the same

23Slide24

Multiple dimensions (3)We can write three general categories of indifference curves in matrix notation:

Ai = k I, where

k is a constant: Circular indifference curves

= sI

, where s is a vector of weights: Elliptical indifference curves; separable Ai not diagonal: nonseparable preferences

24Slide25

Example: separable WED in two dimensionsReturn to committee example. Consider Member B

B prefers y to z iff

Let’s assume that A

looks like this:

Note that B’s preferences are separable, but the salience terms are not equal.

25Slide26

Example: separable WED in two dimensions (2)Let’s go through the calculations:

In this example, B prefers

, because

26Slide27
Slide28

Nonseparable preferencesNonseparable preferences have an additional interaction term:

Positive

a12

indicates a negative complementarity (greater distance)

Negative

12 indicates positive complementarity (lesser distance)

28Slide29

Nonseparable preferences (2)Effects of the interaction term:

For positive complementarity: if issue j is fixed higher than the ideal point, then the conditionally ideal

value will be

higher

than the ideal point.

For negative complementarity: if issue j is fixed

higher

than the ideal point, then the conditionally ideal

value will be

lower than the ideal point.For positive or negative complementarity: if issue j is fixed at the ideal point, then the interaction term will be zero, and complementarity will have no effect.Slides Produced by Jeremy Spater, Duke University. All rights reserved.

29Slide30

Nonseparable preferences (3)Calculating conditional preference: Say Project 1 budget is fixed at

We write out the WED:Minimize the WED to find the Project 2 budget that maximizes the voter’s utility:

Note that the conditionally ideal Project 2 budget is

not

the same as the voter’s

unconditionally

ideal Project 2 budget.

30Slide31

Generalized Median Voter Theorem (GMVT)Assumptions

:N voters; k issuesSeparable and symmetric preferences “Separating hyperplane”: k-1 dimensions (e.g., a line in 2-space) that divides the ideal points of the voters into two separate groups.

Theorem

An alternative

is the median if every hyperplane containing y has at least half the ideal points on either side of the hyperplane.