Week 1 - Introduction

Question 1

How we represent an indiviuals preferences

Answer 1

a ⪰i b ⪰i c

Question 2

Plurality Rule

Answer 2

• Winner is the candidate who receives the most votes, but does not necessarily receive more the 50% of votes.
• Plurality rule would select a as has recieved most no 1 votes.

Question 3

Simple Majority Rule

Answer 3

Winner is the candidate who is most preferred by > 50% of voters
candidate

Question 4

Condorcet Voting (Pairwise majority rule)

Answer 4

The candidate who wins a majority of the vote in every head-to-head election against each of the other candidates – that is, a candidate preferred by more voters than any others – is the Condorcet winner, although Condorcet winners do not exist in all cases.

Question 5

Condorcet paradox

Answer 5

Collective preferences can be cyclic, even if the preferences of individual voters are not cyclic.

Question 6

Binary relation

Answer 6

• A binary relation over the set X describes the relative merits of any two
  outcomes in X with respect to some criterion.
• A simple example is letting X = {1, 2, 3, 4}, and choosing the binary relation ≥. We can compare
  any two elements of X with this relation. For example, choosing elements 1 and 2, we can say that
  2 ≥ 1.

Question 7

Completeness

Answer 7

For any individual i and any two alternatives a, b, either a ≻i b or
b ≻i a

Question 8

Transitivity

Answer 8

For any individual i and any three alternatives a, b, and c, if a ≻i b
and b ≻i c then a ≻i c

Question 9

Social Wellfare Function

Answer 9

• Mapping individual preferences to social preferences.
• Maps an admissible profile P = (≻1, ≻2, . . . , ≻n) ∈ Ln
  into a social “preference” at this profile, F(P) ⊆ A × A.

Question 10

Desirable properties of SWF we can logically get at the same
time?

Answer 10

1. Universal domain/ unrestricted - We’re not ruling out any possible preferences
   - SWF to be defined at any logically possible n-tuple of
   individual preferences (≻1, ≻2, . . . , ≻n) ∈ Ln
2. Linear order/ transitivity - We want ≻∗
   to be a linear order (i.e. complete and transitive) at any preference profile
   Pairwise majority voting would violate transitivity.
3. Unanimity - If every individual agrees that a is better than b, then the social preference
   produced by the SWF should also strictly rank a over b
4. Independence of irrelevant alternatives - if the society is trying to decide between a and b, then what people think
   of c ̸= a, b shouldn’t matter for the decision
   Borda count would violate IIA

Question 11

SWF dicatatorship

Answer 11

SWF is a dictatorship, if it ranks alternatives the same as one particular individual’s preference ordering
In other words:
There is an individual i such that a ≻∗ b if and only if a ≻i b, for any
a, b ∈ A, regardless of the preferences of the remaining individuals

Question 12

Arrows Theorem

Answer 12

As long as |A| ≥ 3, any SWF that satisfies transitivity, unanimity, and IIA
is a dictatorship

Question 13

What are we trying to do?

Answer 13

We’re looking for a way to aggregate preferences – that is, we want a way to turn each possible set of individual preferences {>1, >2, . . . , >N } into a preference relation weak preference ∗ for “society”
We won’t require %∗
to be strict

Question 14

Extremal Lemma

Answer 14

For any policy b, if every individual i ranks b either strictly best or strictly worst, then SWF must rank b either strictly best or strictly worst as well.

• take the example:
• We can see that every individial ranks c, either best or worst.
• Therefore SWF would rank c best or worst.
• However it tells us that SWF has ranked C last. Therefore a must be socially preffered to c.