under Metric Preferences Piotr Skowron TU Berlin based on joint - PowerPoint PPT Presentation

Social Choice under Metric Preferences Piotr Skowron TU Berlin based on joint work with Elliot Anshelevich (RPI), Onkar Bhardwaj (RPI), Edith Elkind (Univ. of Oxford), and John Postl (RPI)

Voting: the Basic Model • Input: – a set of alternatives (candidates) C = {c 1 , ..., c m } – a set of voters V = {1, ..., n} – for each voter, a total order (ranking) over C • Output: – a winner (possibly a tied set of winners) • Goal: maximize joint satisfaction of the voters

Where do Preferences Come From? • Sometimes it is natural for voters to order candidates – Mexican food > Indian food > Chinese food • Sometimes voters associate a cardinal utility with each candidate, and order them according to cardinal utility benefits – taxes if Tories win > benefits – taxes if Labour wins • This work: utilities that come from a metric

Facility Location

Voting with Metric Preferences z x M y i • (pseudo-)metric d on V  C • voter i prefers x to y if d(i, x) < d(i, y)

How Restrictive Are Metric Preferences? • Can any collection of preferences be generated by a metric? – yes, even by the Euclidean metric in R k for large enough k – but not by Euclidean metric in R or R 2 • 1d-Euclidean preferences can be recognized in polynomial time [Doignon & Falmagne’94, Knoblauch’08, Elkind&Faliszewski’14] • Recognizing 2d-Euclidean preferences is ∃ R -hard [Peters’17]

What is a Good Alternative? • We may want to select an alternative that minimizes the sum of distances to the voters • ... or the maximum distance • ... or the distance of the median voter • These tasks are easy if voters are able to report distances (or locations) • However, typically voters are unable to provide distance information, and find it easier to submit rankings

What Can We Do With Rankings? • We assume that voters report rankings of candidates: – i: x > y > z preference profile – j: y > x > z • Given rankings only, we cannot find a candidate that minimizes social cost y x j i 1 - e 1 + e

What Can We Do With Rankings? • We assume that voters report rankings of candidates: – i: x > y > z preference profile – j: y > x > z • Given rankings only, we cannot find a candidate that minimizes social cost y x j i 1 - e 1 + e • i: x > y; j: y > x • cost(x)  3, cost(y)  1

Distortion • We have seen that every deterministic voting rule may be off by a factor of 3 – and every randomized voting rule may be off by a factor of 2 • This holds even for very simple metric spaces ( R ) • But can we match this lower bound? • Definition: distortion of a voting rule f on a profile P wrt a metric d is dist(f, P, d): cost(w) max w  f(P) , where x is optimal for P wrt d cost(x)

Distortion Bounds? • P: a profile • D(P): the set of all pseudo-metrics d that can generate P • Question: is there a voting rule f such that dist(f, P, d) is bounded by a (small) constant for all profiles P and all d in D(P)? • I.e., can we bound the loss caused by having ordinal rather than cardinal information? • For the general utilitarian setting the answer is “no” [Procaccia&Rosenschein’06, Caragiannis et al.’15]

Optimal Voting Rule? • Given a profile P, we can compute the worst-case distortion of a given alternative, over all pseudo-metrics in D(P) – linear programming • Distortion-optimal voting rule: select the alternative with the best worst-case distortion – cumbersome to work with • This work: distortion of common voting rules

Single-Winner Rules: Plurality • Plurality: – each voter names her favorite candidate – candidates with the largest number of votes win m-1 voter/candidate pairs 1 w e x – cost(x) ≤ 2 + 2 e (m - 1) lower bound: 2m - 1 – cost(w)  m - 1 upper bound: 2m - 1

Single-Winner Rules: Borda • Borda: – each candidate gets m-i points from each voter who ranks her in position i – candidates with the largest number of points win 2m-3 voters: x > w > ... w 2 voters: w > ... > x x lower bound: 2m - 1 upper bound: 2m - 1 – score(x) = (2m-3)(m-1) – score(w) = (2m-3)(m-2) + 2(m-1) > score(x)

Scoring Rules • A scoring rule for an election with m candidates is given by a vector (s 1 , ..., s m ), s 1 ≥ ... ≥ s m – each candidate gets s i points from each voter who ranks him i-th – candidate with the maximum number of points wins • Plurality: (1, 0, ..., 0) • Borda: (m-1, m-2, ..., 2, 1, 0) • Harmonic rule: (1, 1/2, 1/3, ... 1/m) • k-approval: (1, ..., 1, 0, ..., 0) k – equivalent to allowing voters to vote for k candidates

Distortion of Scoring Rules • Proposition: the worst-case distortion of k-approval with k > 1 is unbounded y x • Theorem: the worst-case distortion of every scoring rule for m candidates is at least (log m) 1/2 – essentially 1d-construction; Plurality and Borda are special cases • Theorem: harmonic rule has sublinear worst-case distortion m/log m – this bound is tight

Condorcet Winners • Definition: a candidate c (weakly) defeats a candidate d if more than half (at least half) of the voters rank c above d • A candidate is a (weak) Condorcet winner if he (weakly) defeats all other candidates a b b c d a d a a a b c c d b c b d b c d a d c a is the Condorcet winner

Distortion of Condorcet Winners • Theorem: fix a metric d and a profile P for d. Let x be optimal for P wrt d. If P has a weak Condorcet winner w then cost(w) ≤ 3cost(x) – wlog d(x, w) = 1 x w – if w > i x then d(i, x) ≥ 1/2 – cost(x) ≥ n/4 – if x > i w then d(i, w) ≤ d(i, x) + 1 – cost(w) ≤ cost(x) + n/2 – cost(w)/cost(x) ≤ 1 + (n/2)/(n/4) = 3

Do Elections Always Have Condorcet Winners? a • 2 voters rank a above b • 2 voters rank b above c a c b • 2 voters rank c above a b a c c b c b a • No Condorcet winner! • Definition: G is a (weak) majority graph for a profile P over a candidate set C if its vertex set is C and there is an edge from a to b iff a (weakly) defeats b – chaos theorem: every tournament can be realized as a majority graph for some profile

Voting Rules: Copeland • Copeland rule: – each candidate gets 1 point for each candidate he defeats – the candidate with the largest number of points wins • The Copeland rule selects the Condorcet winner when it exists: – in an m-candidate election, the Condorcet winner gets m-1 point, all other candidates get at most m-2 points

Distortion of Copeland • Theorem: the distortion of the Copeland rule does not exceed 5, and this bound is tight • Proof idea: – a Copeland winner is a king in the weak majority graph: it has a path of length at most 2 to every other vertex – in particular, it has path of length at most 2 to the optimal alternative – if it defeats the optimal alternative, distortion is ≤ 3 – otherwise, the argument is similar

Single Transferable Vote • Idea: simulate multiple rounds of voting – check if there is a candidate with > n/2 1 st -place votes – if yes, declare him a winner – if not, select a candidate with min # of 1 st -place votes and delete him from all votes – repeat a > b > c a > c a > c > b a > c b > a > c a > c a wins c > b > a c > a c > a > b c > a

Distortion of STV • Theorem: the distortion of STV for m-candidate profiles is at most log m • Theorem: the distortion of STV for m-candidate profiles can be as high as (log m) 1/2 • E.g., STV is – worse than Copeland – but better than popular scoring rules, and – no worse than any scoring rule

Not in this talk... • A complex (but poly-time!) rule called Ranked Pairs has distortion at most 3 if the weak majority graph does not have long cycles – but not in general! [Goel et al.’16] • Randomized voting rules [Anshelevich , Bhardwaj, Postl’15] • Strategic issues [Feldman, Fiat, Golomb’16] • Restricted metric spaces – e.g., R – e.g., candidates are vertices of a simplex • Other measures of social cost

Open Problems • Close the gap between the lower bound of 3 and the upper bound of 5 • Understand the distortion-minimizing rule • Identify the scoring rule with the best worst-case distortion • Better understanding of the randomized scenario • Beyond voting: making decisions based on ordinal information only