Tuomas Sandholm Professor Computer Science Department Carnegie Mellon University

1. Social choice theory= preference aggregation= voting assuming agents tell the truth about their preferences Tuomas Sandholm Professor Computer Science Department Carnegie Mellon University

2. 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 Webarticles [Avery,Resnick&Zeckhauser], rating CDs...

3. Condorcet paradox [year 1785] • x > z > y • y > x >z • z > y > x • Majority rule • Three voters: 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.

4. x x y z y z x y z z x y z y y x z x Agenda paradox • x > z > y (35%) • y > x > z (33%) • z > y > x (32%) • Binary protocol (majority rule) = cup • Three types of agents: • 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

5. Pareto dominated winner paradox Voters: • x > y > b > a • a > x > y > b • b > a > x > y

6. Inverted-order paradox • Borda rule with 4 alternatives • Each agent gives 4 points to best option, 3 to second best... • Agents: • x=22, a=17, b=16, c=15 • Remove x: c=15, b=14, a=13 • 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

7. Borda rule also vulnerable to irrelevant alternatives • Three types of agents: • Borda winner is x • Remove z: Borda winner is y • x > z > y (35%) • y > x > z (33%) • z > y > x (32%)

8. Majority-winner paradox • a > b > c • a > b > c • a > b > c • b > c > a • b > c > a • b > a > c • c > a > b • Agents: • Majority rule with any binary protocol: a • Borda protocol: b=16, a=15, c=11

9. Is there a desirable way to aggregate agents’ preferences? • Set of alternatives A • Each agent iin {1,..,n} has a complete, transitive (not necessarily strict) ranking <iof 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 < inL, 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 inL , 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.

10. 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.

11. 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. ■

12. 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)