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
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