Part 2: Individual To Social Preferences Flashcards
What measures democracy
B) what has happened on average globally?
Economist intelligence unit democracy index
Global average score been falling
Political institutions are not neutral. If they were neutral, what would be the effect on economic policy
No effect
2 key challenges of turning individual preferences into social preferences
Aggregation problem
Agency problem
Aggregation problem
How can individual preferences be aggregated to group preferences (will majority preference be applied)
Agency problem
Information problem - does political process (voting) ensure majority preference will be achieved
Stable match
When no unmatched partner each prefers the other to their partner (not mutually profitable to break away)
E.g a married couple, NEITHER wants to join up with someone from another pair to make them mutally better off.
Stable roommate problem with 4 agents
Pick from other 3 in order.
Look at table pg 16:
A) is a room with Henry and Mary, and a second room with Peter and Jack stable?
B) a room with M+P, and room with J+H
C) room with M+J and room with P+H
Not stable, as Mary would prefer to be with Peter, as well as Peter prefer to be with Mary than Jack. Thus Mary and Peter are the blocking pair (mutually profitable: thus unstable)
B)
Not stable as Henry would prefer to be with Peter than Jack, and Peter would rather be with henry than Mary.
Thus Henry and Peter are blocking pair, deviation is mutually profitable: thus unstable!
C) stable as even though not all got first choices, no mutually profitable deviation
Arrow’s impossibility theorem
No mechanism is able to aggregate individual preferences consistently i.e satisfy all 5 properties , thus political institutions cannot be neutral
Hence why called impossibility theorem
So no system that aggregates individual preferences consistently and satisfies 5 properties
What are the 5 properties for social welfare function
Unrestricted domain
Rationality
Unanimity
Independence of irrelevant alternatives (IIA)
Non-dictatorship
3 policy options Ω = [A, B, C]
Each individuals has preferences which are complete and transitive and strict: Meaning.
Completeness: either A>B or B>A
Transitivity: if A>B and B>C, A>C
Strict: no indifference
What is the mapping of individual prerences into social preference called
Social welfare function (SWF)
Explain the first 4 properties
Unrestricted domain (universality) - consider all preferences
Rationality (complete and transitive) (agenda-setting voting method violates!)
Unanimity i.e if everyone agrees A>B policy, SWF preference should prefer A>B
Independence of irrelevant alternatives: if we are trying to figure out whether society prefers A to B, what people think of C shouldn’t matter (irrelevant - just concentrate on issue at stake) (Borda count can violate this!)
Sad result Arrow’s impossibility theorem of SWF satisfying all 4 properties:
If satisfies all 4, it must be a dictatorship (SWF only accounts for a same certain individual’s preferences, regardless of anyone elses’)
(5th property was non-dictatorship, but satisfying all 4 before means 5th isn’t possible)
So what other choices are there to avoid dictatorship
Ignore satisfying one/more of the other 4 properties
E.g violating unrestricted domain (where we ignore some individuals preferences)
Which one is the strongest restriction
IIA , since we ignores cardinal aspect of preferences, only ordinal pair-wise.
E.g if 100 policy options, and A and B ranked 45 and 47, you near indifferent between them.
However if ranked 1 and 100, you probably like A a lot more. IIA treats the 2 cases as the same
So this stuff looked at aggregation problem.
Now look at voting rules. What is a condorcet winner
Policy that beats any other feasible policy in a pair-wise vote
Steps to condorcet
Rank candidates in order of preferences (position of order, not scores!)
Then pairwise comparison where winner is determined by majority
(When a voter does not include full preferences, assumed to prefer candidates they have ranked over all other ones they omitted)
Condorcet paradox:
Some voting rules can fail to produce a clear cut winner
Condorcet paradox ice cream example pg37
pg 38 ice cream example of concordet winner
no clear winner in a pairwise
3 citizens with the following preferences:
A>B>C;
B >C>A;
C>A>B
Cycling outcome - there is no social transitivity!
A vs B: A wins
B vs C: B wins
C vs A: C wins
concorcet paradox! no clear winnerh
When there is a cycling outcome i.e no transivity;
What do agents have an incentive to do
agents may not truthfully reveal their preference since a cycle, so vote strategically to get next best
So we have chocolate vanilla or strawberry. How many strong preferences (no indifferences) ordering possibilities are there
b) as a result, how many possible socieites
6
b) 6 x 6 x 6 = 216 possible societies (since 3 agents)
when would the cycle occur
If one of the alternatives is ranked once first by one voter, 2nd by another voter, and 3rd by the last voter
(like 3 citizens with the following preferences:
A>B>C;
B >C>A;
C>A>B)
given 6 strong preference orderings, and 216 possible societies, how many times will lead to a cycle?
b) what can we conclude; what is the probability of a condorcet cycle (not finding a winner)
12
b) 12 is low out of 216! so condorcet voting (pairwise comparisons) mostly works to find a clear winner
probability of not finding a winner is 12/216 x 100 = 5.56%
(we see how this probability changes with the number of alternatives and voters!)
What does probability of finding a condorcet cycle (no clear winner) changes with (2)
And state the relationship with them
Number of alternatives (policy options - only had 3 in example) POSITIVE REL
Number of voters - POSITIVE REL
this is bad for voting organisers, since more candidates and more voters lower chance of finding a clear winner
Pg 44 voting exercise q1
show majority voting yield intransitivity:
A vs B: A wins. B vs C: B wins, C vs A: C wins
thus no clear winner; condorcet cycle
What is odd: Tim’s are ordered strange, right>left>middle
Q2 pg 45
Tim can vote strategically;
B vs C: he prefers C, but he knows B vs C, B wins! thus now he must try get his 2nd best option, A.
So stage 1 vote B (vote strategically not sincerely)
This is so in stage 2 it is the winner vs A, so A vs B.
Which , A wins!
So he votes untruthfully to get his 2nd best option rather than B
pg 45 q3
Vote strategically and vote A over C, despite preferring C.
since A can win, which is pitted against B, and then A beats B
so Tim gets his 2nd best option again. (if he stayed true to preference, C wins, and then loses to B, which his worst outcome
issues of condorcet (pairwise) voting) (2)
Fair, but time consuming to find winner in a large set (thus people have modifications of it to implement it on a large set)
Doesn’t always lead to a winner (increases probability of not finding winner with number of candidates and voters). which is problem as doesn’t give incentive to tell truth (seen in cycles)
Now explore different voting rules and their outcome
Simple majority voting. Look at graph pg 47;
what is this system, and what is problem
Rule: everyone indicates most preferred option. the policy with most votes wins. so, A would win!
problem: given public info…
Incentives for voters 4 5 6 and 7, prefer at least 2 other policies to A, e.g 4 and 5 would rather C than A, so vote C as their most preferred option to get 4 votes for C, so C wins! (so 4 and 5 get their 2nd best option of C)
another voting method:
Agenda setting
b) issue
Policies voted in pairwise comparison, in a pre-established order
b) violates rationality as no transitivity
Borda count
Weights different preferences
Most preferred alternative gets k points, next most preferred gets k-1
Winning alternative is one with max number of points
Borda count example: if k=2, using pg 47 preferences, who wins and how many points
B is voted 1st by voter 4 and 5, so 2 give 2 point = 4
then voted 2nd best by voters 123 so 3 points
3+4 = 7 points for B
A has 3 votes for 1st place, so 6 points
What if Borda k=3?
Which assumption is violated? and why??
Issue, C then wins!
Violates IIA, as ranking has not changed, only the scoring rule, yet the winner did!
So different voting rules have different policy otucomes (condorcet, simple majority, agenda setting, borda)
