Definitions 1-4 Flashcards
3 preference relations of consumers?
1) Complete (ie. x must be weakly preferred to y or other way round, must be able to weakly rank all options ITO preference)
2) Reflexive (ie. x is weakly preferred to x (is at least as good as itself))
3) Transitive (if a is preferred to b and b to c then a is preferred to c)
What is convexity of a set?
A convex set is a set of points such that, given any two points A and B within the set, the line which joins them lies entirely within the set.
What is the Samuelson rule?
Samuelson rule states that for socially efficient provision of public goods, MRS1+MRS2+…+MRSn=MRT
What happens when there is competitive eq in public and private goods situ?
CE will not, in general, be PE since MRS are unlikely to add to 1
See
groves clarke mechanism L7 p3
Explain the 4 axioms of voting rules?
1) Universal domain (all possible preference selections are allowed)
2) Anonymity (the identity of each voter is irrelevant to the outcome of the vote)
3) Neutrality (if the ranking of 2 alternatives is reversed for all voters, then the social ranking is reversed)
4) Positive responsiveness (For a preferences profile, if some voters change their votes in favour of one alternative, and the rest do not change their vote, then the social ranking does not reverse in the other direction)
What is May’s Theorem?
An aggregation rule satisfies the 4 axioms U, A, N and PR if and only if it is the majority (plurality) rule
What is the majority rule?
Voters vote for their most favoured outcome and the one that gets the most votes wins
2 advantages of Majority rule?
satisfies May’s theorem
easy to use
2 advantages of Majority rule?
1) not very democratic; ignores lots of information
2) Lots of wasted votes (UK 2005 election, 52% votes were for losing candidates)
3) open to manipulation
(also only considers SPPs)
(see notes L8 p1s2)
Explain what plurality with run-off is? Pro of it?
If no candidate gets an absolute majority (ie. above 50%) in first stage, then all but top 2Cs are eliminated; there is then comparison between them again to decide the winner
Minimises the problem of wasted votes
3 problems with plurality with run-off?
1) open to manipulation still
2) is not monotonic (ie. one may win sub elections and lose grand elections)
3) See point one in notes
What is sequential run off?
Sequential voting rounds, each time eliminating the lowest voted outcome IF there is no abs. majority formed; if no majority is formed each round it continues until last round 1v1 and there will be an abs majority
2 advantages of sequential run-off?
Less manipulable (not immune to it tho) can encourage voters to vote for marginal (ie. unpopular) Cs
2 problems with SRF?
not monotonic
‘spoiler’ effect (if several close political candidates, vote-splitting may occur -> overall winner actually being a big distance from others!)
What is the Condorcet paradox?
Collective preferences can be irrational even when individual preferences are rational
See
unit 2 condorcet stuff
A Condorcet winner will not…
lose any pairwise vote
When does a voting rule satisfy the Condorcet criterion?
If it selects the Condorcet winner whenever one exists
Do plurality/plurality with run off satisfy the Condorcet criterion?
no
4 problems with the Condorcet method?
No winner may actually exist
open to manipulation
does not agree with plurality method
a CW may not be ‘ranked highly’
How does a Borda Count work?
It considers relative ranking, produces score
2 adv’s of BC?
Simple, yields a winner
3 problems with BC?
Open to manipulation, not consistent with withdrawals (ie. people choosing not to vote) (failure of IIA), not purely ordinal
IIA
see notes p2s2L8
What does non-manipulability mean?
A voting rule is strategy-proof if no voter can benefit from lying about their preferences, and manipulable if it is not strategy-proof
What is a Dictatorial voting rule?
One that always selects one candidates most preferred option
When is a plurality rule strategy proof?
When there are two voters
What is the Gibbard-Satterthwaite Theorem?
Given there are at least three alternatives, and that for each individual strict ranking of their preferences is permissable. Then the only U, SP social choice function is a dictatorship
What will a non-manipulable voting rule do?
It will make truth-telling the dominant strategy tf indi’s have no incentive to misrepresent preferences
What did Arrow propose?
That any SWF should satisfy:
1) Unrestricted domain
2) Weak pareto principle
3) IIA
4) Non-dictatorship
What is UD ITO SWFs?
the domain of f must include all possible combinations of individual preference relation on X
What is the Weak Pareto Principle?
For any pair of alternatives, x,y, in X, if x is preferred to y for all i, then x is preferred to y
What is IIA?
SEE NOTES L9s1p1 and the three properties of IIA!!!
What is non-dictatorship?
There is no individual whose preferences always solely determine the outcome of the SWF
What is Arrow’s impossibility theorem?
If there are at least 3 social states in X, then there is no SWF that simultaneously satisfies U, WP, IIA and ND
Solution to AIT?
Drop IIA; tf could use BC rule (violates localism and tf IIA)
Tf have to know all possible alternatives before preceding
BUT BC satisfies 2 other conditions: anonymity and neutrality (no outcome is ‘special’)
See
notes L9 weakening IIA bit and dropping UD (important to learn!!!)
See
last 2 pages of L9 if needed
What are the three properties of IIA? Explain them?
Ordinalism = it depends only on preference orderings
Individualism = it depends only on the individuals tastes
Localism = it depends only on the choices at hand
