Efficient Voting via the T op-k Elicitation Scheme: A Probabilistic Approach

1. Efficient Voting via the Top-k Elicitation Scheme: A Probabilistic Approach Joel Oren, University of Toronto Joint work with Yuval Filmus, Institute of Advanced Study

2. Motivation • Common theme: communication-efficient group decision-making. • How to pick the single “best” item/candidate, without extracting too much information from the customers/committee members. • Hiring committees: members can’t rank all of the candidates. • Too many candidates. • Committee members may only be familiar with just subsets of candidates in their own fields.

3. The Basic Setting • Candidates (alternatives/products): • Agents with preferences over the possible outcomes. A canonical task: select the “best” outcome. What is the information need for doing so. Social choice: methods for aggregating preferences. Efficient preference elicitation.

4. Basic Definitions • – candidates, – set of all orderings over . • Voter set with preferences . • Voting rule: . • Score-based voting rules: assign each candidate a score, pick the one with maximal score (most voting rules). • Positional scoring rules (PSR):score vector , ’th ranked candidate receives a score of . • Question:do we really need to know all of to pick Harmonic Borda Geometric

5. Top- Voting – The General Elicitation Scheme • Given a prescribed voting rule , ask each voter to report only his the length- prefix of her preferences. • Voter reports only . • The top- elicitation scheme: an algorithm that • Elicits • Computes a function that picks a winner `based on and the voting rule . • Question: how large does need to be so that ?

6. Previous Work • Theoretically: many common voting rules require a lot of information from the voters, in the worst case [Conitzer & Sandholm’02,05, Xia & Conitzer’08]. • Top- voting: Complexity of finding possible/necessary winners [Baumeister et al.’12]. • Empirically: in practice, top- voting performs well [Kalech et al.’11]. • Bridging the gap: assume a distributional model of preferences. • Lu & Boutilier’11: studied the performance of partial preference elicitation for regret-minimization a objective, under distributional assumptions. • Recently: A lower-bound on , for selecting the Borda winner in an impartial culture [O, Filmus, Boutilier’13].

7. A Distributional Approach to Top- Voting • Input: • Scoring rule: . Focus: positional scoring rules. • Top- algorithm :given top- votes, selects a winning candidate. • Each preference : drawn i.i.d. from distribution over . • Output:. Q: for which , , w.h.p.

8. Three Distributional Models (and high-level Results) • Three distributional models (increasing order of hardness): • Biased Distributions: distributions that “favor” one particular candidate over all else. (positionally-biased, pairwise-biased). • Overall results: As , is sufficient for many scoring rules. See paper for details. • Impartial Culture: is the uniform distribution over . • Main results: • PSRs: an almost tight threshold theorem. • Additional scoring rule: Copeland. • Adversarial: is set by an adversary, but is fully known to us. • Main results: • Harmonic PSR: A worst-case distribution requiring . • Geometric PSR: for any – sufficient; this is tight.

9. Top- for PSRs: under an Impartial Culture • Goal: Given a score vector , s.t., determine a LB and an UB in the limit. • Possible algorithms the upper bound: • OPT alg.: compute the winning prob. of candidates – not clear how to rigorously analyze such an algorithm. • Naïve: ∀, , if , , otherwise. • FairCutOff: like Naïve, but if , .

10. Empirical Results for the three algorithms • , sample populations. FairCutoff overlaps with Opt FairCutoff overlaps with Opt Borda Harmonic

11. A Threshold Theorem for PSRs • The top- scores according to FairCutoff: for , • We consider a measure depending on the amount of “noise” in each of the two parts of the score vector, in terms of : • , • Set – the -partition variability ratio. • Theorem: in the limit • LB: if , no top- can determine the right winner w.p. . • UB: if , then FairCutoff determines the right winner w.p. .

12. Applications Theorem: in the limit • LB: if , no top- can determine the right winner w.p. . • UB: if , then FairCutoff determines the right winner w.p. . • Borda: assuming that , . Gives a LB of (improvement over the previous bound of ). • Harmonic: UB – is sufficient,LB – . • Geometric: () UB – -- sufficient,LB – .

13. Proof Sketch – Order Statistics of Correlated RVs • Upper bound: FairCutoff guesses the right winner w.h.p. • For each candidate, partition voter scores based on top-/bottom-( partition: + + + + + + + + +

14. Proof Sketch (continued) • Each candidate’s score is the sum: . • Score according to top-. • First and second order statistics: . • First step: estimate the distribution of . • Complication: the scores are correlated – diagonalize covariance matrix via a linear transformation, without affecting the distribution of the difference. • Approximate via CLT + order statistics analysis of Gaussian RVs. • Difference grows as variance of grows (). • Second step: switch from to loses the “added noise” – has a bounded effect (use – above gap won’t be closed, w.h.p.

15. Top- elicitation under Copleland’s rule • The scoring rule: for every two distinct candidates , set( if beats [loses to] in a pairwise election. If there’s a tie, set . • A candidate’s score is: . • Theorem: for , no top- algorithm predicts the correct winner w.p. ().

16. The Adversarial Model • The preference distribution is set by an adversary – the full specifications are given to the decision maker. • How bad can the bounds be for that case? • Borda: already got a lower bound of . • Harmonic: we have a polylog UB for the impartial culture. • Theorem: For a fixed , there is a distribution , such that any top- algorithm requires , under the harmonic scoring rule. • Geometric: a UB under an impartial culture. • Theorem: if the decay factor is fixed, then issufficient under the geometric scoring rule, under any distribution, and this is tight.

17. Conclusions & Future Directions • A rigorous, probabilistic analysis of the top- elicitation scheme. • Under impartial culture: a principled method of analysis with applications: • Result for Copeland: extension to weighted majority graph based rules? • Can we extend our results to other rules? • More involved distributions: mixture models. • Top-k for proportional representation.

18. Contact:oren@cs.toronto.eduHomepage: www.cs.toronto.edu/~oren Thank you!