Probability of Violating Arrow's Conditions

32 Pages·2017·0.3 MB·English

Partial lecture notes
THE PROBABILITY OF VIOLATING
ARROW’S CONDITIONS
1
Probability of Violating Arrow's Conditions
B. Preference Aggregation Rules
3. Anti-Plurality
a. Assign zero points to a voter's last preference and one point to
all other preferences. Sum the points across voters and rank
alternatives in descending order of the sums.
Ex: 4 voters 2 voters 4 voters
A A C
C B A
B C B
Probability of Violating Arrow's Conditions
B. Preference Aggregation Rules
3. Anti-Plurality
a. Assign zero points to a voter's last preference and one point to
all other preferences. Sum the points across voters and rank
alternatives in descending order of the sums.
Ex: 4 voters 2 voters 4 voters
A (1) A (1) C (1) A = 7
B = 2
C (1) B (1) A (1)
C = 8
B (0) C (0) B (0)
Rank: C > A > B
b. Note: anti-plurality can violate P and IIA, but never T or ND.
Probability of Violating Arrow's Conditions
B. Preference Aggregation Rules
4. Hare (alternative vote)
a. Begin with a plurality vote. Eliminate the alternative with the
fewest plurality points (eliminating tieing alternatives if there is a
tie for fewest points). Repeat the process on the remaining
alternatives until all alternatives are eliminated. Alternatives are
ranked in reverse order of elimination (i.e., the alternative(s)
eliminated last is preferred most in the social ranking).
Ex: 4 voters 2 voters 4 voters
Plurality Vote:
A A C A = 6
B = 0
C B A
C = 4
B C B
Therefore eliminate B.
Probability of Violating Arrow's Conditions
B. Preference Aggregation Rules
4. Hare (alternative vote)
a. Begin with a plurality vote. Eliminate the alternative with the
fewest plurality points (eliminating tieing alternatives if there is a
tie for fewest points). Repeat the process on the remaining
alternatives until all alternatives are eliminated. Alternatives are
ranked in reverse order of elimination (i.e., the alternative(s)
eliminated last is preferred most in the social ranking).
Ex: 4 voters 2 voters 4 voters
Plurality Vote:
A A C A = 6
C = 4
C C A
Therefore eliminate C.
• Note: Hare can violate P and IIA, but never T or ND.
Rank: A > C > B.
Probability of Violating Arrow's Conditions
B. Preference Aggregation Rules
5. Nanson
• skip
Probability of Violating Arrow's Conditions
B. Preference Aggregation Rules
6. Pairwise Majority (a particular type)
a. Consider all possible two-alternative comparisons. For any pair
of alternatives {a; b}, a is socially preferred to b if more voters
prefer a to b. Alternatives a and b are socially indifferent if they
tie. Rankings are then derived from the pairwise comparisons.
Ex: 4 voters 2 voters 4 voters
A vs B -> A wins
A A C A vs C -> A wins
B vs C -> C wins
C B A
B C B
Rank: A > B > C.
• Note: Pairwise Majority can violate T, but never P, ND, or IIA.
Probability of Violating Arrow's Conditions
B. Preference Aggregation Rules
7. Copeland
a. Conduct a vote using pairwise majority. For each pair assign 1
point to the alternative that wins the pairwise comparison and .5
points if they tie. Rank alternatives in descending order of total
points.
Ex: 4 voters 2 voters 4 voters
A vs B -> A wins. 1 point A.
A A C A vs C -> A wins. 2 points A.
B vs C -> C wins. 1 point B.
C B A
B C B
Rank: A > B = C.
• Note: Copeland can violate IIA, but never P, T, or ND.
Probability of Violating Arrow's Conditions
C. Assumptions
• Notation:
A is the number of alternatives (e.g., candidates).
N is the number of voters.
ρ is a preference profile.
is the set of all possible preference profiles with strict orderings among N
individuals.
• We assume individuals only have strict preferences.
Note: individuals cannot
p is the proportion
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be indifferent,
of individuals with
but society can.
preference I>R>D.
Probability of Violating Arrow's Conditions
D. Theorems
1. For A ≥ 3 and N > 1, plurality, anti-plurality, and Hare will
violate IIA for all
a. This means that for all populations, with any number of
candidates and at least three alternatives, plurality, anti-
plurality, and Hare will always violate IIA.
b. Hence, these rules will always violate violate Arrow’s conditions.
Ex: 4 voters 2 voters 4 voters Plurality: A>C>B.
With one exception we can
A A C
always get plurality to violate
C B A
IIA by moving the middle
B C B ranked alternative to the top,
C in this case, in every
preference order.