Week 1 - Introduction Flashcards
How we represent an indiviuals preferences
a ⪰i b ⪰i c
Plurality Rule
- 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.
Simple Majority Rule
Winner is the candidate who is most preferred by > 50% of voters
candidate
Condorcet Voting (Pairwise majority rule)
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.
Condorcet paradox
Collective preferences can be cyclic, even if the preferences of individual voters are not cyclic.
Binary relation
- 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.
Completeness
For any individual i and any two alternatives a, b, either a ≻i b or
b ≻i a
Transitivity
For any individual i and any three alternatives a, b, and c, if a ≻i b
and b ≻i c then a ≻i c
Social Wellfare Function
- 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.
Desirable properties of SWF we can logically get at the same
time?
- 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 - 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. - 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 - 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
SWF dicatatorship
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
Arrows Theorem
As long as |A| ≥ 3, any SWF that satisfies transitivity, unanimity, and IIA
is a dictatorship
What are we trying to do?
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
Extremal Lemma
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.