5: Social Choice Theory: three or more options Flashcards
What are Arrows Axioms
- Unrestricted Domain
- Rational
- Pareto Principle
- Non Dictatorship
- Independence of irrelevant alternatives
Arrows Unterhosen reichen Papa nicht immer
Unrestricted Domain
(for decision rules)
Applicable to any set of preferences
Rational
(for decision rules)
Provides Complete and Transitive Ranking
Pareto Principle
(for decision rules)
if everyone individually prefers a over b, the SWF should prefer a over b
Non dictatorship
(for decision rules)
There exists no dic, such that whenever dic prefers a over b, the SWF prefers a over b
Independence of irrelevant alternatives
(for decision rules)
The social preference between a and b does not depend on the individual preferences on c
Arrows Impossibility theorem:
if there are at least three options, no SWF can satisfy all axioms
Condorcets Paradox
Pairwise Majority Voting is not transitive
Pairwise Majority Voting
Repeatedly compare two options
→ no consistent ranking or winner
Plurality rule
Everyone Votes for their favorite
→ limited information
Borda Count
Everyone ranks their options, SWF counts points
→ sensitive to irrelevant info
Condorcets rule (for info gathering)
Assumptions:
- there exists a correct order
- everyones pairwise decision is correct by probability p>0.5
________________________________________
Support: (eg three alternatives)
- the support of a>b>c is defined as
S_{abc} =
p^(n_ab+n_ac+n_bc) * (1-p)^(n_ba+n_ca+n_cb)
with n_xy being the number of people who prefere x over y
_______________________________________
The ranking with maximum support has the highest probability of beeing correct
Condorcets Winner
An option that does not loose any pairwise majority vote
→ always places first in condorcets rule
