Fairness with Indivisible Goods: Solu6on Concepts and...
Transcript of Fairness with Indivisible Goods: Solu6on Concepts and...
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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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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Apropor6onalandenvy-freealloca6on
Inourexample:
ThediscreteseAng
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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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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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