Transcript Slides on social choice.

Social choice theory
= preference aggregation
= voting assuming agents tell the truth about their preferences
Tuomas Sandholm
Professor
Computer Science Department
Carnegie Mellon University
Social choice
• Collectively choosing among outcomes
– E.g. presidents
– Outcome can also be a vector
• E.g. allocation of money, goods, tasks, and resources
• Agents have preferences over outcomes
• Center knows each agent’s preferences
– Or agents reveal them truthfully by assumption
• Social choice function aggregates those preferences & picks outcome
• Outcome is enforced on all agents
• CS applications: Multiagent planning [Ephrati&Rosenschein],
computerized elections [Cranor&Cytron], accepting a joint project,
collaborative filtering, rating Web articles
[Avery,Resnick&Zeckhauser], rating CDs...
Condorcet paradox [year 1785]
• Majority rule
• Three voters:
1. x > z > y
2. y > x >z
3. z > y > x
x>z>y>x
Unlike in the example above, under some preferences there is a Condorcet winner,
i.e., a candidate who would win a two-candidate election against each of the other candidates.
Agenda paradox
• Binary protocol (majority rule) = cup
• Three types of agents:
1. x > z > y
2. y > x > z
3. z > y > x
x
x
x
y
z y
z x
z
y z
(35%)
(33%)
(32%)
y
y y
x z
z
x
• Power of agenda setter (e.g. chairman)
• Vulnerable to irrelevant alternatives (z)
• Plurality protocol
• For each agent, most preferred outcome gets 1 vote
• Would result in x
Pareto dominated winner paradox
Voters:
1. x > y > b > a
2. a > x > y > b
3. b > a > x > y
x
x
x
a
a
b
y b
y
a
b
y b
y
Inverted-order paradox
• Borda rule with 4 alternatives
– Each agent gives 4 points to best option, 3 to second best...
• Agents:
1.
2.
3.
4.
5.
6.
7.
x>c>b>a
a>x>c>b
b>a>x>c
x>c>b>a
a>x>c>b
b>a>x>c
x>c>b>a
• x=22, a=17, b=16, c=15
• Remove x: c=15, b=14, a=13
Borda rule also vulnerable to
irrelevant alternatives
• Three types of agents:
1. x > z > y
2. y > x > z
3. z > y > x
• Borda winner is x
• Remove z: Borda winner is y
(35%)
(33%)
(32%)
Majority-winner paradox
• Agents:
1.
2.
3.
4.
5.
6.
7.
a>b>c
a>b>c
a>b>c
b>c>a
b>c>a
b>a>c
c>a>b
• Majority rule with any binary protocol: a
• Borda protocol: b=16, a=15, c=11
Is there a desirable way to aggregate agents’ preferences?
•
Set of alternatives A
•
Each agent i in {1,..,n} has a complete, transitive (not necessarily strict) ranking <i of A
•
Complete, transitive (not necessarily strict) social welfare function F: Ln -> L
•
Some (weak) desiderata of F
– 1. Unanimity: If all voters have the same ranking, then the aggregate ranking
equals that. Formally, for all < in L, F(< ,…,<) =<.
– 2. Nondictatorship: No voter is a dictator. Voter i is a dictator if for all <1 ,…,<n ,
F(<1 ,…,<n) = <i
– 3. Independence of irrelevant alternatives: The social preference between any
alternatives a and b only depends on the voters’ preferences between a and b.
Formally, for every a, b in A and every <1 ,…,<n ,
< ’1 ,…,< ’n in L ,
if we denote < = F(<1 ,…,<n) and < ’ = F(< ’1 ,…,< ’n),
then (a <i b <=> a < ’i b for all i) implies that (a < b <=> a < ’ b).
•
Arrow’s impossibility theorem [1951]: If |A| ≥ 3, then no F satisfies desiderata 1-3.
Proof (from “A One-shot Proof of Arrow's Impossibility Theorem”, by Ning Neil Yu)
F
Because k can be anywhere
in the others’ preferences once
we ignore i.
Stronger version of Arrow’s theorem
• In Arrow’s theorem, social welfare function F outputs a ranking of the outcomes
• The impossibility holds even if only the highest ranked outcome is sought:
• Thm. Let |A| ≥ 3. If a social choice function f: Ln -> A is monotonic and
Paretian, then f is dictatorial.
– Definition. f is monotonic if [ x = f(>) and x maintains its position in >’ ] => f(>’) = x
– Definition. x maintains its position whenever x >i y => x >i’ y
• Proof. From f we construct a social welfare function F that satisfies the conditions
of Arrow’s theorem
– For each pair x, y of outcomes in turn, to determine whether x > y in F, move x and y to
the top of each voter’s preference order
• don’t change their relative order
• (order of other alternatives is arbitrary)
• Lemma 1. If any two preference profiles >’ and >’’ are constructed from a preference profile
> by moving some set X of outcomes to the top in this way, then f(>’) = f(>’’)
– Proof. Because f is Paretian, f(>’)  X. Thus f(>’) maintains its position in going from >’ to >’’.
Then, by monotonicity of f, we have f(>’) = f(>’’)
–
–
–
–
–
Note: Because f is Paretian, we have f = x or f = y (and, by Lemma 1, not both)
F is transitive (total order) (we omit proving this part)
F is Paretian (if everyone prefers x over y, then x gets chosen and vice versa)
F satisfies independence of irrelevant alternatives (immediate from Lemma 1)
By earlier version of the impossibility, F (and thus f) must be dictatorial. ■
Voting rules that avoid Arrow’s impossibility
(by changing what the voters can express)
• Approval voting
– Each voter gets to specify which alternatives he/she approves
– The alternative with the largest number of approvals wins
– Avoids Arrow’s impossibility
• Unanimity
• Nondictatorial
• Independent of irrelevant alternatives
• Range voting
– Instead of submitting a ranking of the alternatives, each voter gets to assign a
value (from a given range) to each alternative
– The alternative with the highest sum of values wins
– Avoids Arrow’s impossibility
• Unanimity
• Nondictatorial
• Independent of irrelevant alternatives (one intuition: one can assign a value to an
alternative without changing the value of other alternatives)
– More information about range voting available at www.rangevoting.org
• These still fall prey to strategic voting (e.g., Gibbard-Satterthwaite
impossibility, discussed in the next lecture)