Preference Aggregation on - PowerPoint Presentation

Slide1

Preference Aggregation on

Structured

P

reference Domains

Edith Elkind

University of

OxfordSlide2

Voters and Their Preferencesn voters,

m

candidates

Each voter has a complete ranking of the candidates (his preference order)We may want to select:a single winnera fixed-size subset of winners (a committee)a ranking of the candidates

ABCD

BCDA

CABD

DABC

BCDA

CDAB

ABCD

BCAD

CABDSlide3

Applications

elections

hiring

budget

allocation

collaborative filtering

admissions

abcd

abcd

abcd

abcd

abcd

abcd

abcd

mechanism

group decision

paper selectionSlide4

DifficultiesProblem:

with no assumption on preference structure

counterintuitive behavior

may occurmajority of voters prefer A to B, B to C, C to Acomputational problems are often hard e.g., selecting the most representative committee

ABCD

BCDA

CABD

DABC

BCDA

CDAB

ABCD

BCAD

CABDSlide5

Example: Movie Selection

?Slide6

A

B C D

E F

Single-Peaked Preferences

Definition

: a preference profile is

single-peaked (SP)

wrt

an ordering

<

of candidates (axis) if for each voter

v

there exists a candidate

C

such that:v ranks C first if C < D < E, v prefers D to Eif A < B < C, v prefers B to AExample: voter 1: C > B > D > E > F > A voter 2: A > B > C > D > E > Fvoter 3: E > F > D > C > B > A Slide7

SP Preferences: Condorcet Winners Claim: in

single-peaked

elections,

the majority relation is transitiveWeaker claim: there exists a candidate preferred to every other candidate by a majority of voters (the Condorcet winner)suppose we have n = 2k+1 votersorder them according to their top choicepick the top choice of voter vk+1Slide8

Single-Crossing Preferences Definition

: a profile is

single-crossing (SC)

wrt an ordering of voters (v1, …, vn) if for each pair of candidates A, B there exists an i  {0, …, n}

such that

voters v1, …, v

i prefer A to B

, and voters vi+1

, …, vn prefer

B to A

AB

CD

BACD

B

C

A

DCBADCBDACDBADCBASlide9

SC Preferences: Majority is TransitiveClaim: in

single-crossing

elections,

the majority relation is transitivesuppose we have n=2k+1 votersconsider the ranking of voter vk+1if vk+1 prefers A to B, so do k

other voters

AB

CD

BACD

B

CAD

CB

AD

CBDA

C

D

B

ADCBASlide10

SP and SC Preferences: AlgorithmsMany

NP-hard

problems become easy if we assume that preferences are

SP or SCComputing Dodgson, Young, and Kemeny winnerscoincide with Condorcet winners when they existVarious forms of manipulation [Faliszewski, Hemaspaandra, Hemaspaandra

, Rothe’11, …]Computing the most representative

committee (Chamberlin-Courant’s rule)[Betzler,

Slinko, Uhlmann’13, Skowron

, Yu, Faliszewski, E.’13]Computing Plurality election equilibria under random tie-breaking

[E., Markakis, Obraztsova

, Skowron’?]dynamic programming

algorithms Slide11

Recognizing SP Preferences It is

easy

to check whether an election is

single-peaked wrt a given axisbut what if the axis is not known? Theorem: SP elections can be recognized in poly-time [Bartholdi, Trick’86, Doignon, Falmagne’94, Escoffier

, L

ang, Ozturk’08]Observation: if v ranks

C last, then C is either the leftmost candidate or the

rightmost candidateCorollary: in a SP

election at most 2 candidates are ever ranked last Slide12

Recognizing SC PreferencesTheorem:

SC

elections can be recognized in

poly-time [E., Faliszewski, Slinko’12, Bredereck, Chen, Woeginger’12]Theorem’: for each vote u, can decide in poly-time if there is a SC ordering where u appears firstD

swap(

x, y): |{(A, B):

x prefers A to B

, y prefers B to

A}|Lemma: if u < v < w

, then Dswap(u,

v) < Dswap(

u, w)

…A…

B

……

……

B…A……B……A…u:v:w:Corollary: SC order is unique (up to reversals and duplicates)Slide13

SP

SCSlide14

Single-Peaked Profile That Is Not Single-Crossingv

1

and

v2 have to be adjacent (because of B, C)v3 and v4 have to be adjacent (because of B, C)

v1

and v3 have to be adjacent (because of

A, D)v

2 and v4

have to be adjacent (because of A, D) a contradiction

BC

AD

BCDA

CBAD

CB

DA

D

ACBSlide15

Single-Crossing Profile That Is Not Single-PeakedEach candidate is ranked last

exactly once

1

2...……n-2

n-1

n

nn-1n-2

……

…2

1

nn-1

n-2…

……

1

2

n

12………n-2n-1nn-112………n-2…Slide16

SP

SCSlide17

1D-Euclidean PreferencesBoth

voters a

nd

candidates are points in R v prefers A to B if |v - A| < |v - B|

Observation: 1D-Euclidean preferences are

single-peaked (wrt ordering of candidates on the line)

single-crossing (wrt ordering of voters

on the line)

D

A

C

B

E

v

1

v

2v4v3BACDECBDAEDECBAEDCBASlide18

1-Euc = SP ∩ SC?Proposition [EFS’14, Lackner’14]: There exists

a preference profile that is

SP

and SC, but not 1-Euclidean v1: 2 3 4 5

1 6

v2:

4 5 3 2 1 6 v3:

4 5 6 3 2 1

SC wrt v

1 < v2 < v

3, SP wrt

1 < 2 < 3 < 4 < 5 < 6Not 1-Euclidean: (x(1)+x(

5))/2 < x(v1) < (x(

2

)+x(

3

))/2 (x(3)+x(4))/2 < x(v2) < (x(1)+x(6))/2 (x(2)+x(6))/2 < x(v3) < (x(4)+x(5))/2 413256Slide19

SP

SC

1-EucSlide20

Recognizing 1-Euclidean PreferencesQuestion: can we recognize 1-Euclidean preferences in polynomial time?

Observation

: if the order of candidates is known, it suffices to solve an LP:

variables x(c1), …, x(cm), x(v1), …, x(vn)for each voter v and each pair of candidates

a,

b with a < b,

if a >v

b, add inequality x(v) < (x(a

)+x(b))/2, andif b

>v a,

add inequality x(v) > (x(a)+x(

b))/2Slide21

Ordering Candidates – 1st Attempt

[Knoblauch’10]

: there exists a poly-time algorithm for recognizing

1-Euclidean preferencesIdea: check that the input election is SPif yes, use a SP ordering of the candidatesDifficulty:there can be many SP orderingssome work, others do notAn initial SP ordering needs to be

tweaked…Slide22

Ordering Candidates – 2nd Attempt

[EF’14]

: there exists a poly-time algorithm for recognizing

1-Euclidean preferencesIdea: use the (unique) SC order of voters It works, but…Bad news: this was discovered by Doignon and Falmagne in 1994

v

1

v

n

a

bSlide23

Eliminating LP and … Observation

: when showing that an

SP

SC profile is not 1-Euclidean, we had a very simple infeasibility certificateCan we identify a simple feasibility criterion that does not involve solving the LP?Slide24

Dichotomous PreferencesSo far, we assumed that votes = ordersWhat if voters have binary preferences?

voter

i

approves candidates in Ai, disapproves candidates in V\AiWhat are the analogues of SP/SC preferences in this setting? Can we recognize the preferences in these restricted domains?Can we exploit then to get efficient algorithms?[E. Lackner, IJCAI’15]Slide25

Restricted Binary Domains: Examples Candidate Interval (CI):

candidates can be ordered so that each voter’s approved candidates form an interval

a

b

d

e

f

g

c

u

v

wSlide26

Restricted Binary Domains: ExamplesVoter Interval (VI):voters can be ordered so that for each candidate the set of voters who approve her form an interval

u

v

x

y

z

t

w

a

b

cSlide27

Euclidean PreferencesDichotomous Euclidean (DE): voters and candidates can be placed on the line so that

for each voter v there is a radius r(v)

s.t

. v’s approval set is {c: d(c, v) ≤ r(v)}Dichotomous Uniformly Euclidean (DE): voters and candidates can be placed on the line so that there is a radius r s.t. for each voter v his approval set is {c: d(c, v) ≤ r}Slide28

Refinement-Based ApproachesRefinement: a total order a > b > c > d is a refinement of approval vote {a, b}

Possibly single-peaked (PSP): there is a single-peaked profile of total orders that is a refinement of the given profile

Possibly

single-crossing (PSC)Possibly Euclidean (PE)Slide29

Relationships and ComplexityCI = DE = PSP = PEDUE implies CI and VIVI and CI are incomparable

VI and CI are easy to detect (consecutive 1s)

DUE can be recognized in polynomial time

[Nederlof, Woeginger, May’15]Slide30

ApplicationsPAV: a voting rule to select committees under dichotomous preferencesComputing the output of PAV is NP-hard

[Aziz et al.’15]

even if each voter approves at most 2 candidates and each candidate is approved by at most 3 voters

Our contribution: easiness results for PAV under VI and CI preferences FPT wrt max size of approval setXP wrt max number of approvalsSlide31

Open Problems

Higher dimensions: can we recognize preferences that are

d-Euclidean

for d>1 (voters and candidates are points in Rd)?is this problem even in NP? even for d=2Trees: can we recognize preferences that are

1-Euclidean on

trees (or other median graphs)?Can we decide if a profile can be made

1-Euclidean by deleting k voters or k candidates

?voter deletion: easy for SC, NP-hard for

SPcandidate deletion: easy for SP, NP-hard for

SC