Chapter 2 of Computational Social Choice by William Zwicker Introduction If we assume every two voters play equivalent roles in our voting rule every two alternatives are treated equivalently by the rule ID: 707653
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Slide1
Introduction to Theory of Voting
Chapter 2 of Computational Social Choice
by William
ZwickerSlide2
Introduction
If we assume
every two voters play equivalent roles in our voting rule
every two alternatives are treated equivalently by the rule
there are only two alternatives to choose from
then May’s Theorem
tells us that the only reasonable voting method is
majority rule
.
This Talk (Chapter): Relax only the third condition and focus on ranked ballots!Slide3
Introduction
Definition: A voting rule is called a social choice function or SCF.
Prominent theorems about social choice functions:
Existence of majority cycles
Independence of Irrelevant Alternatives principle (IIA) → usually violated except for dictatorships
Gibbard
-Satterthwaite Theorem (GST) → every SCF other than dictatorship fails to be
strategyproofSlide4
Social Choice Functions:
Plurality, Copeland, and
BordaSlide5
Social Choice Functions:
Plurality, Copeland, and
Borda
a
is the unique plurality winner, or “social choice”.
It is difficult to see how
a
could win under
any
reasonable rule that use the second versus third place information in the ballots.
pairwise majoritySlide6
Social Choice Functions:
Plurality, Copeland, and
Borda
asymmetric
Borda
vs symmetric
Borda
affinely equivalent
b wins!Slide7
Social Choice FunctionsSlide8
Social Choice Functions
→ Copland and
Borda
announces “e” as the winner
↑Slide9
Axioms I: Anonymity, Neutrality, and the Pareto Property
Axioms
—precisely defined properties of voting rules as functions (phrased without referring to a particular mechanism)
Axioms often have
normative
content, meaning that they express, in some precise way, an intuitively appealing behavior we would like our voting rules to satisfy (such as a form of fairness).Slide10
Axioms I: Anonymity, Neutrality, and the Pareto Property
Axioms
—sorted in 3 groups
axioms of minimal demands
axioms of middling strength – controversial
strategyproofnessSlide11
Axioms I: Anonymity, Neutrality, and the Pareto Property
Legislative voting rules are often neither anonymous nor neutral.Slide12
Axioms I: Anonymity, Neutrality, and the Pareto PropertySlide13
Voting Rules I: Condorcet Extensions, Scoring Rules and Run-OffsSlide14
Voting Rules I: Condorcet Extensions, Scoring Rules and Run-OffsSlide15
Voting Rules I: Condorcet Extensions, Scoring Rules and Run-OffsSlide16
Voting Rules I: Condorcet Extensions, Scoring Rules and Run-OffsSlide17
Voting Rules I: Condorcet Extensions, Scoring Rules and Run-OffsSlide18
An Informational Basis for Voting Rules:
Fishburn’s
Classification
C1: SCFs corresponding to
tournament solutions, e.g.
Copeland, Top Cycle and Sequential Majority Comparison
C2: need the additional information in the
weighted tournament
—the net preferences
, e.g.
BordaC3: need “more” informationSlide19
Axioms II: Reinforcement and Monotonicity Properties
How does SCF respond when:
One or more voters change their ballots
One or more voters are added to a profileSlide20
Axioms II: Reinforcement and Monotonicity PropertiesSlide21
Axioms II: Reinforcement and Monotonicity PropertiesSlide22
Axioms II: Reinforcement and Monotonicity PropertiesSlide23
Axioms IISlide24
Voting Rules II:
Kemeny
and Dodgson
No social choice function is both reinforcing and a Condorcet extension but John
Kemeny
defined a neutral, anonymous, and reinforcing Condorcet extension that escapes this limitation via a change in context: his rule is a
social preference function.
Kemeny
measures the distance between two linear orderings,
by counting pairs of alternatives on which they disagree. For any profile P, the Kemeny Rule returns the ranking(s) minimizing the distance.If
a were a Condorcet winner for the profile P and did not rank
a
on top, then lifting
a
simply to top would strictly decrease the distance. Thus all rankings in the
Kemeny
outcome place
a
on top and in this sense
Kemeny
is a Condorcet extension.Slide25
Voting Rules II:
Kemeny
and DodgsonSlide26
Voting Rules II:
Kemeny
and Dodgson
Every preference function that can be defined by minimizing distance to unanimity is a ranking scoring rule, hence is reinforcing in the preference function sense.
We can convert a preference function into an SCF by selecting all top-ranked alternatives from winning rankings, but this may transform a reinforcing preference function into a non-reinforcing SCF, as happens for
Kemeny
.
A large variety of voting rules are “distance rationalizable”—they fit the minimize distance from consensus scheme.
Topic of Chapter 8