Introduction to Theory of Voting 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 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!

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 strategyproof

Social Choice Functions: Plurality, Copeland, and Borda

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 majority

Social Choice Functions: Plurality, Copeland, and Borda asymmetric Borda vs symmetric Borda affinely equivalent b wins!

Social Choice Functions

Social Choice Functions → Copland and Borda announces “e” as the winner ↑

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).

Axioms I: Anonymity, Neutrality, and the Pareto Property Axioms—sorted in 3 groups axioms of minimal demands axioms of middling strength – controversial strategyproofness

Axioms I: Anonymity, Neutrality, and the Pareto Property Legislative voting rules are often neither anonymous nor neutral.

Axioms I: Anonymity, Neutrality, and the Pareto Property

Voting Rules I: Condorcet Extensions, Scoring Rules and Run-Offs

Voting Rules I: Condorcet Extensions, Scoring Rules and Run-Offs

Voting Rules I: Condorcet Extensions, Scoring Rules and Run-Offs

Voting Rules I: Condorcet Extensions, Scoring Rules and Run-Offs

Voting Rules I: Condorcet Extensions, Scoring Rules and Run-Offs

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. Borda C3: need “more” information

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 profile

Axioms II: Reinforcement and Monotonicity Properties

Axioms II: Reinforcement and Monotonicity Properties

Axioms II: Reinforcement and Monotonicity Properties

Axioms II

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.

Voting Rules II: Kemeny and Dodgson

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