Introduction to Social Choice

Question 0

The cardinality of a set…

Answer 0

The number of elements in that set.

Question 1

A set

Answer 1

A collection of elements

Example: A={x, y, z} where A is a set containing the elements x,y,z

Question 2

What is R, I and P

Answer 2

Preference relations

Question 3

What does xRy imply?

Answer 3

That x is at least as good as y

Question 4

What does xIy imply?

Answer 4

The the individual thinks alternatives x and y are equally good.

Question 5

How can xIy also be represented?

Answer 5

xRy & yRx

Question 6

What does xPy imply?

Answer 6

The individual thinks that the alternative x is preferable to y

Question 7

How else can xPy be written?

Answer 7

xRy & not(yRx)

Question 8

Assumptions of R

Answer 8

R is:
Reflexive
Connected
Transitive

Question 9

What do we say R is if it is reflexive, transitive and connected?

Answer 9

Ordering

Question 10

The majority decision rule

Answer 10

[|{xRy}|≥|{yRx}|]

Question 11

What is the main problem with the majority decision rule?

Answer 11

Can generate social preference cycles

Question 12

Agenda setting (manipulation)

Answer 12

A single individual has the power to set a voting agenda. Assuming there are three elements:
First he would specify voting on two individual elements and in doing so eliminate one.
Second the society would vote on the two remaining elements and by majority vote eliminate another. In doing so there is one left so society chooses the remaining element

Question 13

An alternative to the majority decision rule

Answer 13

The Borda Count

Question 14

The Borda Count

Answer 14

Attaches points to first(3) , second(2) and third(1) then adds them up

Question 15

Problem with the Borda Count

Answer 15

An alteration in an individuals preferences over two elements can alter society’s ranking and choice

Question 16

Unrestricted Domain and Collective Rationality

Answer 16

The social decision rule R should be defined for all possible preference profiles, and R should be an ordering.

Question 17

The Pareto Criterion

Answer 17

For all preference profiles and for all x,y

Question 18

Independence of Irrelevant alternatives

Answer 18

For all pairs of preference profiles, if no individual’s ranking of x and y changes (even if their rankings of alternatives changes) then society’s ranking of x and y should not change

Question 19

Non-dictatorship

Answer 19

There should not exist an individual i e N such that, for all preference profiles and for all x,y e X, if xPy then xPy

Question 20

Arrow’s Impossibility Theorem

Answer 20

Theorem which states if |X| ≥ 3 there does not exist any social decision rule R satisfying conditions (1), (2), (3) and (4)

Question 21

Which condition does the Borda count violate?

Answer 21

Independence of Irrelevant Alternatives

Question 22

Which social decision rule does the majority decision rule violate?

Answer 22

Unrestricted domain and collective rationality

Question 23

Is there are way to get around Arrow’s Impossibility Theorem?

Answer 23

The Median Voter Theorem, The Jury Decision rule and Individual rights.

Question 24

Assumptions of the Median Voter Rule

Answer 24

1. Issues are defined along a single dimension

2. Each voter’s preferences are ‘single peaked’

Question 25

The median voter rule

Answer 25

If x is a single dimensional issue and if all voters have single peaked preferences, then xm cannot lose under the majority decision rule.

xi* is individual i’s ideal point iff Ui(xi)>Ui(xi), for all xi not equal xi

Question 26

Which criterion is relaxed under the median voter theory?

Answer 26

The unrestricted Domain

Question 27

What is weakened under the jury decision rule?

Answer 27

The transitivity of R so that it is quasi-transitive

Question 28

What happens if R becomes quasi-transitive?

Answer 28

P becomes transitive

Question 29

Individual rights

Answer 29

For some pair of alternatives an individual is a dictator

Question 30

Acyclicity is…

Answer 30

weaker than quasi-transitivity which is weaker than transitivity

Question 31

Sen’s conditions

Answer 31

R is reflexive, connected and acyclic

It i indviduals xPy then society xPy

Is the Pareto Criterion

Question 32

Sen’s theorem

Answer 32

there does not exist a social condition rule which will satisfy all of the (Sen’s) conditions 1,2&3

Question 33

What does Sen’s theorem show?

Answer 33

It shows that the Pareto Criterion is inconsistent with individual rights.

Question 34

Non-Paternalism

Answer 34

social welfare depends only on an individual’s utility

Question 35

Paretian

Answer 35

social welfare is increasing in each individual’s utility

Question 36

Symmetry

Answer 36

all individuals are treated the same

Question 37

Strict Concavity

Answer 37

if the social welfare function is concave it displays an aversion to inequality. Redistributive from rich to poor