Introduction to Social Choice Flashcards
The cardinality of a set…
The number of elements in that set.
A set
A collection of elements
Example: A={x, y, z} where A is a set containing the elements x,y,z
What is R, I and P
Preference relations
What does xRy imply?
That x is at least as good as y
What does xIy imply?
The the individual thinks alternatives x and y are equally good.
How can xIy also be represented?
xRy & yRx
What does xPy imply?
The individual thinks that the alternative x is preferable to y
How else can xPy be written?
xRy & not(yRx)
Assumptions of R
R is:
Reflexive
Connected
Transitive
What do we say R is if it is reflexive, transitive and connected?
Ordering
The majority decision rule
[|{xRy}|≥|{yRx}|]
What is the main problem with the majority decision rule?
Can generate social preference cycles
Agenda setting (manipulation)
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
An alternative to the majority decision rule
The Borda Count
The Borda Count
Attaches points to first(3) , second(2) and third(1) then adds them up
Problem with the Borda Count
An alteration in an individuals preferences over two elements can alter society’s ranking and choice
Unrestricted Domain and Collective Rationality
The social decision rule R should be defined for all possible preference profiles, and R should be an ordering.
The Pareto Criterion
For all preference profiles and for all x,y
Independence of Irrelevant alternatives
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
Non-dictatorship
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
Arrow’s Impossibility Theorem
Theorem which states if |X| ≥ 3 there does not exist any social decision rule R satisfying conditions (1), (2), (3) and (4)
Which condition does the Borda count violate?
Independence of Irrelevant Alternatives
Which social decision rule does the majority decision rule violate?
Unrestricted domain and collective rationality
Is there are way to get around Arrow’s Impossibility Theorem?
The Median Voter Theorem, The Jury Decision rule and Individual rights.
Assumptions of the Median Voter Rule
- Issues are defined along a single dimension
2. Each voter’s preferences are ‘single peaked’
The median voter rule
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
Which criterion is relaxed under the median voter theory?
The unrestricted Domain
What is weakened under the jury decision rule?
The transitivity of R so that it is quasi-transitive
What happens if R becomes quasi-transitive?
P becomes transitive
Individual rights
For some pair of alternatives an individual is a dictator
Acyclicity is…
weaker than quasi-transitivity which is weaker than transitivity
Sen’s conditions
R is reflexive, connected and acyclic
It i indviduals xPy then society xPy
Is the Pareto Criterion
Sen’s theorem
there does not exist a social condition rule which will satisfy all of the (Sen’s) conditions 1,2&3
What does Sen’s theorem show?
It shows that the Pareto Criterion is inconsistent with individual rights.
Non-Paternalism
social welfare depends only on an individual’s utility
Paretian
social welfare is increasing in each individual’s utility
Symmetry
all individuals are treated the same
Strict Concavity
if the social welfare function is concave it displays an aversion to inequality. Redistributive from rich to poor
