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WorkingPaper () Random assignment with multi-unit demands HarisAziz Received:date/Accepted:date 5 1 0 Abstract We consider the multi-unitrandom assignment problem in which agents 2 expresspreferencesoverobjectsandobjectsareallocatedtoagentsrandomlybased n onthepreferences.Themostwell-establishedpreferencerelationtocomparerandom u allocationsofobjectsisstochasticdominance(SD)whichalsoleadstocorresponding J notionsofenvy-freeness,efficiency,andweakstrategyproofness.Weshowthatthere 9 1 existsnorulethatisanonymous,neutral,efficientandweakstrategyproof.Forsingle- unitrandomassignment,weshowthatthereexistsnorulethatisanonymous,neutral, ] efficient and weak group-strategyproof. We then study a generalization of the PS T (probabilisticserial)rulecalledmulti-unit-eatingPS andprovethatmulti-unit-eating G PS satisfiesenvy-freeness,weakstrategyproofness,andunanimity. . s c Keywords Fair division · probabilistic serial rule · strategyproofness· Pareto [ optimality 3 v JELClassification:C70·D61·D71 0 0 7 1 Introduction 7 . 1 Intheassignmentproblem,agentsexpresslinearpreferencesoverobjectsandanob- 0 ject is assigned to each agent keeping in view the agents’ preferences. The prob- 4 1 lem models one of the most fundamental setting in computer science and eco- : nomicswithnumerousapplications(Ga¨rdenfors,1973;Wilson,1977;Young,1995; v Svensson, 1994, 1999; Bouveretetal., 2010; Abrahametal., 2005). Depending on i X the application setting, the objects could be car-park spaces, dormitory rooms, re- r placementkidneys,schoolseats, etc. Theassignmentproblemis also referredto as a HarisAziz NICTAandUNSW2033Sydney,Australia Tel.:+61-2-83060490 Fax:+61-2-83060405 E-mail:[email protected] 2 HarisAziz house allocation (Abrahametal., 2005; Abdulkadirog˘luandSo¨nmez, 1999). If the outcome of the assignment problem is deterministic then it can be inherently unfair. Take the exampleof two agentshaving identicalpreferencesovertwo objects. Thenanyreasonablenotionoffairnessdemandsthatbothagentshaveequalrightto each of the two objects. Since randomization is one of the oldest tools to achieve fairness, we consider the random assignment problem (HyllandandZeckhauser, 1979; Young, 1995; BogomolnaiaandMoulin, 2001; KattaandSethuraman, 2006; GuoandConitzer, 2010; Bhalgatetal., 2011; Budishetal., 2013) in which objects are allocated randomly to agents according to their preferences. The outcome is a randomassignmentwhichspecifiestheprobabilityofeachobjectbeingallocatedto eachoftheagents.Incontrasttosomeoftheearlierworkonrandomassignment,we focusontherandomassignmentprobleminwhichtherecanbemoreobjectsthanthe numberofagents(Kojima,2009). Whenagentsexpressordinalpreferencesoverobjectsbuttheoutcomesarefrac- tional or randomized allocations, then there is a need to use lottery extensions to extend preferencesover objects to preferencesover random allocations. In random settings, the most established preference relation between random allocations is stochastic dominance (SD). SD requires that one random allocation is preferred to another one if and only if the former first-order stochastically dominates the latter. This relation is especially important because one random allocation stochastically dominates another one if and only if the former yields at least as much expected utility as the latter for any von-Neumann-Morgenstern (vNM) utility representa- tionconsistentwiththeordinalpreferences(Azizetal.,2013c).TheSDrelationcan be used to define correspondingnotionsof envy-free,efficiency,and strategyproofness (BogomolnaiaandMoulin, 2001; KattaandSethuraman, 2006). In this paper, wecheckwhichlevelsoffairness,efficiency,andstrategyproofnesscanbesatisfied simultaneously. Fortherandomassignmentproblemwithoutmulti-unitdemands,themostcom- mon and well-known way to assign objects is random priority (RP) in which a permutation of agents is chosen uniformly at random and agents successively take their most preferred available object (Abdulkadirog˘luandSo¨nmez, 1998; BogomolnaiaandMoulin, 2001; Cre`sandMoulin, 2001). Although RP is strate- gyproofandresultsinaParetooptimalassignment,BogomolnaiaandMoulin(2001) inaremarkablepapershowedthatRPdoesnotsatisfythestrongerefficiencynotion ofstochasticdominance(SD)efficiencyandalsoafairnessconceptcalledSD-envy- freeness.1Furthermore,theypresentedanelegantalgorithmcalledPS (probabilistic serial) that is not only SD-efficient and SD-envy-free but also satisfies weak SD- strategyproofness.InPS,agents‘eat’themostfavouredavailableobjectatthesame rateuntilalltheobjectsareconsumed.Thefractionofobjectconsumedbyanagent istheprobabilityoftheagentgettingthatobject.2 Sinceitsinception(BogomolnaiaandMoulin,2001), PS hasreceivedconsider- able attention and has been extended in a number of ways (KattaandSethuraman, 1 Another drawback of RP is that the resultant fractional allocation is #P-complete to com- pute(Azizetal.,2013a). 2 BytheBirkhoff-von Neumanntheorem, anyfractional assignment canberepresented byaconvex combinationoverdiscreteassignments. Randomassignmentwithmulti-unitdemands 3 2006; AthanassoglouandSethuraman, 2011; Yilmaz, 2009). In particular, it can be naturally extended to the more general case with multi-unit demands in which there are nc objects and c ≥ 1 objects are allocated to each of the agents (BogomolnaiaandMoulin, 2001; Heo, 2011; Kojima, 2009). The extension doesnotrequireanymodificationtothespecificationof PS:agentscontinueeating their most preferred available object until all the objects have been consumed. Al- though this one-at-a-time extension (which we will refer to as OPS) still satisfies SD-efficiencyandSD-envy-freeness,itisnotweakSD-strategyproof(Kojima,2009). IncidentallythereisanotherextensionofPS calledthemulti-unit-eatingprobabilis- tic serial that was briefly described by CheandKojima (2010) but has received no attentionintheliterature.Inmulti-unit-eatingPS,eachagenttriestoeathiscmost preferredobjectsthatarestillavailableatauniformspeeduntilallobjectshavebeen consumed.Weshowthatmulti-unit-eatingPS satisfiesdesirableproperties:itisweak SD-strategyproof,SD-envy-free,andunanimous. We point out that the problem of discrete assignment with multi-unit de- mandshasattractedconsiderableattention(BouveretandLang,2011;Budish,2011; EhlersandKlaus, 2003; Hatfield, 2009; Kalinowskietal., 2013; Bouveretetal., 2010). In this paper, we focus on random assignments with multi-unit demands. Multi-unit demand is a natural requirement in settings such as course allocation (Budish, 2011). Moreover,we will require that each agents gets equal number of objects (Hatfield, 2009). This is a natural requirement in settings such as paper assignmenttoreferees. ApartfromRPandPS,twoothernaturalassignmentrulesareuniformandprior- ity.Intheuniformrule,eachagentgets1/nofeachobject(Chambers,2004;Kojima, 2009).Intheprioritymechanism,thereisapermutationofagents,andeachagentin thepermutationisassignedthecmostpreferredavailableobjects.Theprioritymech- anism is also referredto as serial dictator in the literature (Svensson, 1994, 1999). Whereasuniformdoesnottakeintoaccountthepreferencesofagentsandishighly inefficient, priority is highly unfair to the agents at the end of the permutation. In more recent work, Nguyenetal. (2015) proposed two mechanisms for the random assignment problem that also handle limited complementarities. Hashimoto (2013) presentedageneralizationofRPformoregeneralsettings. Contributions Wefirstprovethatformulti-unitdemands,thereexistsnoanonymous, neutral,weakSD-strategyproofandSD-efficientrandomassignmentrule.Thestate- mentissomewhatsurprisingconsideringthatallthefouraxiomsusedinthestatement areminimalrequirements.Incidentally,wehavenotusedSDenvy-freenessthatisof- tenusedtoobtaincharacterizationsorimpossibilitystatementsintheliterature(Heo, 2011; BogomolnaiaandMoulin, 2001; EhlersandKlaus, 2003; Kojima, 2009) and isaverydemandingrequirement.Theresultisthenextendedtorandomassignment withoutmulti-unitdemandsifrequiringweakSDgroup-strategyproofnessinsteadof weakSDstrategyproofness.Oursecondresultcarriesovertothesettingrandomized voting in which agents express weak orders over alternatives and the outcome is a lotteryoverthealternatives. We then conduct an axiomatic analysis of the multi-unit-eating PS. It is first highlighted that the definition of multi-unit-eating PS in the literature is not en- 4 HarisAziz tirely correct. A proper definition of multi-unit-eating PS is formulated. We show thatformulti-unitdemands,incontrasttoOPS,multi-unit-eatingPS satisfiesweak SD-strategyproofness.Weprovethatmulti-unit-eatingPS satisfiesSDenvy-freeness whichisoneofthestrongestnotionsoffairness.Ontheotherhand,multi-unit-eating PS does not fare well in terms of efficiency. We prove that multi-unit-eating PS does not even satisfy ex post efficiency although it does satisfy unanimity. There- fore when we generalize PS for multi-unitdemands, OPS is the right extension if thefocusisonefficiency.Ontheotherhandmulti-unit-eatingPS istherightexten- sion, if the aimis to maintainweak SD-strategyproofness.The argumentsforweak SD-strategyproofnessandSDenvy-freenessof MPS multi-unit-eatingPS alsosim- plifytheproofsforPS forsingle-unitdemandsin(BogomolnaiaandMoulin,2001). Thestudyhelpsclarifytherelativemeritsofdifferentassignmentrulesformulti-unit demands. The relative merits of prominentrandom assignment rules are then sum- marizedinTable1inthefinalsection. 2 Preliminaries Random assignment problem The model we consider is the random assignment problem which is a triple (N,O,%) where N is the set of n agents {1,...,n}, O = {o ,...,o }isthesetofobjects,and%= (% ,...,% )specifiesstrict, complete,and 1 m 1 n transitivepreferences% ofagentioverO.Wewillassumethatmisamultipleofn i i.e.,m = ncwherecisaninteger.WewilldenotebyR(O)asthesetofallcomplete andtransitiverelationsoverthesetofobjectsO. A random assignment p is a (n × m) matrix [p(i)(o )] such that for j 1≤i≤n,1≤j≤m alli ∈ N, ando ∈ O, p(i)(o ) ∈ [0,1]; p(i)(o ) = 1forall j ∈ {1,...,m};and j j i∈N j oj∈Op(i)(oj)=cforalli∈ N.ThevaluePp(i)(oj)representstheprobabilityofobject Poj beingallocatedtoagenti.Eachrow p(i) = (p(i)(o1),...,p(i)(om))representsthe allocation of agent i. The set of columns correspond to probability vectors of the objectso ,...,o .Afeasiblerandomassignmentisdiscreteif p(i)(o)∈{0,1}forall 1 m i ∈ N and o ∈ O. A random assignmentrule specifies for each preferencesprofile a randomassignment.Twominimalfairnessconditionsforrulesareanonymityand neutrality.Informally,theyrequirethatthe rule shouldnotdependonthe namesof theagentsorobjectsrespectively. We define the SD (stochastic dominance) relation which is an incomplete relation that extends the preferences of the agents over objects to preferences over random allocations. Given two random assignments p and q, p(i) %SD q(i) i.e., i a player i SD prefers allocation p(i) to allocation q(i) if p(i)(o ) ≥ oj∈{ok:ok%io} j oj∈{ok:ok%io}q(i)(oj)forallo ∈ O. Since SD is incomplete, iPt can be that two allo- cPations p(i)andq(i)areincomparable:p(i)(cid:31)SD q(i)andq(i)(cid:31)SD p(i). i i Next,wedefinetheDL(downwardlexicographical)relationwhichisacomplete relation.Letp(i)andq(i)betworandomallocations.Leto∈Obethemostpreferred objectsuchthat p(i)(o),q(i)(o).Then, p(i)≻DL q(i) ⇐⇒ p(i)(o)>q(i)(o). i Randomassignmentwithmulti-unitdemands 5 Example1 ConsidertherandomassignmentproblemfortwoagentsN = {1,2}and fourobjectsO={o ,o ,o ,o }withthefollowingpreferences: 1 2 3 4 1:o ,o ,o ,o 1 2 3 4 2:o ,o ,o ,o 2 1 3 4 Letusassume thatagent1getso with probabilityone,andobjectso ando with 1 3 4 probability half. Then the random assignment can be represented by the following matrix. 101/21/2 p= . 011/21/2! Note that agent 1’s preference is o ≻ o ≻ o ≻ o . Based on the preferences 1 1 2 1 3 1 4 overobjects,onecanconsiderpreferencesoverallocations: p(1) ≻SD p(2)andalso 1 p(1)≻DL p(2). 1 Envy-freeness AnassignmentpsatisfiesSDenvy-freenessifeachagent(weakly)SD prefers his allocation to that of any other agent: p(i) %SD p(j)foralli, j ∈ N. An i assignmentpsatisfiesweakSDenvy-freenessifnoagentstrictlySDpreferssomeone else’s allocation to his: ¬[p(j) ≻SD p(i)]foralli, j ∈ N. For fairness concepts, SD i envy-freenessimpliesweakSD-envy-freeness(BogomolnaiaandMoulin,2001). Economicefficiency Anassignmentisperfectifeachagentsgetshismostpreferred c objects.An assignment p is SD-efficient is there existsno assignmentq such that q(i) %SD p(i) for all i ∈ N and q(i) ≻SD p(i) for some i ∈ N. An assignment is i i ex post efficient if it can be representedas a probabilitydistribution over the set of SD-efficientdiscreteassignments.PerfectionimpliesSD-efficiencywhichimpliesex postefficiency. AnassignmentruleisSD-efficient(expostefficient)ifitalwaysreturnsanSD- efficient(ex post efficient) assignment. An assignmentrule satisfies unanimity,if it returnstheperfectassignmentifaperfectassignmentexists. SD-efficiency implies ex post efficiency which implies unanimity. The first im- plication was shown by (BogomolnaiaandMoulin, 2001). For the second implica- tion, assume that an assignment does not satisfy unanimity, there exists a perfect assignmentpbutthemechanismreturnssomeimperfectassignmentq.TheonlySD- efficient assignment that gives c units to each agent is p. However since q , p, it cannot be achieved by a probability distribution over SD-efficient discrete assignments. Strategyproofness A random assignment function f is SD-strategyproof if f(%)(i) %SD f(%′,% )(i)forall%′and% . A random assignment i i −i i −i function f is weak SD-strategyproof if ¬[f(%′,% )(i) ≻SD f(% i −i i )(i)]forall%′∈R(O)and%′∈R(O)n−1. It is easy to see that SD-strategyproofness i i implies weak SD-strategyproofness (BogomolnaiaandMoulin, 2001). A random assignment function f is weak SD-group-strategyproof if there never exists an S ⊂ N and %′∈ R(O)|S| such that f(%′,% )(i) ≻SD f(%)(i)foralli ∈ S and S S −S i % ∈R(O)n−|S|. −S 6 HarisAziz 3 Generalimpossibilities Fortherandomassignmentproblemforwhichthenumberofobjectsisnotmorethan thenumberofagents,thereexistsarule(PS)thatisanonymous,neutral,SD-efficient and weak SD-strategyproof.Howeverwhen the numberof objectsis more than the numberofagents,wegetthefollowingimpossibility(Theorem1). Theorem1 Fortherandomassignmentproblemwithc > 1,thereexistsnoanonymous,neutral,SD-efficient,andweakSD-strategyproofrule. Proof We consider a random assignment setting with two agents and four objects withtherequirementthateachagentsgetstwounitsofhouses. % : a,b,c,d 1 % : b,c,a,d 2 %′: b,a,c,d 1 %′: b,a,c,d 2 Letuscompute f(% ,%′).Byanonymityandneutralityof f 1 2 w x y z f(% ,%′)= . 1 2 x wy z! BySD-efficiencyof f, 10y z f(% ,%′)= . 1 2 01y z! Byanonymityandneutralityof f, 101/21/2 f(% ,%′)= . 1 2 011/21/2! Byusingsimilararguments,SD-efficiency,anonymity,andneutralityof f implies that 11/20 1/2 f(%′,% )= . 1 2 01/21 1/2! Nowletusconsider x x x x f(% ,% )= 11 12 13 14 . 1 2 x21 x22 x23 x24! For f(% ,% )tobefeasible, 1 2 x ,x ,x ,x ,x ,x ,x ,x ≥0 11 12 13 14 21 22 23 24 x +x +x +x =2 11 12 13 14 x +x +x +x =2 21 22 23 24 x +x = x +x = x +x = x +x =1 11 21 12 22 13 23 14 24 Randomassignmentwithmulti-unitdemands 7 Next,weshow thatif f(% ,% ) = f(%′,% ) or f(% ,% ) = f(% ,%′), then f is 1 2 1 2 1 2 1 2 notweakSD-strategyproof. If f(% ,% )= f(%′,% ),then 1 2 1 2 f(% ,%′)(2)≻SD f(% ,% )(2). 1 2 2 1 2 Hence, f isnotweakSD-strategyproof. If f(% ,% )= f(% ,%′),then 1 2 1 2 f(%′,% )(1)≻SD f(% ,% )(1). 1 2 1 1 2 Hence, f isnotweakSD-strategyproof. Thereforethe only way f can still be weak SD-strategyproofif both of the fol- lowingconditionshold. – f(% ,% )(1)isincomparablefor1with f(%′,% )(1). 1 2 1 2 – f(% ,% )(2)isincomparablefor2with f(% ,%′)(2). 1 2 1 2 Thismeansthatthefollowingconstraintsshouldhold. Given that agent 2 reports % , agent 1 should not benefit by misreporting %′ 2 1 insteadof% .Thisimpliesthatx +x +x >1.5. 1 11 12 13 Given that agent 1 reports % , agent 2 should not benefit by misreporting %′ 1 2 insteadof% .Thisimpliesthatx +x +x >1.5. 2 22 23 21 Addingboththeseinequalitiesyields x +x +x +x +x +x >3. 11 12 13 22 23 21 Butthisisacontradictionsince x +x +x +x +x +x = (x +x )+ 11 12 13 22 23 21 11 21 (x +x )+(x +x )=3.Henceif f isSD-efficient,andanonymous,neutral,then 12 22 13 23 itcannotbeweakSD-strategyproof. The same argument can be extended to arbitrary number of agents where each agent requires two objects from among o ,...,o . Each new agent i ∈ {3,...,n} 1 2n most prefers objects o ,o and least prefers objects o ,o ,o ,o . Hence in each 2i−1 2i 1 2 3 4 SD-efficientassignmenteachagenti∈{3,...,n}isallocatedo ando completely. 2i−1 2i Thesameargumentsforthecaseoftwoagentsapplytothemoregeneralcase.Sim- ilarly, the same argumentscan also be extended to the case where c > 2. One can addmoreobjectstoendofthepreferencelistsofbothagentsandeachagentgetsa uniformfractionoftheseobjectsattheendofthepreferencelists. ⊓⊔ Theorem 1 complements an earlier impossibility result of Kojima (2009) that statesthereexistsnoSD-efficient,SDenvy-free,andweakSD-strategyproofrandom assignment rule for multi-unit demands. In Theorem 1, the property of SD envy- freenessisreplacedbyanonymity. Theproofabovecanbeextendedbycloningagents1and2toprovethefollowing statementforthebasicassignmentsettingwithsingle-unitdemand. Theorem2 Fortherandomassignmentproblem,thereexistsnoanonymous,neutral, SD-efficient, and weak SD group-strategyproofness rule even for equal number of agentsandobjects. 8 HarisAziz Proof Weconsiderarandomassignmentsettingwithfouragentsandfoursobjects. Therearetwoagentsthatareoftype1andtwoagentsoftype2.Lettherealprefer- encesoftheagents{1,2}oftype1be% andlettherealpreferencesofagents{3,4} 1 oftype2be% . 2 % : a,b,c,d 1 % : b,c,a,d 2 %′: b,a,c,d 1 %′: b,a,c,d 2 Letuscompute f(% ,% ,%′,%′). 1 1 2 2 Byanonymityandneutrality,weknowthat w/2 x/2 y/2z/2 w/2 x/2 y/2z/2 f(% ,% ,%′,%′)= . BySD-efficiency,wek1now1 tha2t 2 xx//22 ww//22 yy//22zz//22 1/2 0 y/2z/2 1/2 0 y/2z/2 f(% ,% ,%′,%′)= . Duetoanonymityandn1eut1rali2tyo2f f,  00 11//22yy//22zz//22 1/2 0 1/41/4 1/2 0 1/41/4 f(% ,% ,%′,%′)= . Byusingsimilarargum1ents1,SD2-effi2 cienc00y,a11n//o22ny11m//44ity11,//a44ndneutralityof f implies that 1/21/4 0 1/4 1/21/4 0 1/4 f(%′,%′,% ,% )= . Nowletusconsider 1 1 2 2  00 11//4411//2211//44 x /2 x /2 x /2 x /2 11 12 13 14 x /2 x /2 x /2 x /2 f(% ,% ,% ,% )= 11 12 13 14 . For f(% ,% ,% ,%1)to1be2feas2ible,xx2211//22 xx2222//22 xx2233//22 xx2244//22 1 1 2 2 Randomassignmentwithmulti-unitdemands 9 x ,x ,x ,x ,x ,x ,x ,x ≥0 11 12 13 14 21 22 23 24 x +x +x +x =2 11 12 13 14 x +x +x +x =2 21 22 23 24 x +x = x +x = x +x = x +x =1 11 21 12 22 13 23 14 24 Next,weshowthatif f(% ,% ,% ,% ) = f(%′,%′,% ,% )or f(% ,% ,% ,% )= 1 1 2 2 1 1 2 2 1 1 2 2 f(% ,% ,%′,%′),then f isnotweakSDgroup-strategyproof. 1 1 2 2 If f(% ,% ,% ,% )= f(%′,%′,% ,% ),then 1 1 2 2 1 1 2 2 f(% ,% ,%′,%′)(3)≻SD f(% ,% ,% ,% )(3). 1 1 2 2 2 1 1 2 2 Hence, f isnotweakSDgroup-strategyproof. If f(% ,% ,% ,% )= f(% ,% ,%′,%′),then 1 1 2 2 1 1 2 2 f(%′,%′,% ,% )(1)≻SD f(% ,% ,% ,% )(1). 1 1 2 2 1 1 1 2 2 Hence, f isnotweakSDgroup-strategyproof. Giventhatagentoftype% report% ,thenagentoftype% shouldnotbenefitby 2 2 1 misreporting%′ insteadof% .Thisimpliesthatx +x +x >1.5. 1 1 11 12 13 Giventhatagentsoftype2report% ,thenagentsoftype2shouldnotbenefitby 1 misreporting%′ insteadof% .Thisimpliesthatx +x +x >1.5. 2 2 22 23 21 Hence, x +x +x +x +x +x >3 11 12 13 22 23 21 Butthisisacontradictionsince x +x +x +x +x +x = (x +x )+ 11 12 13 22 23 21 11 21 (x +x )+(x +x )=3.Henceif f isSD-efficientandanonymous,andneutral, 12 22 13 23 thenitcannotbeweakSDgroup-strategyproof. Thesameargumentcanbeextendedtoarbitrarynumberofagents. ⊓⊔ Theorem 2 (that holds for single-unit demands) complements Theorem 1 in (Kojima, 2009) that only holds for multi-unit demands. The assignment problem in which m = n can be viewed as a subdomain of voting in which each alterna- tive is a discrete assignment and preferences of an agent over assignments simply dependon his allocated object(AzizandStursberg, 2014). As a corollaryof Theo- rem 2, we get that when agents may express indifference, there exists no random- izedsocialchoicerulethatisanonymous,neutral,SD-efficient,andweakSDgroup- strategyproof. This proves a weaker version of the conjecture that there exists no randomized social choice rule that is anonymous, neutral, SD-efficient, and weak SD-strategyproof(Azizetal.,2013b). We now show that if one of SD-efficiency, anonymity, or weak SD- strategyproofnessis dropped,then there exist rules that satisfy the other properties mentioned in the two impossibility theorems respectively even for multi-unit demands. If SD-efficiency is dropped or is replaced by ex post efficiency, then RP satisfies strategyproofness, anonymity, neutrality and ex post efficiency. If 10 HarisAziz anonymityisdropped,thentheprioritymechanismachievesSD-efficiencyandgroup SD-strategyproofness.If weak SD-strategyproofnessis dropped, then OPS satisfies the other properties. It remains open whether neutrality is necessarily required to obtainthetwoimpossibilitytheorems. 4 Multi-unit-eating PS Inthissection,weexaminethepropertiessatisfiedbymulti-unit-eatingPS (MPS). Before we proceed, we will try to get a better understanding of how multi-unit- eating PS works. CheandKojima (2010) defined multi-unit-eating PS as the rule inwhicheachagenteatshisc mostpreferredobjectsatspeed1duringthetimein- tervalt ∈ [0,1].Theyassumedthatateachpointeachagenthascobjectsavailable forconsumptionduringtherunningofmulti-unit-eatingPS andhencealltheobjects areconsumedattime1.Wefirstshowthatitmaybethecasethatlessthancobjects are available for consumption. Consider the illustration of multi-unit-eating PS in Figure1.Attimet=7/8,onlyo isremaining.Hencethefirstgoalistodecidehow 4 todefinemulti-unit-eatingPS whenagentshavelessthancobjectstoeat.Weresort tothefollowingdefinitionofmulti-unit-eatingPS. Let rem(t) be the number of objects that have not been completely eaten at time t. In multi-unit-eating PS, each agenteats his min(c,rem(t)) mostpre- ferredavailableobjectswithspeed1ateverytimepointuntilalltheobjects havebeenconsumed. Agent1 o1,o2 o1,o3 o1,o4 o4 o4 Agent2 o3,o2 o3,o4 o1,o4 o4 o4 0 1/2 3/4 7/8 1 9/8 1: o1,o2,o3,o4 2: o3,o2,o4,o1 3/4 1/21/4 1/4 p= . 1/4 1/23/4 3/4! Fig.1 Illustrationofmulti-unit-eatingPS withagentseatingtheirpreferredobjectsovertime.Theeven- tualassignmentisp.