Transcript 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 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
Under some preferences there is a Condorcet winner.
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?
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Set of alternatives A
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Each agent i in {1,..,n} has a ranking <i of A
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Social welfare function F: Ln -> L
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To avoid unilluminating technicalities in proof, assume <i and < are strict total orders
•
Some possible (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 (follows logic of Nisan’s proof in Algorithmic Game Theory book, but fixes errors and includes missing part)
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Assume F satisfies unanimity and independence of irrelevant alternatives
Lemma. Any function F: Ln -> L that satisfies unanimity and independence of irrelevant
alternatives also satisfies pairwise unanimity. That is, if for all i, a <i b, then a < b.
– Proof. Let <* in L be such that a <* b. By unanimity, a < b in F(<* ,..,<*). If
<1,...,<n are all such that a <i b, then we have a <i b <=> a <* b, and so by
independence of irrelevant alternatives, for <' = F(<1,...,<n), we have a <' b. ■
Lemma (pairwise neutrality). Let >1 ,…,>n and >’1 ,…, >’n be two player profiles such
that a >i b <=> c >’i d. Then a > b <=> c >’ d.
– Proof. Assume wlog that a > b and c not equal to d. We merge each >i and >’i into
a single preference >i by putting c just above a (unless c = a) and d just below b
(unless d = b) and preserving the internal order within the pairs (a,b) and (c,d).
– By pairwise unanimity, c > a and b > d. Thus, by transitivity, c > d. ■
Take any a not equal to b in A, and for every i in {0,…,n} define a preference profile Pi
in which exactly the first i players rank b above a. By pairwise unanimity, in F(P0) we
have a > b, while in F(Pn) we have b > a. Thus, for some i* the ranking of a and b flips:
in F(Pi*-1) we have a > b, while in F(Pi*) we have b > a.
We conclude the proof by showing that i* is a dictator:
Lemma. Take any c not equal to d in A. If c >i* d then c > d.
– Proof. Take some alternative e that is different from c and d. For i < i* move e to
the top in >i, for i > i* move e to the bottom in >i, and for i* move e so that c >i* e >i*
d. By independence of irrelevant alternatives, we have not changed the social
ranking between c and d.
– Notice that the players’ preferences for the ordered pair (d,e) are identical to their
preferences for (a,b) in Pi*, but the preferences for (c,e) are identical to the
preferences for (a,b) in Pi*-1, and thus using the pairwise neutrality claim, socially e
> d and c > e, and thus by transitivity c > d in the preferences where e was moved.
By independence of irrelevant alternatives, moving e does not affect the relative
ranking of c and d; thus c > d also under the original preferences. ■ ■
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(>’’)
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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)