FairnesswithIndivisibleGoods:Solu6onConceptsandAlgorithms

VangelisMarkakisAthensUniversityofEconomicsandBusiness(AUEB)

Dept.ofInforma<cs

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Cake-cuAngproblemsInput:

!  Asetofresources

!  Asetofagents,withpossiblydifferentpreferences

Goal:Dividetheresourcesamongtheagentsinafairmanner

Mathema6calformula6ons:Ini6atedby[Steinhaus,Banach,Knaster’48]

Empirically:sinceancient6mes

•  AncientEgypt:!  LanddivisionaroundNile(i.e.,ofthemostfer6leland)

•  AncientGreece:!  Sponsorshipsoftheatricalperformances

•  Undertakenbymostwealthyci6zens

•  Mechanismusedwasgivingincen6vessothatwealthierci6zenscouldnotavoidbecomingsponsors

•  Firstreferencesofthecut-and-chooseprotocol!  Theogony(Hesiod,8thcenturyB.C.):runbetweenPrometheusand

Zeus

!  Bible:runbetweenAbrahamandLot

Someearlyreferences

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•  hZp://www.spliddit.org!  JonathanGoldman,ArielProcaccia

!  Algorithmsforvariousclassesofproblems(rentdivision,divisionofgoods,etc)

•  hZp://www.nyu.edu/projects/adjustedwinner/!  StevenBrams,AlanTaylor

!  Implementa6onofthe“adjustedwinner”algorithmfor2players

•  hZps://www.math.hmc.edu/~su/fairdivision/calc/!  FrancisSu!  Implementa6onofalgorithmsforalloca6nggoodswithanynumberof

players

Availableimplementa6ons

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Preferences:•  Modeledbyavalua6onfunc6onforeachagent

•  vi(S)=valueofagentiforobtainingasubsetS

Typeofresources:1.  Con6nuousmodels

!  Infinitelydivisibleresources(usuallyjusttheinterval[0,1])!  Valua6onfunc6ons:definedonsubsetsof[0,1]

2.  Discretemodels!  Setofindivisiblegoods!  Valua6onfunc6ons:definedonsubsetsofthegoods

ModelingFairDivisionProblems

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Forthistalk:•  Resources=asetofindivisiblegoodsM={1,2,…,m}•  Setofagents:N={1,2,…,n}•  Analloca6onofMisapar66onS=(S1,S2,...,Sn),Si⊆M

! ∪iSi=MandSi∩Sj=∅

ThediscreteseAng

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Allvalua6onsweconsidersa6sfy:•  vi(∅)=0(normaliza6on)•  vi(S)≤vi(T),foranyS⊆T(monotonicity)Specialcasesofinterest:!  Addi6ve:vi(S ∪ T) = vi(S) + vi(T),foranydisjointsetsS,T

!  Assumedinthemajorityoftheliterature

!  Sufficestospecifyvij foranygoodj:vi(S) = Σj ∈ S vij, foranyS ⊆ M !  Addi6vewithiden6calrankingsonthevalueofthegoods!  Iden6calagents:Samevalua6onfunc6onforeveryone

!  Submodular:vi(S ∪ {j}) - vi(S) ≥ vi(T ∪ {j}) - vi(T),foranyS⊆T,andj∉T!  Subaddi6ve:vi(S ∪ T) ≤ vi(S) + vi(T), foranyS,T⊆M

Valua6onfunc6ons

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Valua6onfunc6ons

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Addi$ve

Submodular

Subaddi$ve

Examplewithaddi6vevalua6ons

ThediscreteseAng

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30 40 35 5 40

30 20 40 30 0

Charlie

Franklin

Marcie

Part1:Ahierarchyofsomesolu6onconceptsinfairdivision

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1.  Propor6onality

Analloca6on(S1,S2,...,Sn)ispropor6onal,ifforeveryagenti,vi(Si)≥1/n⋅vi(M)

Historically,thefirstconceptstudiedintheliterature[Steinhaus,Banach,Knaster’48]

Solu6onConcepts

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2.Envy-freenessAnalloca6on(S1,S2,...,Sn)isenvy-free,ifvi(Si)≥vi(Sj)foranypairofplayersiandj•  Suggestedasamathpuzzlein[Gamow,Stern’58]•  Moreformallydiscussedin[Foley’67,Varian’74]Astrongerconceptthanpropor6onality(aslongasvalua6onsaresubaddi6ve):Envy-freeness⇒n⋅vi(Si)≥vi(M)⇒Propor6onality

Solu6onConcepts

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Inourexample:

ThediscreteseAng

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30 40 35 5 40

30 20 40 30 0

Apropor6onalandenvy-freealloca6on

Inourexample:

ThediscreteseAng

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35 5 25 0 35

30 40 35 5 40

30 20 40 30 0

Apropor6onalbutnotenvy-freealloca6on

3.Compe66veEquilibriumfromEqualIncomes(CEEI)Supposeeachagentisgiventhesame(virtual)budgettobuygoods.ACEEIconsistsof•  Analloca6onS=(S1,S2,...,Sn)•  Apricingonthegoodsp=(p1,p2,...,pm)suchthatvi(Si)ismaximizedsubjecttothebudgetconstraintAnalloca6onS=(S1,S2,...,Sn)iscalledaCEEIalloca6onifitadmitsapricingp=(p1,...,pm),suchthat(S,p)isaCEEI

Solu6onConcepts

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•  Awellestablishedno6onineconomics[Foley’67,Varian’74]•  Combiningfairnessandefficiency•  Quotefrom[Arnsperger’94]:“Tomanyeconomists,CEEIis

thedescrip6onofperfectjus6ce”

Solu6onConcepts

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Claim:ACEEIalloca6onis• envy-free(duetoequalbudgets)• Pareto-efficientinthecon6nuousseAng• Pareto-efficientinthediscreteseAngwhenvalua6onsarestrict(no2bundleshavethesamevalue)

ContainmentRela6onsinthespaceofalloca6ons

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CEEI

Envy-freeness

Propor$onality

"  All3defini6onsare“toostrong”forindivisiblegoods

"  Noguaranteeofexistence

"  Moreappropriateforthecon6nuousseAng(existenceisalwaysguaranteed)

"  Needtoexplorerelaxedversionsoffairness

Someissues

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4.Envy-freenessupto1good(EF1)Analloca6on(S1,S2,...,Sn)sa6sfiesEF1,ifforanypairofagentsi,j,thereexistsagooda∈Sj,suchthatvi(Si)≥vi(Sj\{a})•  i.e.,foranyplayerwhomayenvyagentj,thereexistsanitem

toremovefromSjandeliminateenvy•  Definedby[Budish’11]

Solu6onConcepts

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5.Envy-freenessuptoanygood(EFX)Analloca6on(S1,S2,...,Sn)sa6sfiesEFX,ifforanyplayersiandj,andanygooda∈Sj,wehavevi(Si)≥vi(Sj\{a})•  Removinganyitemfromeachplayer’sbundleeliminatesenvy

fromotherplayers•  Definedby[Caragiannisetal.’16]Fact:Envy-freeness⇒EFX⇒EF1

Solu6onConcepts

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6.MaximinShareAlloca6ons(MMS)Athoughtexperiment:•  Supposewerunthecut-and-chooseprotocolfornagents.•  Sayagentiisgiventhechancetosuggestapar66onofthe

goodsintonbundles•  Therestoftheagentsthenchooseabundleandichooseslast•  Worstcasefori:heislexwithhisleastdesirablebundle

Solu6onConcepts

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•  GivennagentsandS⊆M,then-maximinshareofiw.r.t.Mis-maxisoverallpossiblepar66onsofM-minisoverallbundlesofapar66onS=(S1,S2,...,Sn)

Introducedby[Budish’11]

Solu6onConcepts

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Analloca6on(S1,S2,...,Sn)isamaximinshare(MMS)alloca6onifforeveryagenti,vi(Si)≥μi

Fact:Propor6onality⇒MMS

Solu6onConcepts

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μ1=30μ2=40μ3=30

Maximinshares

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HowdoMMSalloca6onscomparetoEF1andEFX?

!  ThereexistEFXalloca6onsthatarenotMMSalloca6ons

!  ThereexistMMSalloca6onsthatdonotsa6sfyEF1(hencenotEFXeither)

MMSvsEF1(andvsEFX)

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MMSvsEF1(andvsEFX)

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30 40 35 5 40

30 20 40 30 0

AMMSalloca6onthatdoesnotsa6sfyEF1•  CharlieenviesFranklinevenaxerremovingany

itemfromFranklin’sbundle

μ1=30μ2=40μ3=30

Rela6onsbetweenfairnesscriteria

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CEEI⇒EF

EFX⇒EF1

Propor6onality⇒MMS

Forsubaddi6vevalua6onfunc6ons!  Upperpartholdsforgeneralmonotonevalua6ons

Rela6onsbetweenfairnesscriteria

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CEEI

Envy-freeness

Propor$onality

MMS

EFX

EF1

Pictorially:

Part2:ExistenceandComputa6on

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Mostlybadnews:

"  Noguaranteeofexistenceforeitherpropor6onalityorenvy-freeness

"  NP-hardtodecideexistenceevenforn=2(equivalenttomakespanfor2iden6calprocessors)

"  NP-hardtocomputedecentapproxima6ons

"  E.g.Forapproxima6ngtheminimumenvyalloca6on[Lipton,Markakis,Mossel,Saberi’04]

"  S6llopentounderstandifthereexistsubclassesthatadmitgoodapproxima6ons

"  Ontheposi6veside:Existencewithhighprob.onrandominstances,whenn=O(m/logm)[Dickersonetal.’14]

Envy-freenessandPropor6onality

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Part2a:EF1andEFX

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ExistenceofEF1alloca6ons?

Theorem:Formonotonevalua6onfunc6ons,EF1alloca6onsalwaysexistandcanbecomputedinpolynomial6me

EF1

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Existenceestablishedthroughanalgorithm

Algorithm1-GreedyRound-Robin•  Fixanorderingoftheagents•  Whilethereexistunallocateditems

•  Letibethenextagentintheround-robinorder•  Askitopickhismostdesirableitemamongtheunallocatedones

EF1forAddi6veValua6ons

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Algorithm1worksforaddi6vevalua6onsProof:Throughoutthealgorithm,eachplayermayhaveanadvantageonlyby1itemw.r.t.otherplayers⇒EF1

•  Fornon-addi6vevalua6ons,moreinsigh|ultolookatagraph-theore6crepresenta6on

•  LetSbeanalloca6on(notnecessarilyofthewholesetM)•  Theenvy-graphofS:

•  Nodes=agents

•  Directededge(i,j)ifienviesjunderS

•  Howdoesthishelp?

EF1forGeneralValua6ons

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•  Anitera6vealgorithm6llwereachacompletealloca6on•  Supposewehavebuiltapar6alalloca6onthatisEF1

•  Ifthereexistsanodewithin-degree0:givetothisagentoneofthecurrentlyunallocatedgoods

•  Ifthisisnotthecase:

!  Thegraphhascycles!  Startremovingthembyexchangingbundles,asdictatedbyeach

cycle

!  Un6lwehaveanodewithin-degree0

EF1forGeneralValua6ons

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Algorithm2–TheCycleElimina<onAlgorithm•  Fixanorderingofthegoods,say,1,2,...,m•  Atitera6oni:

•  Findanodejwithin-degree0(bypossiblyelimina6ngcyclesfromtheenvy-graph)

•  Givegooditoagentj

EF1forGeneralValua6ons

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Proofofcorrectness:•  Removingcyclesterminatesfast

•  Numberofedgesdecreasesaxereachcycleisgone•  Ateverystep,wecreateenvyonlyforthelastitem•  Thealloca6onremainsEF1throughoutthealgorithm

ExistenceofEFXalloca6ons?-  forn=2

!  YES(forgeneralvalua6ons)-  forn≥3

!  Greatopenproblem!!  Guaranteedtoexistonlyforagentswithiden6cal

valua6ons

EFX

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[Rawls’71]Theleximinsolu6onisthealloca6onthat

!  MaximizestheminimumvalueaZainedbyanagent

!  Iftherearemul6plesuchalloca6ons,picktheonemaximizingthe2ndminimumvalue

!  Thenmaximizethe3rdminimumvalue

!  Andsoon...

Adetour:theleximinsolu6on

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•  Thisinducesatotalorderingoveralloca6ons•  Leximinisaglobalmaximumunderthisordering

[Plaut,Roughgarden’18]:aslightlydifferentversionAleximin++alloca6on

!  MaximizestheminimumvalueaZainedbyanagent

!  Maximizesthebundlesizeoftheagentwiththeminimumvalue

!  Thenmaximizesthe2ndminimumvalue

!  Followedbymaximizingthebundlesizeofthe2ndminimumvalue

!  Andsoon...

ExistenceresultsforEFXalloca6ons

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Theorem:Forgeneralbutiden6calagents,theleximin++solu6onisEFX

[Plaut,Roughgarden’18]:Separa6onbetweengeneralandaddi6vevalua6onsTheorem:1.  exponen6allowerboundonquerycomplexity

•  Evenfor2agentswithiden6calsubmodularvalua6ons2.  Polynomial6mealgorithmfor2agentsandarbitraryaddi6ve

valua6ons3.  Polynomial6mealgorithmforanyn,andaddi6vevalua6ons

withiden6calrankings•  Allagentshavethesameorderingonthevalueofthegoods

Algorithmicresults

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Algorithmforaddi6vevalua6onswithiden6calrankings:Runthecycleelimina6onalgorithm,byorderingthegoodsindecreasingorderofvalue

!  Ateverystepofthealgorithmweallocatethenextitemtoanagentno-oneenvies

!  Envywecreateisonlyfortheitematthecurrentitera6on

!  Butthishaslowervaluethanallthepreviousgoods!  Hencethealloca6onremainsEFXthroughoutthealgorithm

Algorithmicresults

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Algorithmfor2agentsandarbitraryaddi6vevalua6onsVaria6onofcutandchoose!  Agent1runsthepreviousalgorithmwith2copiesofherself!  Agent2picksherfavoriteoutofthe2bundlescreated!  Agent1picksthelexoverbundle

Algorithmicresults

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Part2b:MMSalloca6ons

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Existence?-  forn=2

!  YES(viaadiscreteversionofcut-and-choose)

-  forn≥3!  NO[Procaccia,Wang’14]

!  Knowncounterexamplesbuildonsophis6catedconstruc6ons

-  Howoxendotheyexistforn≥3?!  Actuallyextremelyoxen

!  Extensivesimula6ons[Bouveret,Lemaitre’14]withrandomlygenerateddatadidnotrevealnega6veexamples

MMSalloca6ons

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Wewillagainstartwithaddi6vevalua6ons

Computa6on

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ApproximateMMSalloca6onsQ:Whatisthebestαforwhichwecancomputeanalloca6on(S1,S2,...,Sn)sa6sfyingvi(Si)≥αμiforeveryi?

•  NP-hardtoevencomputethequan6tyμiforagenti•  ExistenceproofofMMSalloca6onsyieldsanexponen6al

algorithm1.  Letplayer1computeapar66onthatguaranteesμ1tohim

!  i.e.,apar66onthatisasbalancedaspossible2.  Player2picksthebestoutofthe2bundles

•  ConvertStep1topoly-6mebylosingε,e.g.usingthePTASof[Woeginger’97]

Corollary:Forn=2,wecancomputeinpoly-6mea(1-ε)-MMSalloca6on

Approxima6onAlgorithmsforAddi6veValua6ons

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Forn=2

•  Startwithanaddi6veapproxima6on•  Recallthegreedyround-robinalgorithm(Algorithm1)Theorem:GreedyRound-Robinproducesanalloca6on(S1,S2,...,Sn)suchthat

vi(Si)≥μi–vmax,wherevmax=maxvij

Approxima6onAlgorithmsforAddi6veValua6ons

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Forn≥3

WhendoesGreedyRound-Robinperformbadly?•  Inthepresenceofgoodswithveryhighvalue

•  BUT:eachsuchgoodcansa6sfysomeagent

•  Suggestedalgorithm:GetridofthemostvaluablegoodsbeforerunningGreedyRound-Robin

Algorithm3:•  LetS:=M,andαi:=vi(S)/n

•  While∃i,j,suchthatvij≥αi/2,

•  allocatejtoi

•  n:=n-1,S:=S\{j},recomputetheαi’s•  RunGreedyRound-Robinonremaininginstance

Approxima6onalgorithmsforaddi6vevalua6ons

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AllweneedistoensureamonotonicitypropertyLemma:Ifweassignagoodjtosomeagent,thenforanyotheragenti≠j:

μi(n-1,M\{j})≥μi(n,M)

Theorem:Algorithm2producesanalloca6on(S1,S2,...,Sn)suchthatforeveryagenti:

vi(Si)≥1/2μi(n,M)=1/2μi

A½-approxima6onforAddi6veValua6ons

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•  Algorithm2is6ght•  Whatifwechangethedefini6onof“valuable”byconsidering

vij≥2αi/3insteadofαi/2?•  NotclearhowtoadjustGreedyRound-Robinforphase2•  Bea6ng1/2needsdifferentapproaches

Beyond1/2...

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2/3-approxima6onguarantees:•  [Procaccia,Wang’14]

!  2/3-ra6o,exponen6aldependenceonn

•  [Amana6dis,Markakis,Nikzad,Saberi’15]!  (2/3-ε)-ra6oforanyε>0,poly-6meforanynandm

•  [Barman,Murty’17]!  2/3-ra6o,poly-6meforanynandm

Beyond1/2...

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Recursivealgorithmsof[Procaccia,Wang’14],[Amana6dis,Markakis,Nikzad,Saberi’15]Basedon:•  Exploi6ngcertainmonotonicityproper6esofμi(⋅,⋅)

!  Tobeabletomovetoreducedinstances

•  Resultsfromjobscheduling

!  TobeabletocomputeapproximateMMSpar66onsfromtheperspec6veofeachagent

•  Matchingarguments(perfectmatchings+findingcounterexamplestoHall’stheoremwhennoperfectmatchingsexist)

!  Tobeabletodecidewhichagentstosa6sfywithineachitera6on

2/3-approxima6onalgorithms

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Recursivealgorithmsof[Procaccia,Wang’14],[Amana6dis,Markakis,Nikzad,Saberi’15]Highleveldescrip6on:•  Eachitera6ontakescareof≥1person,un6lno-onelex•  Duringeachitera6on,

Let{1,2,...,k}=s6llac6veagents

1.  Askoneoftheagents,sayagent1,toproduceaMMSpar66onwithkbundlesaccordingtohisvalua6onfunc6on

2.  Findasubsetofagentssuchthat:

a)  theycanbesa6sfiedbysomeofthesebundles

b)  theremaininggoodshave“enough”valuefortheremainingagents

2/3-approxima6onalgorithms

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Thealgorithmof[Barman,Murty’17]Lemma1:Itsufficestoestablishtheapproxima6onra6oforaddi6vevalua6onswithiden6calrankings

Lemma2:Foraddi6vevalua6onswithiden6calrankings,thecycleelimina6onalgorithm(axerorderingthegoodsindecreasingorderofvalue)achievesa2/3-approxima6on

2/3-approxima6onalgorithms

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•  Anintriguingcase...

•  Forn=2,MMSalloca6onsalwaysexist

•  Theproblemsstartatn=3!

•  S6llunclearifthereexistsaPTAS

Thecaseofn=3agents

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Progressachievedsofar:Algorithms Approx.ra<o

[Procaccia,Wang’14] 3/4

[Amana6dis,Markakis,Nikzad,Saberi’15] 7/8

[Gourves,Monnot’17] 8/9

•  Noneofthealgorithmsgothroughwithnon-addi6vevalua6ons

•  Noposi6veresultsknownforarbitraryvalua6ons

Non-addi6vevalua6ons

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Theorem[Barman,Murty’17]:Foragentswithsubmodularvalua6ons,thereexistsapolynomial6me1/10-approxima6onalgorithm

[Ghodsi,Hajiaghayi,Seddighin,Seddighin,Yami’17]:Posi6veresultsforvariousclassesofvalua6onfunc6ons:•  Addi6ve:Polynomial6me¾-approxima6on•  Submodular:Polynomial6me1/3-approxima6on•  Subaddi6ve:ExistenceofO(logm)-approxima6on

Andsomemorerecentprogress

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Part3:Relatedopenproblemsandotherresearchdirec6ons

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Canwethinkofalterna6verelaxa6onstoenvy-freenessand/orpropor6onality?[Caragiannisetal.’16]:•  PairwiseMMSalloca6ons

!  Consideranalloca6onS=(S1,S2,...,Sn),andapairofplayers,i,j!  LetB:=allpar66onsofSi∪Sjintotwosets(B1,B2)!  Fairnessrequirementforeverypairi,j:

•  AstrongercriterionthanEFX•  RelatedbutincomparabletoMMSalloca6ons•  Existenceofφ-approxima6on(goldenra6o)

!  OpenproblemwhetherpairwiseMMSalloca6onsalwaysexist

Otherfairnessno6ons

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Canwethinkofalterna6verelaxa6onstoenvy-freenessand/orpropor6onality?Fairnessinthepresenceofasocialgraph[Chevaleyre,Endriss,Maudet’17,Abebe,Kleinberg,Parkes’17,Bei,Qiao,Zhang’17]

•  Evaluatefairnesswithregardtoyourneighbors!  Mostdefini6onseasytoadapt

!  E.g.,graphenvy-freeness:sufficestonotenvyyourneighbors

[Caragiannisetal.’18]:

•  Moreextensions,withoutcompletelyignoringthegoodsallocatedtonon-neighbors

Otherfairnessno6ons

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•  Sofarweassumedagentsarenotstrategic•  Canwedesigntruthfulmechanisms?•  [Amana6dis,Birmpas,Christodoulou,Markakis’17]:

!  Mechanismdesignwithoutmoney

!  Tightresultsfor2playersthroughacharacteriza6onoftruthfulmechanisms

!  Besttruthfulapproxima6onforMMS:O(1/m)!  TruthfulmechanismsforEF1:onlyifm≤4

•  Characteriza6onresultsfor≥3players?

Mechanismdesignaspects

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•  Cake:M=[0,1]•  Setofagents:N={1,2,…,n}•  Valua6onfunc6ons:

!  Givenbyanon-atomicprobabilitymeasurevion[0,1],foreachi

•  Accesstothevalua6onfunc6ons:!  Valuequeries:askanagentforhervalueofagivenpiece!  Cutqueries:askanagenttoproduceapieceofagivenvalue

Thecon6nuousseAng

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"  Envy-free(andhencepropor6onal)alloca6onsalwaysexistComputa6on?"  n=2:cut-and-choose(2queries)"  n=3:[Selfridge,Conwaycirca60s](lessthan15queries)"  n=4:[Aziz,Mckenzie’16a](closeto600queries)"  Generaln:

"  [Brams,Taylor’95]:Finiteprocedurebutwithnoupperboundonnumberofqueries

"  [Aziz,Mackenzie’16b]:Firstboundedalgorithmbutwithexcep6onallyhighcomplexity

Envy-freealloca6onsinthecon6nuousseAng

63#queries≤

Lowerbounds"  Con6guouspieces:therecanbenofiniteprotocolthat

producesenvy-freealloca6on"  Non-con6guouspieces:Ω(n2)[Procaccia’09]

"  Separa6ngenvy-freenessfrompropor6onality"  Canwedoshortenthegapbetweentheupperandlower

bound?

Envy-freealloca6onsinthecon6nuousseAng

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Summarizing...Arichareawithseveralchallengingwaystogo•  Conceptual

–  Defineorinves6gatefurthernewno6ons

•  Αlgorithmic–  Bestapproxima6onforMMSalloca6ons?–  EFXforarbitraryaddi6vevalua6ons?–  Algorithmsforthecon6nuousseAng?

•  Game-theore6c–  Mechanismdesignaspects?

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