MITHUN CHAKRABORTY, WARUT SUKSOMPONG, YAIR ZICK, …

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Weighted Envy-Freeness in Indivisible Item Allocation

MITHUN CHAKRABORTY, University of Michigan, USA

AYUMI IGARASHI, National Institute of Informatics, Japan

WARUT SUKSOMPONG, National University of Singapore, Singapore

YAIR ZICK, University of Massachusetts, Amherst, USA

We introduce and analyze new envy-based fairness concepts for agents with weights that quantify their en-titlements in the allocation of indivisible items. We propose two variants of weighted envy-freeness up toone item (WEF1): strong, where envy can be eliminated by removing an item from the envied agent’s bun-dle, and weak, where envy can be eliminated either by removing an item (as in the strong version) or byreplicating an item from the envied agent’s bundle in the envying agent’s bundle. We show that for additivevaluations, an allocation that is both Pareto optimal and strongly WEF1 always exists and can be computedin pseudo-polynomial time; moreover, an allocation that maximizes the weighted Nash social welfare maynot be strongly WEF1, but always satisfies the weak version of the property. Moreover, we establish that ageneralization of the round-robin picking sequence algorithm produces in polynomial time a strongly WEF1allocation for an arbitrary number of agents; for two agents, we can efficiently achieve both strongWEF1 andPareto optimality by adapting the adjusted winner procedure. Our work highlights several aspects in whichweighted fair division is richer and more challenging than its unweighted counterpart.

1 INTRODUCTION

The fair allocation of resources to interested parties is a central issue in economics and has in-creasingly received attention in computer science in the past few decades [Brams and Taylor, 1996,Markakis, 2017, Moulin, 2003, 2019, Thomson, 2016]. The problem has a wide range of applications,from reaching divorce settlements [Brams and Taylor, 1996] and dividing land [Segal-Halevi et al.,2017] to sharing apartment rent [Gal et al., 2017]. Envy-freeness is one of the most commonly stud-ied fairness criterion in the literature; it stipulates that all agents find their assigned bundle to bethe best among all bundles in the allocation [Foley, 1967, Varian, 1974].Envy-freeness is a compelling notion when all agents have equal entitlements—indeed, in a stan-

dard envy-free allocation, no agent would rather take the place of another agent with respect tothe assigned bundles. However, in many division problems, agents may have varying claims on theresource. For instance, consider a facility that has been jointly funded by three investors—Alice,Bob, and Charlie—where Alice contributed 3/5 of the construction expenses while Bob and Char-lie contributed 1/5 each. One could then expect Alice to envy either Bob or Charlie if she doesnot value her share at least three times as much as each of the latter two investors’ share whenthey divide the usage of the facility. Besides this interpretation as the cost of participating in theresource allocation exercise, the weights may also represent other publicly known and acceptedmeasures of entitlement such as eligibility or merit. A prevalent example is inheritance division,wherein closer relatives are typically more entitled to the bequest than distant ones. Likewise,different countries have differing entitlements when it comes to apportioning humanitarian aid.Envy-freeness can be naturally extended to the general setting in which agents have weights des-ignating their entitlements. When the resource to be allocated is infinitely divisible (e.g., time touse a facility, or land in a real estate), it is known that a weighted envy-free allocation exists forany set of agents’ weights and valuations [Robertson and Webb, 1998, Zeng, 2000].In this article, we initiate the study of weighted envy-freeness for the ubiquitous setting where

the resource consists of indivisible items. Indeed, inheritance division usually involves discreteitems such as real estate, cars, and jewelry; similarly, facility usage is often allocated in fixed timeslots (e.g., hourly). Since envy-freeness cannot always be fulfilled even in the canonical setting

http://arxiv.org/abs/1909.10502v7

Mithun Chakraborty, Ayumi Igarashi, Warut Suksompong, and Yair Zick 2

without weights, for example when all agents agree that one particular item is more valuablethan the remaining items combined, recent works have focused on identifying relaxations of envy-freeness that can be satisfied in the case of equal entitlements. The most salient of these approxi-mations is perhaps envy-freeness up to one item (EF1): for any two agents 8 and 9 , if agent 8 enviesagent 9 , then we can eliminate this envy by removing a single item from 9 ’s bundle [Budish, 2011].Lipton et al. [2004] showed that an EF1 allocation exists and can be computed efficiently for anynumber of agents with monotone valuations.1 Our goal in this work is to extend EF1 to the generalcase with arbitrary entitlements, and explore the relationship of these extensions to other impor-tant justice criteria such as proportionality and Pareto optimality. The richness of the weightedsetting will be evident throughout our work; in particular, we demonstrate that while some pro-tocols from the unweighted setting can be generalized to yield strong guarantees, others are lessrobust and cease to offer desirable properties upon the introduction of weights.

1.1 Our Contributions

We assume that agents have positive (not necessarily rational) weights representing their entitle-ments and, with the exception of Propositions 3.2, 8.1, and 8.2, that they are endowed with additivevaluation functions.We begin in Section 2 by proposing two generalizations of EF1 to theweightedsetting: (strong) weighted envy-freeness up to one item (WEF1) and weak weighted envy-freenessup to one item (WWEF1). While WEF1 may appear as the more natural extension, we argue that itcan impose a highly demanding constraint when the weights vastly differ, so that WWEF1 is a use-ful alternative. In Section 3, we focus on two classical EF1 protocols. On the one hand, we showthat the envy cycle elimination algorithm of Lipton et al. [2004] does not extend to the weightedsetting except in the special case of identical valuations. On the other hand, we construct a weight-based picking sequence which allows us to compute a WEF1 allocation efficiently—this generalizesa folklore result from the unweighted setting. The analysis of this algorithm is significantly moreinvolved than for the unweighted version and requires making intricate algebraic manipulations.Nevertheless, the algorithm itself is simple both to explain and to implement, so we believe that itis suitable for maintaining fairness in practice.In Sections 4 and 5, we examine the interplay between fairness and Pareto optimality. For two

agents, we exhibit that a weighted variant of the adjusted winner procedure allows us to computean allocation that is both WEF1 and Pareto optimal in polynomial time—our algorithm provides anatural discretization of the classical procedure, which was designed for the divisible item setting.We then show by adapting an algorithm of Barman et al. [2018] that a Pareto optimal and WEF1

allocation is guaranteed to exist and can be found in pseudo-polynomial time for any number ofagents. Furthermore, we prove that while an allocation with maximum weighted Nash welfaremay fail to satisfy WEF1, such an allocation is both Pareto optimal and WWEF1, thereby generalizingan important result of Caragiannis et al. [2019]. Our proof for the WWEF1 result follows a similaroutline as that of Caragiannis et al., but we need tomake a case distinction based on the comparisonbetweenweights.We continue our investigation in Section 6 by exploring the relationship betweenweighted envy-freeness and the weighted versions of other fairness concepts—in particular, wefind that several relationships from the unweighted setting break down—and illustrate throughexperiments in Section 7 that envy-freeness is often harder to satisfy in the presence of weightsthan otherwise. Finally, we conclude in Section 8 by discussing some obstacles that we faced when

1EF1 is also remarkable for its robustness: it can be satisfied under cardinality constraints [Biswas and Barman, 2018]and connectivity constraints [Bilò et al., 2019, Suksompong, 2019], has a relatively low price in terms of social welfare[Barman et al., 2020, Bei et al., 2019], and is computable using few queries [Oh et al., 2019].


trying to extend our ideas and results beyond additive valuation functions; specifically, we showthat a WWEF1 allocation may not exist when agents have non-additive (submodular) valuations.

1.2 Related Work

There is a long line of work on the fair division of indivisible items; see, e.g., the surveys byBouveret et al. [2016] and Markakis [2017] for an overview. Prior work on the fair allocation ofindivisible items to asymmetric agents has tackled fairness concepts that are not based on envy.Farhadi et al. [2019] introduced weighted maximin share (WMMS) fairness, a generalization of anearlier fairness notion of Budish [2011]. Aziz et al. [2019b] explored WMMS fairness in the alloca-tion of indivisible chores—items that, in contrast to goods, are valued negatively by the agents—where agents’ weights can be interpreted as their shares of the workload. Babaioff et al. [2019]studied competitive equilibrium for agents with different budgets. Recently, Aziz et al. [2020] pro-posed a polynomial-time algorithm for computing an allocation of a pool of goods and chores thatsatisfies both Pareto optimality and weighted proportionality up to one item (WPROP1) for agentswith asymmetric weights. Unequal entitlements have also been considered in the context of divisi-ble itemswith respect to proportionality [Barbanel, 1995, Brams and Taylor, 1996, Cseh and Fleiner,2020, Robertson and Webb, 1998, Segal-Halevi, 2019]. We remark here that (weighted) proportion-ality is a strictly weaker notion than (weighted) envy-freeness under additive valuations. However,while PROP1 is also implied by EF1 in the unweighted setting, the relationship between the corre-sponding weighted notions is much less straightforward, as we demonstrate in Section 6.In addition to expressing the entitlement of individual agents, weights can also be applied to

settings where each agent represents a group of individuals [Benabbou et al., 2019, 2020a]—here,the size of a group can be used as its weight.2 Specifically, in the model of Benabbou et al. [2020a],groups correspond to ethnic groups (namely, the major ethnic groups in Singapore, i.e., Chinese,Malay, and Indian). Maintaining provable fairness guarantees amongst the ethnic groups is highlydesirable; in fact, it is one of the principal tenets of Singaporean society.

2 PRELIMINARIES

Throughout the article, given a positive integer A , we denote by [A ] the set {1, 2, . . . , A }. We aregiven a set # = [=] of agents, and a set$ = {>1, . . . , ><} of items or goods. Subsets of$ are referredto as bundles, and each agent 8 ∈ # has a valuation function E8 : 2$ → R≥0 over bundles; thevaluation function for every 8 ∈ # is normalized (i.e., E8 (∅) = 0) and monotone (i.e., E8 (() ≤ E8 () )whenever ( ⊆ ) ). We denote E8 ({>}) simply by E8 (>) for any 8 ∈ # and > ∈ $ .

An allocation � of the items to the agents is a collection of = disjoint bundles �1, . . . , �= suchthat

⋃8 ∈# �8 ⊆ $ ; the bundle �8 is allocated to agent 8 and E8 (�8) is agent 8’s realized valuation

under�. Given an allocation�, we denote by�0 the set$ \(⋃

8 ∈# �8 ), and its elements are referredto as withheld items. An allocation � is said to be complete if �0 = ∅, and incomplete otherwise.In our setting with different entitlements, each agent 8 ∈ # has a fixed weight F8 ∈ R>0; these

weights regulate how agents value their own allocated bundles relative to those of other agents,and hence bear on the overall (subjective) fairness of an allocation. More precisely, we define

the weighted envy of agent 8 towards agent 9 under an allocation � as max{0,

E8 (� 9 )F9− E8 (�8 )

F8

}.

An allocation is weighted envy-free (WEF) if no agent has positive weighted envy towards anotheragent. Intuitively, agent 8 beingweighted envy-free towards agent 9 means that 8’s valuation for her

2Note that in this model, each group has a valuation function that represents the overall preference of its mem-bers. Other group fairness notions do not assume the existence of such aggregate functions and instead take di-rectly into account the preferences of the individual agents in each group [Conitzer et al., 2019, Kyropoulou et al., 2020,Segal-Halevi and Suksompong, 2019, 2021].


share �8 , given that 8’s entitlement isF8 , is at least as high as 8’s valuation for � 9 if 8’s entitlementwere F 9 . Weighted envy-freeness reduces to traditional envy-freeness when F8 = F , ∀8 ∈ #

for some positive real constant F . Since a complete envy-free allocation does not always exist,it follows trivially that a complete WEF allocation may not exist in general. We briefly remarkhere that with indivisible items, it is possible to define variations of weighted envy-freeness—forexample, ifF8 = 1 andF 9 = 2, one could require that agent 9 ’s bundle can be divided into two partsneither of which agent 8 finds more valuable than her own bundle. Nevertheless, the definition thatwe use is mathematically natural and can be directly applied to arbitrary (not necessarily rational)weights.

We now state the main definitions of our article, which naturally extend envy-freeness up to oneitem (EF1) [Budish, 2011, Lipton et al., 2004] to the weighted setting.

Definition 2.1. An allocation� is said to be (strongly) weighted envy-free up to one item (WEF1) iffor any pair of agents 8, 9 with � 9 ≠ ∅, there exists an item > ∈ � 9 such that

E8 (�8 )F8

≥ E8 (� 9 \ {>})F 9

.

More generally, � is said to be weighted envy-free up to 2 items (WEF2) for an integer 2 ≥ 1 if forany pair of agents 8, 9 , there exists a subset (2 ⊆ � 9 of size at most 2 such that

E8 (�8 )F8

≥ E8 (� 9 \ (2 )F 9

.

Definition 2.2. An allocation � is said to be weakly weighted envy-free up to one item (WWEF1) iffor any pair of agents 8, 9 with � 9 ≠ ∅, there exists an item > ∈ � 9 such that

eitherE8 (�8 )F8

≥ E8 (� 9 \ {>})F 9

orE8 (�8 ∪ {>})

F8≥ E8 (� 9 )

F 9.

More generally,� is said to be weakly weighted envy-free up to 2 items (WWEF2) for an integer 2 ≥ 1if for any pair of agents 8, 9 , there exists a subset (2 ⊆ � 9 of size at most 2 such that

eitherE8 (�8 )F8

≥ E8 (� 9 \ (2)F 9

orE8 (�8 ∪ (2)

F8≥ E8 (� 9 )

F 9.

In other words, an allocation is WEF1 if any (weighted) envy from an agent 8 towards anotheragent 9 can be eliminated by removing a single item from 9 ’s bundle. Similarly, WWEF1 requires thatany such envy can be eliminated by either removing an item from 9 ’s bundle or adding a copy ofan item from 9 ’s bundle to 8’s bundle.A valuation function E : 2$ → R≥0 is said to be additive if E (() =

∑>∈( E (>) for every

( ⊆ $ . We will assume additive valuations for most of the article; this is a very common as-sumption in the fair division literature and offers a tradeoff between expressiveness and simplic-ity [Bouveret and Lemaître, 2016, Caragiannis et al., 2019, Kurokawa et al., 2018]. Under this as-sumption, both WEF1 and WWEF1 reduce to EF1 in the unweighted setting. Moreover, one can checkthat under additive valuations, an allocation satisfies WWEF1 if and only if for any pair of agents 8, 9with � 9 ≠ ∅, there exists an item > ∈ � 9 such that

E8 (�8 )F8

≥ E8 (� 9 )F 9

− E8 (>)min{F8 ,F 9 }

.

The criterion WEF1 can be criticized as being too demanding in certain circumstances, when theweight of the envied agent is much larger than that of the envying agent. To illustrate this, considera problem instance where agent 1 has an additive valuation function and is indifferent among all


items taken individually, e.g., E1 (>) = 1 for every > ∈ $ . Now, if F1 = 1 and F2 = 100, theneliminating one item from agent 2’s bundle reduces agent 1’s weighted valuation of this bundleby merely 0.01. As such, we may trigger a substantial adverse effect on the overall welfare of theallocation by aiming to eliminate agent 1’s weighted envy towards agent 2. This line of thinkingwas our motivation for introducing the weak weighted envy-freeness concept. We also note thatWWEF1 can be viewed as a stronger version of what one could refer to as “transfer weighted envy-freeness up to one item”: agent 8 is transfer weighted envy-free up to one item towards agent 9 underthe allocation � if there is an item > ∈ � 9 that would eliminate the weighted envy of 8 towards 9upon being transferred from � 9 to �8 , i.e., E8 (�8 ∪ {>}) ≥ F8

F9· E8 (� 9 \ {>}).

In addition to fairness, we often want our allocation to satisfy an efficiency criterion. One impor-tant such criterion is Pareto optimality. An allocation �′ is said to Pareto dominate an allocation�if E8 (�′8 ) ≥ E8 (�8 ) for all agents 8 ∈ # and E 9 (�′9 ) > E 9 (� 9 ) for some agent 9 ∈ # . An allocation isPareto optimal (or PO for short) if it is not Pareto dominated by any other allocation.Allocations maximizing the Nash welfare—defined as NW(�) :=

∏8 ∈# E8 (�8 )—are known to

offer strong guarantees with respect to both fairness and efficiency in the unweighted setting[Caragiannis et al., 2019]. For our weighted setting, we define a natural extension called weightedNash welfare—WNW(�) := ∏

8 ∈# E8 (�8 )F8 . Since it is possible that the maximum attainable WNW(�) is0, we define amaximumweighted Nash welfare or MWNW allocation along the lines of Caragiannis et al.[2019] as follows: given a problem instance, we find a maximum subset of agents, say#max ⊆ # , towhich we can allocate bundles of positive value, and compute an allocation to the agents in #max

that maximizes3∏

8 ∈#maxE8 (�8 )F8 . To see why the notion of MWNW makes intuitive sense, consider

a setting where agents have a value of 1 for each item; furthermore, assume that the number ofitems is exactly

∑=8=1F8 . In this case, one can verify (using standard calculus) that an allocation

maximizing MWNW assigns to agent 8 exactlyF8 items. Indeed, following the interpretation ofF8 asthe number of members of group 8 (see Section 1.2), the expression E8 (�8 )F8 can be thought of aseach member of group 8 deriving the same value from the set �8 ; the group’s overall Nash welfareis thus E8 (�8 )F8 .We also examine the extent to which weighted envy-freeness relates to the weighted versions

of two other key fairness notions: proportionality and the maximin share guarantee.An allocation� is said to be weighted proportional (WPROP) if for every agent 8 ∈ # , it holds that

E8 (�8 ) ≥ F8∑9∈# F9

E8 ($). For a positive integer 2 , it is weighted proportional up to 2 items (WPROP2) if

for every 8 ∈ # , there exists a subset of items not allocated to 8 , i.e., (2 ⊆ $ \ �8 , of size at most2 such that E8 (�8 ) ≥ F8∑

9∈# F9· E8 ($) − E8 ((2 ); this is a natural extension of the (weighted) PROP1

concept [Aziz et al., 2020, Conitzer et al., 2017].4

Let Π($) denote the collection of all (ordered) =-partitions of the set of items $ , or, in otherwords, the collection of all complete allocations of $ to = agents. Then, the weighted maximinshare [Farhadi et al., 2019] of agent 8 is defined as:

WMMS8 := max(�1,�2,...,�=) ∈Π ($)

min9 ∈#

F8

F 9E8 (� 9 ).

An allocation � is called WMMS if E8 (�8 ) ≥ WMMS8 for every 8 ∈ # . More generally, for any approxi-mation ratio U ∈ (0, 1], � is called U-WMMS if E8 (�8 ) ≥ U · WMMS8 for every 8 ∈ # .

3There can be multiple maximum subsets#max having the same cardinality but different maximum weighted Nash welfare.Our main positive result for MWNW (Theorem 5.1) holds for all such subsets #max .4For any 0 < 1, WPROP0 implies WPROP1 but the converse does not hold. Indeed, the former follows directly from thedefinition. For the latter, consider a problem instance with = = 2 and $ = {>1, . . . , >1 }, weights F1 > 1 − 1 and F2 = 1,and identical, additive valuation functions such that E8 (>) = 1 for all 8 ∈ # and > ∈ $ . It can be verified that the allocationthat gives all items to agent 2 is WPROP1 but not WPROP0.


3 WEF1 ALLOCATIONS

We commence our exploration of weighted envy-freeness by considering extensions of two stan-dard methods for producing EF1 allocations in the unweighted setting: the envy cycle eliminationalgorithm and the round-robin algorithm. As we will see, these two procedures experience contrast-ing fortunes in the presence of weights: while the idea of eliminating envy cycle fundamentallyfails, the round-robin algorithm admits an elegant generalization that can take into account arbi-trary entitlements of the agents.

3.1 Envy Cycle Elimination Algorithm

Before we discuss the envy cycle elimination algorithm of Lipton et al. [2004], let us briefly recaphow it works in the unweighted setting. The algorithm allocates one item at a time in an arbitraryorder. It also maintains an “envy graph”, which captures the envy relation between the agentswith respect to the (incomplete) allocation at each stage. The next item is allocated to an unenviedagent, and any envy cycle that forms as a result is eliminated by letting each agent take the bundleof the agent that she envies. This cycle elimination step allows the algorithm to ensure that thereis an unenvied agent to whom it can allocate the next item.As far as envy in the traditional sense is concerned, what an agent actually “envies” is an allo-

cated bundle regardless of who owns that bundle. However, both the subjective valuations of allo-cated bundles and the relative weights interact in non-trivial ways to determine weighted envy. Itis easy to see that weighted envy of 8 towards 9 does not imply traditional envy of 8 towards 9 , andvice versa. A crucial implication is that even if agent 8’s bundle is replaced with the bundle of anagent 9 towards whom 8 has weighted envy, 8’s realized valuation, and hence the ratio of her real-ized valuation to her weight, may decrease as a result. Indeed, consider a problem instance with= = 2 and $ = {>1, >2, >3}, weights F1 = 3 and F2 = 1, and identical, additive valuation functionssuch that E8 (>) = 1 for all 8 ∈ # and > ∈ $ . Under the complete allocation with �1 = {>1, >2},agent 1 has weighted envy towards agent 2 since E1(�2)/F2 = 1/1 = 1 > 2/3 = E1 (�1)/F1, butagent 1 would not prefer to replace �1 with �2 since that reduces her realized valuation from 2 to1. On the other hand, agent 2 could benefit from replacing �2 with �1 even though she does nothave weighted envy towards agent 1. As such, the natural extension of the envy cycle eliminationalgorithm, where an edge exists from agent 8 to agent 9 if and only if 8 has weighted envy towards9 , does not guarantee a complete WEF1 or even WWEF1 allocation.

Proposition 3.1. The weighted version of the envy cycle elimination algorithm may not produce

a complete WWEF1 allocation, even in a problem instance with two agents and additive valuations.

Proof. Consider a problem instance with = = 2 and< = 12, weights F1 = 1 and F2 = 2, andvaluation functions defined by

E1(>A ) =

1 for A = 1;

0.1 for A = 12;

0.21 otherwise;

and E2(>A ) =

1.1 for A = 1;

0.1 for A = 12;

0.2 otherwise.

Suppose that the weighted envy cycle elimination algorithm iterates over >1, >2, . . . , >12, andstarts by allocating >1 to agent 1 due to, say, lexicographic tie-breaking. At this point, agent 2 hasweighted envy towards agent 1 and not vice versa; moreover, this condition persists until items>2, . . . , >10 have all been allocated to agent 2. At this point, item >11 also goes to agent 2, resultingin valuations E1(�1) = E1(>1) = 1 and E2 (�2) = E2 ({>2, . . . , >11}) = 2. Agent 2 still has weightedenvy towards agent 1 since E2(�2)/F2 = 1 < 1.1/1 = E2 (�1)/F1; on the other hand, agent 1 alsodevelops weighted envy towards agent 2 since E1(�2)/F2 = 1.05 > 1 = E1(�1)/F1. Thus, there


is a cycle in the induced weighted envy graph. For an unweighted envy graph, we would “de-cycle” the graph at this point by swapping bundles over the cycle and that would still maintain theinvariant that the allocation is EF1. However, if we swap the bundles in this example so that thenew allocated bundles are �′1 = �2 = {>2, . . . , >11} and �′2 = �1 = {>1}, agent 2 will end up having(weak) weighted envy up to more than one item towards agent 1 since E2(�′2)/F2 = 1.1/2 = 0.55and E2 (�′1 \ {>})/F1 = (0.2 × 9)/1 = 1.8 for every > ∈ �′1, and this weighted envy persists nomatter how we allocate >12.5 �

By replacing each of the items >2, . . . , >11 with 2 smaller items of equal value, one can check thatthe envy cycle elimination algorithm cannot even guarantee WWEF2 for any fixed 2 . In spite of thisnegative result, the algorithm does work in the special case where the agents all have the samevaluations.

Proposition 3.2. The weighted version of the envy cycle elimination algorithm produces a com-plete WEF1 allocation whenever agents have identical (not necessarily additive) valuations, i.e., E8 (() =E (() for some E : 2$ → R≥0, ∀8 ∈ # , ∀( ⊆ $ .

Proof. By construction of the algorithm, the (incomplete) allocation at the end of each iterationis guaranteed to be WEF1 as long as we can find an agent, say 8 , towards whom no other agent hasweighted envy at the beginning of the iteration: we give the item under consideration to agent 8and thus any resulting weighted envy towards 8 can be eliminated by removing this item. If thereis no unenvied agent, then the weighted envy graph consists of at least one cycle; however, underidentical valuations, the envy graph cannot have cycles. Indeed, suppose that agents 1, 2, . . . , ℓform a cycle (in that order) for some ℓ ∈ [=]. Since agents have identical valuations, it must bethat E (�1)/F1 < E (�2)/F2 < · · · < E (�ℓ )/Fℓ < E (�1)/F1, a contradiction. �

3.2 Picking Sequence Protocols

We now turn our attention to protocols that let agents pick their favorite item according to somepredetermined sequence. When all agents have equal weight and additive valuations, it is well-known that a round-robin algorithm, wherein the agents take turns picking an item in the order1, 2, . . . , =, 1, 2, . . . , =, . . . , produces an EF1 allocation. This is in fact easy to see: If agent 8 is aheadof agent 9 in the ordering, then in every “round”, 8 picks an item that she likes at least as much as9 ’s pick; by additivity, 8 does not envy 9 . On the other hand, if agent 8 picks after agent 9 , then byconsidering the first round to begin at 8’s first pick, it follows from the same argument that 8 doesnot envy 9 up to the first item that 9 picks.We show next that in the general setting with weights, we can construct a weight-dependent

picking sequence which guarantees WEF1 for any number of agents and arbitrary weights. Theresulting algorithm is efficient, intuitive and can be easily explained to a layperson, so we believethat it has a strong practical appeal. Unlike in the unweighted case, however, the proof that thealgorithm produces a fair allocation is much less straightforward and requires making severalintricate arguments.

Theorem3.3. For any number of agents with additive valuations and arbitrary positive real weights,

there exists a picking sequence protocol that computes a complete WEF1 allocation in polynomial time.

To prove Theorem 3.3, we construct a picking sequence such that in each turn, an agent withthe lowest weight-adjusted picking frequency picks the next item (Algorithm 1). We claim that

5Another interesting feature of this example is that the two agents have commensurable valuations, i.e., both agents havethe same valuation for the entire collection of items $ :

∑>∈$ E1 (>) =

∑>∈$ E2 (>) = 3.2. This shows that the negative

result of Proposition 3.1 holds even if we impose the additional restriction of commensurability on the valuation functions.


ALGORITHM 1: Weighted Picking Sequence Protocol

1: Remaining items $̂ ← $ .2: Bundles �8 ← ∅, ∀8 ∈ # .3: C8 ← 0, ∀8 ∈ # . /*number of times each agent has picked so far*/

4: while $̂ ≠ ∅ do5: 8∗ ← argmin8 ∈# C8

F8, breaking ties lexicographically.

6: >∗ ← argmax>∈$̂ E8∗ (>), breaking ties arbitrarily.

7: �8∗ ← �8∗ ∪ {>∗}.8: $̂ ← $̂ \ {>∗}.9: C8∗ ← C8∗ + 1.10: end while

after the allocation of each item, for any agent 8 , every other agent is weighted envy-free towards8 up to the item that 8 picked first.

To this end, first note that choosing an agent who has had the minimum (weight-adjusted)number of picks thus far ensures that the first = picks are a round-robin over all of the agents; inthis phase, the allocation is obviously WEF1 since each agent has at most one item at any point. Wewill show that, after this phase, the algorithm generates a picking sequence over the agents withthe following property:

Lemma 3.4. Consider an agent 8 chosen by Algorithm 1 to pick an item at some iteration C , and

suppose that this is not her first pick. Let C8 and C 9 be the numbers of times agent 8 and some other

agent 9 appear in the prefix of iteration C in the sequence respectively, not including iteration C itself.

ThenC 9C8≥ F9

F8.

Proof. Since agent 8 is picked at iteration C , it must be that 8 ∈ argmin:∈#C:F:

. This means thatC8F8≤ C 9

F9, i.e.,

C 9C8≥ F9

F8since C8 > 0. �

The property guaranteed by Lemma 3.4 is sufficient to ensure that the latest picker does notattract weighted envy up to more than one item towards herself after her latest pick:

Lemma 3.5. Suppose that, for every iteration C in which agent 8 picks an item after her first pick,

the numbers of times that agent 8 and some other agent 9 appear in the prefix of the iteration in the

sequence, not including iteration C itself—call them C8 and C 9 respectively—satisfy the relationC 9C8≥ F9

F8.

Then, in the partial allocation up to and including 8’s latest pick, agent 9 is weighted envy-free towards

8 up to the first item 8 picked.

8 9 8 9 8 8

1 23

1 23

23

Fig. 1. Illustration of the intuition behind the proof of Lemma 3.5. Here, 8 < 9 , F8 = 3, and F 9 = 2. Therectangles represent the agents’ buckets, and the numbers therein correspond to their capacities. Note that

agent 8 does not receive a bucket in her first pick. Agent 9 ’s buckets are filled, while those of agent 8 are

empty.

Before we prove Lemma 3.5, we first provide a high-level intuition. Recall the argument for theunweighted case at the beginning of Section 3.2. One way to visualize this argument is that when


we consider envy from agent 9 towards agent 8 , every time agent 9 picks an item, we give her abucket with 1 unit of water, while every time agent 8 picks an item from the second time onwards,we give her an empty bucket of capacity 1. Agent 9 is allowed to pour water from any of herbuckets into any of 8’s buckets that comes later in the sequence. Since 9 values an item that shepicks at least as much as any item that 8 picks in a later turn, in order to establish EF1, it sufficesto show that 9 can fill up all of 8’s buckets using such operations. A similar idea can be used in theweighted setting, except that in order to account for the weights, every time agent 8 picks after thefirst time, we give her an empty bucket of capacityF 9/F8 units (see Figure 1 for an example whenF8 > F 9 ). Note in particular that this bucket setup is entirely independent of the agents’ valuationsfor the items. However, unlike in the unweighted setting, where agent 9 can accomplish the taskby simply pouring all the water from each of her buckets into 8’s following bucket, in the weightedcase, 9 may need to pour water from a bucket into several of 8’s buckets, even those coming after9 ’s subsequent bucket. The proof below formalizes this intuition.

Proof of Lemma 3.5. Let W :=F9

F8. Consider any iteration C in which agent 8 is chosen after

her first pick. Let agent 9 ’s values for the items allocated to agent 8 in the latter’s second, third,. . . , (C8 + 1)st picks (the last one occurring at the iteration C under consideration) be V1, V2, . . . ,VC8 respectively. If >

∗ is the first item picked by agent 8 and �C the partial allocation up to andincluding iteration C , then clearly E 9 (�C

8 \{>∗}) =∑C8

G=1 VG . Let the number of times agent 9 appearsin the prefix of agent 8’s second pick be g1; that between agent 8’s second and third picks be g2; . . . ;that between agent 8’s C th8 and (C8 + 1)st picks be gC8 . Let agent 9 ’s values for the items she herselfpicked during phase G ∈ [C8 ] be UG1 , UG2 , . . . , UGgG respectively, where the phases are defined as in the

previous sentence. Then, E 9 (�C9 ) =

∑C8G=1

∑gG~=1 U

G~ . Now, for A ∈ [C8 ], since

∑AG=1 gG and A are the

numbers of times agents 9 and 8 appear in the prefix of the latter’s (A + 1)st pick respectively, thecondition of the lemma dictates that

A∑

G=1

gG ≥ AW ∀A ∈ [C8 ] . (1)

Note that g1 ≥ W > 0; however, gG can be zero for G ∈ {2, 3, . . . , C8 }—this corresponds to thescenario where agent 8 picked more than once without agent 9 picking in between. Moreover,every time agent 9 was chosen, she picked one of the items she values the most among thoseavailable, including the items picked by agent 8 later. Hence, if gG > 0 for some G ∈ [C8 ], then

UG~ ≥ max{VG , VG+1, . . . , VC8 } ∀~ ∈ [gG ]

⇒gG∑

~=1

UG~ ≥ gG max{VG , VG+1, . . . , VC8 }. (2)

Note that Inequality (2) holds trivially if gG = 0 since both sides are zero; hence it holds for everyG ∈ [C8 ].

We claim that for each A ∈ [C8 ],A∑

G=1

gG∑

~=1

UG~ ≥ W

A∑

G=1

VG +(

A∑

G=1

gG − AW)max{VA , VA+1, . . . , VC8 }.

To prove the claim, we proceed by induction on A . For the base case A = 1, we have from Inequal-ity (2) that

g1∑

~=1

U1~ ≥ g1 max{V1, V2, . . . , VC8 }


≥ WV1 + (g1 − W)max{V1, V2, . . . , VC8 }.For the inductive step, assume that the claim holds for A − 1; we will prove it for A . We have

A∑

G=1

gG∑

~=1

UG~ =

A−1∑

G=1

gG∑

~=1

UG~ +gA∑

~=1

UA~

≥ W

A−1∑

G=1

VG +(A−1∑

G=1

gG − (A − 1)W)max{VA−1, VA , . . . , VC8 } +

gA∑

~=1

UA~

≥ W

A−1∑

G=1

VG +(A−1∑

G=1

gG − (A − 1)W)max{VA−1, VA , . . . , VC8 } + gA max{VA , VA+1, . . . , VC8 }

≥ W

A−1∑

G=1

VG +(A−1∑

G=1

gG − (A − 1)W)max{VA , VA+1, . . . , VC8 } + gA max{VA , VA+1, . . . , VC8 }

= W

A−1∑

G=1

VG +(

A∑

G=1

gG − (A − 1)W)max{VA , VA+1, . . . , VC8 }

= W

A−1∑

G=1

VG + W max{VA , VA+1, . . . , VC8 } +(

A∑

G=1

gG − AW)max{VA , VA+1, . . . , VC8 }

≥ W

A−1∑

G=1

VG + WVA +(

A∑

G=1

gG − AW)max{VA , VA+1, . . . , VC8 }

= W

A∑

G=1

VG +(

A∑

G=1

gG − AW)max{VA , VA+1, . . . , VC8 },

where the first inequality follows directly from the inductive hypothesis, the second from Inequal-ity (2), and the third from the following facts:

A−1∑

G=1

gG − (A − 1)W ≥ 0 due to Inequality (1),

and max{VA−1, VA , . . . , VC8 } ≥ max{VA , VA+1, . . . , VC8 }.This completes the induction and establishes the claim.

Now, taking A = C8 in the claim, we get

C8∑

G=1

gG∑

~=1

UG~ ≥ W

C8∑

G=1

VG +(

C8∑

G=1

gG − C8W)VC8

≥ W

C8∑

G=1

VG ,

where we use Inequality (1) again for the second inequality. This implies that E 9 (�C9 ) ≥

F9

F8·E 9 (�C

8 \{>∗}), i.e., agent 9 is weighted envy-free towards agent 8 up to one item (specifically, the first itempicked by agent 8), concluding the proof of the lemma and therefore the proof of correctness. �

With Lemmas 3.4 and 3.5 in hand, we are now ready to prove Theorem 3.3.

Proof of Theorem 3.3. It is easy to see that directly after an agent picks an item, her envytowards other agents cannot get any worse than before. Since the partial allocation after the initial


round-robin phase is WEF1 and every agent is weighted envy-free up to one item towards everysubsequent picker due to Lemmas 3.4 and 3.5, the allocation is WEF1 at every iteration, and inparticular at the end of the algorithm. This establishes the correctness of the algorithm.For the time complexity, note that there are$ (<) iterations of thewhile loop. In each iteration,

determining the next picker takes$ (=) time, while letting the picker pick her favorite item takes$ (<) time. Since we may assume that< > = (otherwise it suffices to allocate at most one item toevery agent), the algorithm runs in time $ (<2). �

IfF8 equals a positive constantF for every 8 ∈ # , then Algorithm 1 degenerates into the tradi-tional round-robin procedure which is guaranteed to return an EF1 allocation for additive valua-tions, but may not be Pareto optimal; as such, Algorithm 1 may not produce a PO allocation either.This is easily seen in the following example: = = < = 2, F1 = F2 = 1, E1(>1) = E1(>2) = 0.5,E2(>1) = 0.8, and E2(>2) = 0.2. With lexicographic tie-breaking for both agents and items, our algo-rithm will give us�1 = {>1} and�2 = {>2}, which is Pareto dominated by�′1 = {>2} and�′2 = {>1}.On the other hand, if each agent has the same value for all items, the algorithm is equivalent to anapportionment method called Adams’ method [Balinski and Young, 2001].6 In the apportionmentsetting, agents correspond to states of a country, and items to seats in a parliament. Since all seatsare considered identical, the states can simply “pick” any seat from the remaining seats in appor-tionment, whereas for item allocation it is important that each agent picks her favorite item in herturn.

4 WEF1 AND PO ALLOCATIONS

As the picking sequence that we construct in Section 3.2 yields an allocation that is WEF1 but mayfail Pareto optimality, our next question is whether WEF1 can be achieved in conjunction withthe economic efficiency notion. We show that this is indeed possible, by generalizing the classicadjusted winner procedure for two agents and an algorithm of Barman et al. [2018] for highernumbers of agents.

4.1 Two Agents

When agents have equal entitlements, it is known that fairness and efficiency are compatible:Caragiannis et al. [2019] showed that an allocation maximizing the Nash social welfare satisfiesboth PO and EF1. Unfortunately, this approach is not applicable to our setting—we show that theMWNW allocation may fail to be WEF1. In fact, we prove a much stronger negative result: for anyfixed 2 , the allocation may fail to be WEF2 even for two agents with identical valuations.

Proposition 4.1. Let 2 be an arbitrary positive integer. There exists a problem instance with twoagents having identical additive valuations for which any MWNW allocation is not WEF2 .

Proof. Suppose that = = 2, and the weights are F1 = 1 and F2 = : for some positive integer :

such that(1 + 1

:+2):

> 2; such an integer : exists because lim:→∞(1 + 1

:+2):

= 4 . Let< = : +2 + 2,so $ = {>1, >2, . . . , >:+2+2}. The agents have identical, additive valuations defined by E8 (>) = 1 for

all 8 ∈ # and > ∈ $ . Since (1 + 1

:+2):

> 2, we have 1 · (: + 2 + 1): > 2 · (: + 2): . Moreover, for2 ≤ 8 ≤ : + 2 , we have

(1 + 1

: + 2 + 1 − 8

):>

(1 + 1

: + 2

):> 2 >

8 + 18

,

and so 8 (: + 2 + 2 − 8): > (8 + 1) (: + 2 + 1 − 8): . This means that any MWNW allocation � must giveone item to agent 1, say �1 = {>1}, and the remaining items to agent 2, i.e., �2 = {>2, . . . , >:+2+2}.6We are grateful to Ulrike Schmidt-Kraepelin for pointing out this connection.


However, even if we remove a set (2 of at most 2 items from�2, wewould still have E1(�2\(2 )/F2 ≥1 + 1/: > 1 = E1(�1)/F1, so the allocation is not WEF2 . �

Given that a MWNW allocation may not be WEF1 in our setting, a natural question is whether thereis an alternative approach for guaranteeing the existence of a PO and WEF1 allocation.We first showthat this is indeed the case for two agents: we establish that such an allocation exists and can becomputed in polynomial time for two agents, by adapting the classic adjusted winner procedure[Brams and Taylor, 1996] to the weighted setting.

Theorem 4.2. For two agents with additive valuations and arbitrary positive real weights, a com-plete WEF1 and PO allocation always exists and can be computed in polynomial time.

ALGORITHM 2: Weighted Adjusted Winner Procedure

Require:E1 (>1)E2 (>1) ≥

E1 (>2)E2 (>2) ≥ · · · ≥

E1 (>< )E2 (>< ) w.l.o.g.

1: 3 ← 1.2: while 1

F1

∑3A=1 E1 (>A ) < 1

F2

∑<A=3+2 E1 (>A ) do

3: 3 ← 3 + 1.4: end while

5: �1 ← {>1, . . . , >3 }.6: �2 ← {>3+1, . . . , ><}.

Proof. Assume first that both agents have positive values for all items, i.e., E1(>) > 0 andE2(>) > 0 for every > ∈ $ ; we will deal with the case where this does not hold later. We claimthat theWeighted AdjustedWinner procedure as delineated in Algorithm 2 produces an allocationsatisfying the theorem statement.First note that the left-hand side of the while loop condition is strictly increasing in 3 and

trivially exceeds the right-hand side for 3 =< − 1;7 hence, there always exists 3 ∈ [< − 1] whichsatisfies the stop criterion and the loop terminates at the smallest such 3 .

WEF1 property: If thewhile loop ends with 3 = 3∗, let us denote >3∗ by >∗. Then, by construction,E1 (�1)F1≥ E1 (�2)−E1 (>3∗+1)

F2, which implies that agent 1 is weighted envy-free towards agent 2 up to

one item (specifically, item >3∗+1).On the other hand, by construction, we also get that

E1(�1) − E1(>∗)F1

<

E1(�2)F2

. (3)

Moreover, due to the ordering of the ratios, we have∑3∗

A=1 E1 (>A )∑3∗A=1 E2 (>A )

≥ E1(>∗)E2(>∗)

≥∑<

A=3∗+1 E1 (>A )∑<A=3∗+1 E2 (>A )

⇒ E1(�1)E2(�1)

≥ E1(>∗)E2(>∗)

≥ E1(�2)E2(�2)

⇒ E1 (�1) ≥E1(>∗)E2(>∗)

· E2(�1);

E1 (�2) ≤E1(>∗)E2(>∗)

· E2(�2).

7For 3 ≥< − 1, we set the right-hand side of the while loop condition to zero.


Combining with Inequality (3), dividing through by E1 (>∗)E2 (>∗) , and simplifying, we get

E2(�1) − E2(>∗)F1

<

E2(�2)F2

.

Thus, agent 2 is also weighted envy-free towards agent 1 up to one item (specifically, item >∗).

Pareto optimality: First note that no incomplete allocation can be Pareto optimal since the real-ized valuation of either agent could be strictly improved by augmenting her bundle with a withhelditem (under our assumption that each agent values each item positively). Since the allocation �

produced by Algorithm 2 is complete, it suffices to show that it cannot be Pareto dominated byan alternative complete allocation �′. Any such complete allocation can be thought of as beinggenerated by transferring items between �1 and �2.Suppose that E1(�′1) > E1(�1) for some complete allocation�′ different from�. Since �1∪�2 =

�′1 ∪ �′2 = $ , this inequality implies that

E1 (�′1 ∩�1) + E1 (�′1 ∩�2) > E1 (�′1 ∩�1) + E1 (�′2 ∩�1)⇒ E1 (�′1 ∩�2) > E1 (�′2 ∩�1). (4)

If �′2 ∩ �1 = ∅, then �′2 ⊂ �2 (since �′2 ≠ �2). Hence, E2 (�′2) < E2(�2) so that �′ cannot Paretodominate �. As such, we will assume that �′2 ∩ �1 ≠ ∅. Then, due to the ratio ordering and how�1, �2 are constructed, we must have

E1(>)E2(>)

≥ E1 (>∗)E2 (>∗)

∀> ∈ �′2 ∩ �1

⇒ E1(�′2 ∩ �1)E2(�′2 ∩ �1)

=

∑>∈�′2∩�1

E1(>)∑

>∈�′2∩�1E2(>)

≥ E1 (>∗)E2 (>∗)

.

By similar reasoning, it holds that

E1(>∗)E2(>∗)

≥ E1(�′1 ∩ �2)E2(�′1 ∩ �2)

.

Combining with Inequality (4), we get

E1 (�′2 ∩�1)E2 (�′2 ∩�1)

≥ E1(�′1 ∩ �2)E2(�′1 ∩ �2)

>

E1 (�′2 ∩�1)E2 (�′1 ∩�2)

⇒ E2 (�′1 ∩�2) > E2 (�′2 ∩�1), since E1 (�′2 ∩�1) > 0,

⇒ E2 (�′2 ∩�2) + E2 (�′1 ∩ �2) > E2 (�′2 ∩�2) + E2 (�′2 ∩�1)⇒ E2 (�2) > E2(�′2),

which contradicts the necessary condition for�′ to Pareto dominate�: E2 (�′2) ≥ E2(�2). AssumingE2(�′2) > E2(�2) leads us to an analogous conclusion. Hence, � must be Pareto optimal.

Complexity: The< ratios can be sorted in$ (< log<) time. Each newwhile loop condition canbe checked in $ (1) time, so the total time taken by the while loop is $ (<). Hence, the algorithmruns in $ (< log<) time.

Let us now address the scenario where there are items of zero value to an agent. Of course,items valued at zero by both agents can be safely ignored. We will initialize the bundle �8 withitems valued positively by agent 8 ∈ {1, 2} only, i.e., �0

1 = {> ∈ $ : E1 (>) > 0, E2(>) = 0} and�02 = {> ∈ $ : E2(>) > 0, E1 (>) = 0}. Then we run Algorithm 2 on the remaining items and use its


output (�1, �2) to augment the respective bundles. We will now show that the resulting allocation(�0

1 ∪ �1, �02 ∪ �2) is WEF1 and PO.

By the argument in the previous part, there is an item > ′ ∈ �2 ⊆ �02 ∪ �2 such that

E1(�01 ∪ �1)F1

≥ E1(�1)F1

≥ E1(�2) − E1(> ′)F2

=E1(�0

2 ∪ �2) − E1(> ′)F2

,

since E1 (�02) = 0. Thus, agent 1 is weighted envy-free towards agent 2 up to one item. An analogous

argument shows that agent 2 is also weighted envy-free towards agent 1 up to one item.Since a Pareto optimal allocation cannot be incomplete (because each item has a positive value

to at least one agent), it suffices to show that the (complete) allocation under consideration is notPareto dominated by any complete allocation. Again, any complete allocation can be obtained from(�0

1 ∪ �1, �02 ∪ �2) by swapping items between agents. It is evident that any allocation in which

an item > ∈ �01 (resp., an item > ∈ �0

2) belongs to agent 2 (resp., agent 1) is Pareto dominated bythe allocation wherein this item is given to agent 1 (resp., agent 2), everything else remaining thesame. Hence, it suffices to show that a Pareto improvement cannot be achieved by swapping itemsin �1 ∪ �2 between the agents—but we already know this from the earlier part of the proof. �

4.2 Any Number of Agents

Having resolved the existence question of PO and WEF1 for two agents, we now investigate whethersuch an allocation always exists for any number of agents, answering the question in the affirma-tive. To this end, we employ a weighted modification of the algorithm by Barman et al. [2018],which finds a PO and EF1 allocation in pseudo-polynomial time for agents with additive valua-tions in the unweighted setting. Like Barman et al., we consider an artificial market where eachitem has a price and agents purchase a bundle of items with the highest ratio of value to price,called “bang per buck ratio”. This allows us to measure the degree of fairness of a given allocationin terms of the prices.Formally, a price vector is an<-dimensional non-negative real vector p = (?1, ?2, . . . , ?<) ∈ R$≥0;

we call ?> the price of item > ∈ $ , and write ? (- ) = ∑>∈- ?> for a set of items - . Let � be an

allocation and p be a price vector. For each 8 ∈ # , we call ? (�8) the spending and 1F8? (�8 ) the

weighted spending of agent 8 . We now define a weighted version of the price envy-freeness up toone item (pEF1) notion introduced by Barman et al. [2018].

Definition 4.3. Given an allocation � and a price vector p, we say that � is weighted price envy-free up to one item (WpEF1) with respect to p if for any pair of agents 8, 9 , either� 9 = ∅ or 1

F8? (�8) ≥

1F9

min>∈� 9? (� 9 \ {>}).

The bang per buck ratio of item > for agent 8 is E8 (>)?>

; we write the maximum bang per buck

ratio for agent 8 as U8 (p). We refer to the items with maximum bang per buck ratio for 8 as 8’sMBB items and denote the set of such items by MBB8 (p) for each 8 ∈ # . The following lemma isa straightforward adaptation of Lemma 4.1 in [Barman et al., 2018] to our setting; it ensures thatone can obtain the property of WEF1 by balancing among the spending of agents under the MBBcondition.

Lemma 4.4. Given a complete allocation � and a price vector p, suppose that allocation � satisfies

WpEF1with respect to p and agents are assigned to MBB items only, i.e.,�8 ⊆ MBB8 (p) for each 8 ∈ # .

Then � is WEF1.


Proof. To show that� is WEF1, take any pair of agents 8, 9 ∈ # . If� 9 = ∅, the required conditionholds trivially. Suppose that � 9 ≠ ∅. Since the allocation � is WpEF1 with respect to p,

1

F8? (�8 ) ≥

1

F 9min>∈� 9

? (� 9 \ {>}).

Multiplying both sides by U8 (p), we obtain1

F8U8 (p) · ? (�8 ) ≥

1

F 9min>∈� 9

U8 (p) · ? (� 9 \ {>})

⇒ 1

F8

∑

>∈�8

E8 (>)?>

?> ≥1

F 9min>∈� 9

∑

>′∈� 9 \{> }

E8 (> ′)?>′

?>′

⇒ 1

F8

∑

>∈�8

E8 (>) ≥1

F 9min>∈� 9

∑

>′∈� 9 \{> }E8 (> ′)

⇒ 1

F8E8 (�8 ) ≥

1

F 9min>∈� 9

E8 (� 9 \ {>}).

For the transition from the first to the second inequality in the chain, we use the definition of

U8 (p) and the assumption �8 ⊆ MBB8 (p): by the definition of U8 (p), we have U8 (p) ≥ E8 (>′)?>′

for all

> ′ ∈ $ , and by the assumption �8 ⊆ MBB8 (p), it holds that U8 (p) = E8 (>)?>

for all > ∈ �8 . The lastinequality allows us to conclude that � is WEF1. �

It is also known that if each agent 8 only purchases MBB items, so that 8 maximizes her valuationunder the budget ? (�8 ), then the corresponding allocation is Pareto optimal.

Lemma 4.5 (First Welfare Theorem; Mas-Colell et al. [1995], Chapter 16). Given a com-plete allocation � and a price vector p, suppose that agents are assigned to MBB items only, i.e.,

�8 ⊆ MBB8 (p) for each 8 ∈ # . Then � is PO.

Proof. To show that � is Pareto optimal, suppose towards a contradiction that another alloca-tion �′ Pareto dominates �. This means that E8 (�′8 ) ≥ E8 (�8 ) for all 8 ∈ # , and E 9 (�′9 ) > E 9 (� 9 )for some 9 ∈ # . Since each agent 8 maximizes her valuation under the budget ? (�8 ) in �, we have? (�′8 ) ≥ ? (�8 ) for all 8 ∈ # , and ? (�′9 ) > ? (� 9 ) for some 9 ∈ # . Indeed, if ? (�8 ) > ? (�′8 ) for some8 ∈ # , this would mean that

E8 (�8 ) = U8 (p)? (�8) > U8 (p)? (�′8 ) ≥∑

>∈�′8

E8 (>)?>

?> = E8 (�′8 ),

a contradiction. Thus, ? (�′8 ) ≥ ? (�8 ) for all 8 ∈ # . A similar argument shows that ? (�′9 ) > ? (� 9 )for some 9 ∈ # . However, this implies that

∑

>∈$?> ≥

∑

8 ∈#? (�′8 ) >

∑

8 ∈#? (�8 ) =

∑

>∈$?> ,

where the last equality holds since � is a complete allocation. We thus obtain the desired contra-diction. �

With Lemmas 4.4 and 4.5, the problem of finding a PO and WEF1 allocation reduces to that offinding an allocation and price vector pair satisfying the MBB condition and WpEF1. We showthat there is an algorithm that finds such an outcome in pseudo-polynomial time. Our algorithmfollows a similar approach as that of Barman et al. [2018]; thus the proof of Theorem 4.6 is deferredto Appendix A.


Theorem4.6. For any number of agents with additive valuations and arbitrary positive real weights,there exists a WEF1 and PO allocation. Furthermore, such an allocation can be computed in time

?>;~(<,=, E<0G ,F<0G ) for any integer-valued inputs, where E<0G := max8 ∈#,>∈$ E8 (>) and F<0G :=max8 ∈# F8 .

The outline of the algorithm is as follows. Our algorithm alternates between two phases: thefirst phase involves reallocating items from large to small spenders (where the “spending” of anagent is defined as the ratio between the price for her bundle of items and her weight; see theformal definition in Appendix A), and the second phase involves increasing the prices of the itemsowned by small spenders. We show that by increasing prices gradually, the algorithm convergesto an allocation and price vector pair satisfying the desired criteria when both input weights andvaluations are expressed as integral powers of (1 + n) for some n > 0. Similarly to Barman et al.[2018], we apply our algorithm to the n-approximate instance of the original input and show thatfor small enough n , the output of the algorithm satisfies the original MBB condition and WpEF1.We note that compared to Barman et al. [2018], the analysis becomes more involved due to thepresence of weights. In particular, each price-rise phase takes into account not only the valuationsbut also the weights; as a result, n needs to be much smaller in order to ensure the equivalence.

5 WWEF1 AND PO ALLOCATIONS: MAXIMUM WEIGHTED NASHWELFARE

In the previous section, we saw that MWNW allocations may fail to satisfy WEF1, showing that theresult of Caragiannis et al. [2019] from the unweighted setting does not extend to the weightedsetting via WEF1 (or even WEF2 for any fixed 2). Given that these allocations maximize a naturalobjective, it is still tempting to ask whether they provide any fairness guarantee. The answer isindeed positive: we show that a MWNW allocation satisfies WWEF1, a weaker fairness notion that alsogeneralizes EF1.

Theorem5.1. For any number of agents with additive valuations and arbitrary positive real weights,

a MWNW allocation is always WWEF1 and PO.

The proof of Theorem5.1 follows a similar outline as the corresponding proof of Caragiannis et al.[2019]. PO follows easily from the definition of MWNW. For WWEF1, we assume for contradiction thatan agent 8 weakly envies another agent 9 up to more than one item in a MWNW allocation. If everyagent has a positive value for every item, we pick an item in agent 9 ’s bundle for which the ratiobetween 8’s value and 9 ’s value is maximized. By distinguishing between the cases F8 ≥ F 9 andF8 ≤ F 9 , we show that we can achieve a higher weighted Nash welfare upon transferring this itemto agent 8’s bundle, which yields the desired contradiction. The case where agents may have zerovalue for items is then handled separately.

Proof of Theorem 5.1. Let� be a MWNW allocation, with #max being the subset of agents havingstrictly positive realized valuations under �. If it were not PO, there would exist an allocation �̂

such that E8 (�̂8) > E8 (�8 ) for some 8 ∈ # and E 9 (�̂ 9 ) ≥ E 9 (� 9 ) for every 9 ∈ # \ {8}. If 8 ∈ # \#max,we would have E 9 (�̂ 9 ) > 0 for every 9 ∈ #max ∪ {8}, contradicting the assumption that #max is alargest subset of agents to whom it is possible to give positive value simultaneously. If 8 ∈ #max,then

∏9 ∈#max

E 9 (�̂ 9 )F9 >

∏9 ∈#max

E 9 (� 9 )F9 , which violates the optimality of the right-hand side.This proves that � is PO.

Like Caragiannis et al. [2019], we will start by proving that� is WWEF1 for the scenario#max = #

and then address the case #max ≠ # . Assume that #max = # . If � is not WWEF1, then there existsa pair of agents 8, 9 ∈ # such that 8 has weak weighted envy towards 9 up to more than oneitem. Clearly, there must be at least two items in 9 ’s bundle that 8 values positively. Moreover, 9


must value these items positively as well—otherwise we can transfer them to 8 and obtain a Paretoimprovement.Let�8

9 := {> ∈ � 9 : E8 (>) > 0}. We construct another allocation�′ by transferring an item >∗ (tobe chosen later) from 9 to 8 so that �′8 = �8 ∪ {>∗}, �′9 = � 9 \ {>∗}, and �′A = �A for all A ∈ # \ {8, 9 }.We have

WNW(�′)WNW(�) =

(E8 (�8 ∪ {>∗})

E8 (�8 )

)F8(E 9 (� 9 \ {>∗})

E 9 (� 9 )

)F9

=

(E8 (�8 ) + E8 (>∗)

E8 (�8 )

)F8(E 9 (� 9 ) − E 9 (>∗)

E 9 (� 9 )

)F9

=

(1 + E8 (>∗)

E8 (�8 )

)F8(1 − E 9 (>∗)

E 9 (� 9 )

)F9

.

First, note that E 9 (>) > 0 for all > ∈ �89 ; otherwise the above ratio for >∗ ∈ �8

9 with E 9 (>∗) = 0

equals(1 + E8 (>∗)

E8 (�8 ))F8

> 1, contradicting the assumption that � is a MWNW allocation. However, even

under this condition, we will show that if agents 8, 9 violated the WWEF1 property, the above ratiowould still exceed 1 for some item >∗.

Case I. F8 ≥ F 9 .

Let us pick an item >∗ ∈ argmin>∈�89

E9 (>)E8 (>) specifically to transfer from 9 to 8 for changing the

allocation from � to �′. This is well-defined by the definition of �89 . Consider

[WNW(�′)WNW(�)

] 1F9

=

(1 + E8 (>∗)

E8 (�8 )

) F8F9

(1 − E 9 (>∗)

E 9 (� 9 )

),

where 1 − E9 (>∗)E9 (� 9 ) > 0 since E 9 (� 9 ) > E 9 (>∗) > 0. Moreover, we have

(1 + E8 (>∗)

E8 (�8 )

) F8F9 ≥

(1 + F8

F 9· E8 (>

∗)E8 (�8 )

)

from Bernoulli’s inequality, since E8 (>∗)E8 (�8 ) > 0 and F8

F9≥ 1. Algebraic manipulations show that

(1 + F8

F 9· E8 (>

∗)E8 (�8)

) (1 − E 9 (>∗)

E 9 (� 9 )

)> 1

⇔ E8 (�8 )F8

<

E8 (>∗)E 9 (>∗)

(E 9 (� 9 ) − E 9 (>∗)

F 9

). (5)

The latter inequality is true under our assumptions for the following reasons: Since agent 8 witha larger weight has weak weighted envy towards agent 9 with a smaller weight up to more than

one item, we have E8 (�8 )F8

<E8 (� 9 )−E8 (>∗)

F9; in addition, due to our choice of >∗,

E 9 (>∗)E8 (>∗)

≤∑

>∈�89E 9 (>)

∑>∈�8

9E8 (>)

≤∑

>∈� 9E 9 (>)∑

>∈� 9E8 (>)

=E 9 (� 9 )E8 (� 9 )

,

since∑

>∈� 9 \�89E 9 (>) ≥ 0 and

∑>∈� 9 \�8

9E8 (>) = 0. Plugging E8 (� 9 ) ≤ E8 (>∗)

E9 (>∗) · E 9 (� 9 ) into the above

strict inequality and simplifying, we obtain (5). But chaining all these inequalities together, we get[WNW(�′)WNW(�)

] 1F9

> 1 ⇒ WNW(�′) > WNW(�).


This is a contradiction, which shows that � is WWEF1 in this case.

Case II. F8 < F 9 .

As in Case I, we pick an item >∗ ∈ argmin>∈�89

E9 (>)E8 (>) , so we also have E8 (� 9 ) ≤ E8 (>∗)

E9 (>∗) · E 9 (� 9 ).Consider

[WNW(�′)WNW(�)

] 1F8

=

(1 + E8 (>∗)

E8 (�8 )

) (1 − E 9 (>∗)

E 9 (� 9 )

) F9F8

,

where 1 − E9 (>∗)E9 (� 9 ) > 0 since E 9 (� 9 ) > E 9 (>∗) > 0. Moreover, we have

(1 − E 9 (>∗)

E 9 (� 9 )

) F9F8

≥(1 − F 9

F8· E 9 (>

∗)E 9 (� 9 )

)

from Bernoulli’s inequality, sinceF9

F8> 1 and 0 <

E9 (>∗)E9 (� 9 ) < 1. Algebraic manipulations show that

(1 + E8 (>∗)

E8 (�8 )

) (1 − F 9

F8· E 9 (>

∗)E 9 (� 9 )

)> 1

⇔ E8 (�8 ) + E8 (>∗)F8

<

E8 (>∗)E 9 (>∗)

· E 9 (� 9 )F 9

. (6)

Since agent 8 with a smaller weight has weak weighted envy towards agent 9 with a larger weight

up to more than one item, we have E8 (�8 )+E8 (>∗)F8

<E8 (� 9 )F9

. Plugging E8 (� 9 ) ≤ E8 (>∗)E9 (>∗) · E 9 (� 9 ) into this

strict inequality, we obtain (6), and chaining all inequalities leads us to the contradiction WNW(�′) >WNW(�). This completes the proof for the scenario where #max = # .

The rest of the proof mirrors the corresponding part in the proof of Caragiannis et al. [2019].If #max ( # , it is easy to see that there can be no weighted envy towards any 8 ∉ #max sinceE 9 (�8 ) = 0 for any such 8 and every 9 ∈ # . Also, for any 8, 9 ∈ #max, we can show as in the prooffor#max = # above that there cannot be (weak) weighted envy up to more than one item. Supposefor contradiction that an agent 8 ∈ # \ #max is not weighted envy-free towards some 9 ∈ #max upto one item under �. This means that 8 still has positive value for 9 ’s bundle even after removingany single item; in particular, 8 values at least two items in � 9 positively. Since 9 must also valuethese items positively, we may transfer one of them to 8 and keep both 8 and 9 ’s valuations positive.This contradicts the maximality of #max. Hence, 8 ∈ # \#max must be weakly weighted envy-freeup to one item towards 9 ∈ #max. It follows that � is WWEF1 in all cases. �

6 WEF1 AND OTHER FAIRNESS NOTIONS

An allocation that satisfies multiple fairness guarantees is naturally desirable but often elusive inthe setting with indivisible items. Hence, wewill now explore the implications of the WEF1propertyon the other fairness criteria defined in Section 2.For additive valuations, Aziz et al. [2020] provided a polynomial-time algorithm for computing

a PO and WPROP1 allocation, whereas we proved the existence of PO and WEF1 allocations in Sec-tion 4.2. It is straightforward to verify that, in the unweighted scenario, any complete envy-freeallocation is also proportional for subadditive valuations and any complete EF1 allocation is PROP1for additive valuations (see, e.g., [Aziz et al., 2020]). This begs the question: does the WEF1 propertyalong with completeness also imply the WPROP1 condition? Unfortunately, the answer is no for any= ≥ 3—in fact, we establish a much stronger result in the following proposition.


Proposition 6.1. For any number = ≥ 2 of agents with additive valuations and arbitrary positivereal weights, any complete WEF1 allocation is WPROP(= − 1). However, for any = ≥ 3, there exists aninstance in which no complete WEF1 allocation is WPROP(= − 2).Proof. Let � be a complete WEF1 allocation under additive valuations, and fix an agent 8 . By

definition of WEF1, for each 9 ∈ # \ {8}, there is a set ( 9 ⊆ � 9 with |( 9 | ≤ 1 such thatF9

F8·

E8 (�8 ) ≥ E8 (� 9 ) −E8 (( 9 ). Summing up the respective inequalities for all 9 ∈ # \ {8} and noting that$ = ∪9 ∈#� 9 due to completeness, we have

∑9 ∈# \{8 }F 9

F8· E8 (�8 ) ≥ E8 ($) − E8 (�8 ) −

∑

9 ∈# \{8 }E8 (( 9 ).

If we let ( := ∪9 ∈# \{8 }( 9 , the above inequality implies that∑

9∈# F9

F8· E8 (�8 ) ≥ E8 ($) − E8 ((). It

follows that

E8 (�8 ) ≥F8∑9 ∈# F 9

· E8 ($) −F8∑9 ∈# F 9

· E8 (() ≥F8∑9 ∈# F 9

· E8 ($) − E8 ((),

where the latter inequality follows from F8∑9∈# F9

< 1 and E8 (() ≥ 0. Since |( | ≤ = − 1, this showsthat � is WPROP(= − 1).For the second part, let 0 < n <

1= (=−1) , and consider an instance with = ≥ 3 agents where

F1 = 1 − (= − 1)n and F8 = n for all 8 ∈ # \ {1}. Moreover, assume that there are < = = items,E1(>) = 1

= for every > ∈ $ , and every other agent has an arbitrary positive value for each item.Let � be any complete WEF1 allocation. Note that every agent must receive exactly one item in�—otherwise there would be an agent with no item and another agent with two or more items,and these two agents would violate the WEF1 property. Then, for any set ( ⊆ $ \�1 with |( | ≤ =−2,we have

F1∑9 ∈# F 9

E1 ($) − E1(() ≥1 − (= − 1)n

1· 1 − = − 2

==

2

=− (= − 1)n >

2

=− 1

==

1

== E1(�1).

Hence, � is not WPROP(= − 2). �

For = symmetric (unweighted) agents with additive valuations, Amanatidis et al. [2018, Prop.3.6] showed that any complete EF1 allocation is 1

=-MMS, and this approximation guarantee is tight.

Moreover, as Caragiannis et al. [2019, Thm. 4.1] proved, amaximumNashwelfare allocation, whichis EF1 and PO, is alsoΘ(1/√=)-MMS. This means that, for a small number of agents, the EF1 propertyprovides a reasonable approximation to MMS fairness. Our next proposition stands in stark contrastto these results: For any number of agents with asymmetric weights, it shows that the WEF1 condi-tion does not imply any positive approximation of the WMMS guarantee, even in conjunction withPareto optimality.

Proposition 6.2. For any constant n > 0 and any number = ≥ 2 of agents, there exists an instancewith additive valuations in which some PO and WEF1 allocation is not n-WMMS.

Proof. Suppose < = = and the weights are F1 > F2 > · · · > F= > 0 with F=/F1 < n . Thevaluation functions are defined by

E1(>A ) = FA ∀A ∈ [=];

E8 (>A ) ={1 if A = 8 − 10 otherwise

∀8 ∈ # \ {1}.

One can check that WMMS1 = F1, obtained from the allocation that gives item >8 to agent 8for every 8 ∈ # . Moreover, the allocation � in which agent 1 receives >= and agent 8 receives


>8−1 for every 8 ∈ # \ {1} is PO and WEF1. The WMMS approximation ratio for agent 1 under � isE1(>=)/WMMS1 = F=/F1 < n . �

For the special case of = = 2 agents, the first part of Proposition 6.1 implies that the output ofthe weighted adjusted winner procedure from Section 4.1 is always WPROP1. However, we showthat it comes with no guarantee on the WMMS approximation.

Proposition 6.3. The output of the weighted adjusted winner procedure (Algorithm 2) is alwaysWPROP1. However, for any constant n > 0, it is not necessarily n-WMMS.

Proof. By Theorem 4.2, Algorithm 2 produces a complete WEF1 allocation for two agents; suchan allocation must be WPROP1 due to Proposition 6.1.Consider an instance with< = = = 2 and weights F1 > F2 > 0 with F2/F1 < n . Let X > 0 be

such thatF2/F1 > X/(1−X). Suppose that E1 (>8) = F8 for 8 ∈ {1, 2}, E2 (>1) = 1−X , and E2(>2) = X .Since

E1(>2)E2(>2)

=F2

X>

F1

1 − X =E1(>1)E2(>1)

,

Algorithm 2 considers the items in the order >2, >1. Then agent 1 receives >2 and agent 2 receives>1. Hence, agent 1’s WMMS approximation ratio is F2/F1 < n . �

Farhadi et al. [2019] showed that a 1/= approximation of the WMMS can be obtained using thecanonical (unweighted) round-robin algorithmwith the added requirement that agentswith higherweights go first; it is not hard to check that this algorithm does not guarantee WWEF2 or WPROP2 forany constant 2 . On the other hand, while our weighted picking sequence from Section 3.2 ensuresWEF1, perhaps surprisingly, it does not come with any WMMS approximation guarantee.

Proposition 6.4. For any constant n > 0, the output of the Weighted Picking Sequence protocol

(Algorithm 1) is not necessarily n-WMMS.

Proof. Consider an instance with= = 2 and< = :+1 for a positive integer : > 1/n . Theweightsare F1 = : and F2 = 1, and both agents have identical valuations with value :2 for one item and1 for each of the remaining : items. We have WMMS1 = :2 and WMMS2 = : . The picking sequenceinduced by the algorithm is 1, 2, 1, 1, . . . , 1, so agent 2 only receives value 1, which is 1/: < n ofher WMMS. In fact, even if we let agent 2 pick first, the picking sequence will be 2, 1, 1, . . . , 1. In thiscase, agent 1 only receives value : , which is again 1/: < n of her WMMS. �

7 EXPERIMENTS

Thus far, we have extensively investigated the existence and computational properties of approx-imations to weighted envy-freeness (WEF). While the WEF notion itself cannot always be satisfiedwith indivisible items, an interesting question is how “likely” it is for a problem instance withweighted agents to admit a WEF allocation, and how the results compare to those for envy-freenessin the unweighted setting.In this section, we approach this question experimentally by generating sets of 1000 instances

with = ∈ {2, 3, 4, 5} agents and< ∈ {2, 3, . . . , 9} items wherein each agent’s value for each item isdrawn independently from a distribution. We perform our experiments on three common familiesof distributions—uniform, exponential, and log-normal—and consider two different weight vectors:F8 = 1 (unweighted) and F8 = 8 (weighted) for every 8 ∈ # . For each generated instance, wedetermine by exhaustive search over all allocations whether a WEF allocation exists. The resultsare shown in Figure 2; the main observations are as follows:

(1) For each fixed distribution and number of agents/items, weighted envy-free allocations arealmost always harder to find than their unweighted analogs. Intuitively, we need more items


2 3 4 5 6 7 8 9Number of items

0

20

40

60

80

100

% o

f E

F/W

EF

-ad

mit

tin

g in

stan

ces

2-UW3-UW4-UW5-UW2-W3-W4-W5-W


0

20

40

60

80

100

% o

f E

F/W

EF

-ad

mit

tin

g in

stan

ces

(a) Uniform distribution on [0, 1] (b) Exponential distribution with mean 1


0

10

20

30

40

50

60

70

80

90

100

% o

f E

F/W

EF

-ad

mit

tin

g in

stan

ces


0

20

40

60

80

100%

of

EF

/WE

F-a

dm

itti

ng

inst

ance

s

(c) Exponential distribution with mean 2 (d) Log-normal dist. with parameters (0, 1)


0

10

20

30

40

50

60

70

80

90

100

% o

f E

F/W

EF

-ad

mit

tin

g in

stan

ces


0

10

20

30

40

50

60

70

80

90

100

% o

f E

F/W

EF

-ad

mit

tin

g in

stan

ces

(e) Log-normal dist. with parameters (1, 1) (f) Log-normal dist. with parameters (0, 2)

Fig. 2. Percentages of instances that admit EF and WEF allocations for different valuation distributions in our

experiments; =-UW, depicted by dashed curves (resp., =-W, depicted by solid curves) refers to a scenario with

= agents with equal weights (resp., weights proportional to agent indices) for all figures.


to satisfy agents with larger weights, and this is not sufficiently compensated by the itemsthat we can save through agents with smaller weights.

(2) The more items we have, the more likely it is that a fair allocation exists. This is to beexpected, since more items means higher flexibility in choosing the allocation.

(3) By contrast, the more agents there are, the less likely it is that a fair allocation exists. This isagain reasonable, as we need to satisfy a larger number of preferences when there are moreagents.

(4) Fair allocations are rarer when the distribution is uniform than when it is skewed. This ob-servation aligns with the intuition that in a uniform distribution the values are more evenlydistributed, so it is harder to find items for which one agent has high value whereas theremaining agents have low value.

Our experimental results illustrate the difficulty of achieving weighted envy-freeness and furtherjustify our quest for the (strong and weak) relaxations of the WEF property.

8 DISCUSSION AND FUTUREWORK

In this article, we have introduced and studied envy-based notions for the allocation of indivisibleitems in a general setting where agents can have different entitlements. As most of our resultshold for additive valuation functions, the reader may wonder whether they can be extended tomore general classes—after all, in the absence of weights, an EF1 allocation is known to exist forarbitrary monotone valuations [Lipton et al., 2004]. We therefore point out some hurdles that wefaced while trying to generalize our weighted envy concepts beyond additive valuations. First, weshow that even for simple non-additive valuations, the existence of a WEF1 or WWEF1 allocation canno longer be guaranteed. Since WWEF1 is weaker than WEF1, it suffices to prove the claim for WWEF1.

Proposition 8.1. There exists an instance with = = 2 agents such that one of the agents has a

(normalized and monotone) submodular valuation,8 the other agent has an additive valuation, and a

complete WWEF1 allocation does not exist.

Proof. Consider an instance with two agents who haveweightsF1 = 1 andF2 = 2, and supposethat there are < > 5 items. The valuation functions are given by E1(() = |( | and E2(() = 1 forevery ( ∈ 2$ \ ∅, and E1(∅) = E2(∅) = 0. The functions are normalized and monotone, with E1additive and E2 submodular.For any complete allocation �, if |�1 | ≥ 2, we have E2 (�1 \ {>})/F1 = 1 > 1/2 ≥ E2(�2)/F2 for

every > ∈ �1. Moreover, it holds that E2(�2 ∪ {>})/F2 = 1/2 < 1 = E2(�1)/F1 for every > ∈ �1.Thus, the only way to make agent 2 weakly weighted envy-free up to one item towards agent 1is to ensure that |�1 | ≤ 1. Assume without loss of generality that �1 = {>1} (if �1 = ∅, agent 1will be even worse off in the argument that follows), so �2 = $ \ {>1}. We have E1(�1) = 1 andE1(�2) = |�2 | = < − 1. Since agent 1 has an additive valuation and a smaller weight than agent2, she would be weakly weighted envy-free up to one item towards agent 2 if and only if thereis an item > ∈ �2 such that E1 (�1 ∪ {>})/F1 ≥ E1(�2)/F2. However, for any > ∈ �2, we haveE1(�1 ∪ {>})/F1 = E1({>1, >}) = 2, whereas E1(�2)/F2 = (< − 1)/2 > 2 since< > 5. This meansthat no complete allocation can be WWEF1. �

By increasing the lower bound on the number of items in the instance of this proof to 52 , onecan show that a complete WWEF2 allocation is also not guaranteed to exist for any constant 2 .One of the key ideas in our analysis of the maximum weighted Nash welfare allocation (Theo-

rem 5.1) is what we call the transferability property: If agent 8 has weighted envy towards agent 9

8A valuation function E : 2$ → R≥0 is said to be submodular if for any $1 ⊆ $2 ⊆ $ and any item > ∈ $ \$2, we haveE ($1 ∪ {> }) − E ($1) ≥ E ($2 ∪ {> }) − E ($2) .


under additive valuations, then there is at least one item > in 9 ’s bundle for which agent 8 has pos-itive (marginal) valuation—in other words, the item > could be transferred from 9 to 8 to augment8’s realized valuation.9 Unfortunately, this property no longer holds for non-additive valuations.

Proposition 8.2. There exists an instance such that an agent 8 with a non-additive valuation

function has weighted envy towards an agent 9 under some allocation �, but there is no item in 9 ’sbundle for which 8 has positive marginal valuation—i.e., �> ∈ � 9 such that E8 (�8 ∪ {>}) > E8 (�8 ).Proof. Consider the example in Proposition 8.1. Under any allocation with |�1 | = < − 1 and|�2 | = 1, agent 2 hasweighted envy towards agent 1 since E2(�2) = 1/2 < 1 = E2(�1)/F1. However,E2(�2 ∪ {>}) = 1 = E2(�2) for every > ∈ �1. �

In light of these negative results, an important direction for future research is to identify ap-propriate weighted envy notions for non-additive valuations. Other interesting directions includeestablishing conditions under which WEF allocations are likely to exist (cf. Section 7),10 investigat-ing weighted envy in the allocation of chores (items with negative valuations) [Aziz et al., 2019b] orcombinations of goods and chores [Aziz et al., 2019a, 2020], incorporating connectivity constraints[Bei et al., 2021, Bilò et al., 2019, Bouveret et al., 2017], and considering weighted versions of otherenvy-freeness approximations such as envy-freeness up to any item (EFX) [Caragiannis et al., 2019,Plaut and Roughgarden, 2020]. From a broader point of view, our work demonstrates that fair di-vision with different entitlements is richer and more challenging than its traditional counterpartin several ways, and much interesting work remains to be done.

ACKNOWLEDGMENTS

This work was partly done while Chakraborty and Zick were at the National University of Singa-pore, Igarashi was at the University of Tokyo, and Suksompong was at the University of Oxford.Chakraborty and Zick were supported by the Singapore NRF Research Fellowship R-252-000-750-733. Igarashi was supported by the KAKENHI Grant-in-Aid for JSPS Fellows no. 18J00997 andJST, ACT-X. Suksompong was supported by the European Research Council (ERC) under grantno. 639945 (ACCORD); his visit to NUS was supported by MOE Grant R-252-000-625-133. Prelimi-nary versions of the article appeared in Proceedings of the 19th International Conference on Au-tonomous Agents and Multiagent Systems, May 2020, and Proceedings of the 2nd Games, Agents,and Incentives Workshop, May 2020. We would like to thank the associate editor and the anony-mous reviewers for several helpful comments.

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A PROOF OF THEOREM 4.6

A.1 Preliminaries

Before we proceed to present the algorithm formally, we introduce the following definitions andnotations. We write the weighted version of the least spending as

wmin(�,p) := min8 ∈#

1

F8? (�8 ).

We call 8 ∈ # with 1F8? (�8) = wmin(�,p) a least weighted spender. Also, we say that 9 is a violator

if some agent weighted-price-envies 9 by more than one item, i.e., � 9 ≠ ∅ and

min>∈� 9

1

F 9? (� 9 \ {>}) > wmin(�,p).

Given n ∈ (0, 1), we say that 9 is an n-violator if � 9 ≠ ∅ and

min>∈� 9

1

F 9? (� 9 \ {>}) > (1 + n)wmin(�,p).

We write the maximum value of the left-hand side as

wmax−1(�,p) := max9 ∈# :� 9≠∅

min>∈� 9

1

F 9? (� 9 \ {>}).

The pair (�,p) is said to be n-weighted price EF1 (n-WpEF1) if no agent is an n-violator.


Now, we define the maximum bang per buck (MBB) network that represents how items can beexchanged among agents without losing Pareto optimality. For an allocation � and a price vectorp, we define the MBB network � (�,p) to be a directed graph where the vertices are given by theagents # and items$ , and the arcs are given as follows:

• there is an arc from agent 8 to item > if > ∈ �8 ; and• there is an arc from item > to agent 8 if > ∈ MBB8 (p) \�8 .

We say that 9 ∈ # ∪$ can reach 8 ∈ # ∪$ if there is a directed path from 9 to 8 in � (�,p). For- ⊆ # , we write MBB(p, - ) = ⋃

8 ∈- MBB8 (p).For a directed path % starting from agent 9 in a MBB network, we denote by > ( 9 , %) the item

owned by 9 on the path % , that is, ( 9 , > ( 9 , %)) ∈ % . For each agent 8 ∈ # , we say that 9 ∈ # is ann-path-violator for 8 if there is a directed path % from 9 to 8 in � (�,p) and 9 ’s weighted spending isgreater than the n-approximate weighted spending of 8 even after removing > ( 9 , %) from 9 ’s bundle,i.e.,

1

F 9? (� 9 \ {> ( 9 , %)}) >

1 + nF8

? (�8 ).

Barman et al. [2018] showed that the number of “swap operations”, which reassign items alongpaths, is bounded by a polynomial in the input size based on a potential function argument. LikeBarman et al., we define a potential function Φ8 with respect to each agent 8 . Let 8 ∈ # be a “rootagent”. For each 9 ∈ # \ {8}, we define the level of 9 , denoted by ℎ8 ( 9 ,�,p), to be half of the lengthof a shortest path from 9 to 8 if there is a directed path from 9 to 8 in � (�,p), and = − 1 otherwise.We say that > ∈ � 9 is 9 ’s critical item with respect to 8 if there is a shortest path % in � (�,p) from9 to 8 where > = > ( 9 , %); we let�8 ( 9 ,�,p) be the set of critical items of 9 with respect to 8 . We nowdefine our potential function Φ8 . For each pair of allocation � and price vector p, we let

Φ8 (�,p) :=∑

9 ∈# \{8 }68 ( 9 ,�,p), (7)

where 68 ( 9 , �,p) := <(= − ℎ8 ( 9 , �,p)) + |�8 ( 9 , �,p) | for each 9 ∈ # \ {8}. It is easy to see thatΦ8 is always non-negative and bounded above by <=2 because 1 ≤ ℎ8 ( 9 ,�,p) ≤ = − 1 and0 ≤ |�8 ( 9 , �,p) | ≤ <. Intuitively, 68 ( 9 , �,p) lexicographically orders allocations under prices p:it decreases when (1) agent 9 gets further away from 8 in the MBB network, or (2) the distancebetween 8 and 9 does not change but the number of critical items possessed by 9 gets smaller. Thisleads to the following lemma: each reassignment of a critical item to an agent who is closer to theroot agent results in a decrease in the potential function [Barman et al., 2018].

Lemma A.1 (Proof of Lemma 13 in the extended version of [Barman et al., 2018]). Given

an allocation � and a price vector p, suppose that % = ( 9 , >1, 81, . . . , >: , 8) is a shortest path from 9 to

8 in � (�,p). Let �′ be the allocation resulting from reassigning >1 from 9 to 81, i.e., �′9 = � 9 \ {>1},

�′81 = �81 ∪ {>1}, and �′8′ = �8′ for all 8′ ∈ # \ { 9 , 81}. Then Φ8 (�,p) − 1 ≥ Φ8 (�′,p).

A.2 Algorithm

We are now ready to present the algorithm (Algorithm3). To bound the number of steps, we assumefor the time being that the input valuations as well as weights are integral powers of some positivevalue (1 + n) with n ∈ (0, 1); later, we will show that for the n-rounded version of a given instance,the algorithm returns an allocation that is Pareto optimal and WEF1 for the original instance if n issmall enough.

A.2.1 Overview of Algorithm 3. Initially, the algorithm allocates each item to an agent who valuesit most, and sets the price of each item to the value that the assigned agent derives from the item.At this point, the bang per buck ratio of each item for the assigned agent is 1, and agents receive


ALGORITHM 3: Algorithm for constructing a PO and WEF1 allocation

Require: For each > ∈ $ , ∃8 ∈ # with E8 (>) > 0 (so that initial prices are positive). The valuations areintegral powers of (1 + n) for some n ∈ (0, 1) (i.e., for each > ∈ $ and 8 ∈ # , there exist integers 08> ≥ 0and 18 ≥ 0 such that E8 (>) = (1 + n)08> andF8 = (1 + n)18 ).

1: ?> ← max8 ∈# E8 (>) for each > ∈ $ .2: Assign each item > to an agent who values it most.3: Initialize 8∗ ∈ argmin8 ∈#

1F8? (�8).

4: while � is not 3n-WpEF1 with respect to p do

5: if 8∗ is not a least weighted spender then6: Update 8∗ ∈ argmin8 ∈#

1F8? (�8).

7: end if

/*Swap phase*/

8: if there is an n-path-violator for 8∗ in � (�, ?) then9: Choose an n-path-violator 9 who has a shortest path % = (80, >1, 81, . . . , >: , 8: ) where 80 = 9 and

8: = 8∗ in � (�,p), i.e., 1F80

? (�80 \ {>1}) > 1+nF8:

? (�8: ) and 1F8C

? (�8C \ {>C+1}) ≤ 1+nF8:

? (�8: ) forC = 1, 2, . . . , : − 1.

10: ℎ ← 011: while 1

F8ℎ? (�8ℎ \ {>ℎ+1}) > 1+n

F8∗? (�8∗) and ℎ ≤ : − 1 do

12: �8ℎ ← �8ℎ \ {>ℎ+1} and �8ℎ+1 ← �8ℎ+1 ∪ {>ℎ+1} /*apply the swap operation as long as

agent 8ℎ is an n-path-violator*/

13: ℎ ← ℎ + 114: end while

/*Price-rise phase*/

15: else

16: Set � to be the set of agents who can reach 8∗ in � (�,p) (the set � includes 8∗).17: G ← wmax−1 (�,p)

wmin (�,p)18: ~ ← min

{U8 (p)?>E8 (>)

�� 8 ∈ � ∧ > ∈ $ \MBB(p, � )}

19: I ← min{

? (� 9 )F9wmin(�,p)

�� 9 ∈ # \ �}

20: if wmin(�,p) > 0 and G = min{G,~, I} then21: W ← G /*raise prices of items in MBB(p, � ) until � becomes 3n-WpEF1*/22: else if wmin(�,p) = 0 or ~ = min{G,~, I} then23: W ← ~ /*raise prices of items in MBB(p, � ) until a new MBB edge between � and

$ \MBB(p, � ) appears*/24: else

25: /*wmin(�,p) > 0 and I = min{G,~, I}*/26: W ← (1 + n) /*raise prices of items in MBB(p, � ) by a factor of (1 + n) if some agent

outside � becomes the least weighted spender when raising prices by G or ~*/

27: end if

28: ?> ← W?> for each > ∈ MBB(p, � )29: end if

30: end while

31: return �

their MBB items only. The initial prices are positive if we assume that for each item, there is anagent who has a positive value for it (any item that has zero value to all agents can be thrownaway without loss of generality). The algorithm then iterates between a swap phase (from Line 8to Line 14) and a price-rise phase (from Line 16 to Line 28).


In the swap phase, the algorithm transfers items along a path to the least weighted spenderfrom a corresponding n-path-violator, reducing inequality in the weighted spending under thefixed prices. The transfer continues as long as the agent who gets a new item is an n-path-violator(Line 11). If no item can be transferred from a violator to the least weighted spender in the currentMBB network (namely, the least weighted spender has no n-path-violator), the algorithm entersthe price-rise phase.In the price-rise phase, we aim to increase reachability between the least weighted spender and

items possessed by violators. To this end, the algorithm uniformly raises prices of MBB itemsMBB(p, � ) = ⋃

8 ∈� MBB8 (p) for the set � of agents who can reach the least weighted spender 8∗.Specifically, as long as 8∗ remains the least weighted spender, the algorithm raises prices until oneof the following holds:

• wmin(�,p) exceedswmax−1(�,p) (in the casewhere the least weighted spending is positiveand G = min{G,~, I}); or• a new MBB edge appears between the agents in � and the items in$ \MBB(p, � ) (in the casewhere the least weighted spending is 0 or ~ = min{G,~, I}),

where G , ~, and I are the multiplicative factors defined in Line 17, 18, and 19, respectively. Notethat as all prices are positive, and $ \ MBB(p, � ) and # \ � are nonempty (we will prove thisin Lemma A.7), ~ and I are well-defined. If the identity of 8∗ changes when increasing prices by afactor of G or~ (i.e., the least weighted spending is positive and I = min{G,~, I}), then the algorithmraises prices by (1 + n). The scaling with G ensures that the resulting outcome is 3n-WpEF1 (see thesubsequent proof for Corollary A.13). Thus, up until the final step, the algorithm continues toraise prices by either ~ or (1 + n), both of which are powers of (1 + n) during the execution of thealgorithm. By doing so, the algorithm keeps the invariant that the prices are powers of (1+n). Wewill argue that wmax−1(�,p) never increases while wmin(�,p) increases by a factor of (1 + n)after a polynomial number of steps, thereby showing that the difference between the value ofwmin(�,p) and that of wmax−1(�,p) becomes negligibly small after a finite number of steps.

A.2.2 Swap phase. First, consider the swap phase starting with Line 8 and ending with Line 14 ofAlgorithm 3. Suppose that % = (80, >1, 81, . . . , >: , 8:) is the shortest path selected by the algorithmfrom the n-path-violator 9 = 80 to the least weighted spender 8∗ = 8: . Let �′ be the resultingallocation after swapping items along the path % by applying Lines 11–14; see Figure 3 for anillustration. An immediate observation is that the path-violator loses at least one item after theswap:

Lemma A.2. �′80 ( �80 .

Further, the swap operation preserves the property that each intermediate agent on the path %

is a non-violator but their weighted spending does not go below the minimum weighted spendingwmin(�,p) with respect to �.

Lemma A.3. For each ℎ = 1, . . . , : , we have 1F8ℎ

? (�′8ℎ) ≥ wmin(�,p) and (1 + n)wmin(�,p) ≥min>∈�′8ℎ

1F8ℎ

? (�′8ℎ \ {>}).

Proof. The claim trivially holds for the least weighted spender 8: = 8∗ as she receives at mostone item >: and does not lose any item by the swap operation. Consider agent 8ℎ with 0 < ℎ < : .By the choice of % , agent 8ℎ is not an n-path-violator in �. Thus, removing >ℎ+1 from the bundle of8ℎ decreases its weighted value to or below (1 + n)wmin(�,p), i.e.,

(1 + n)wmin(�,p) ≥ 1

F8ℎ

? (�8ℎ \ {>ℎ+1}). (8)


80 81 82 83 · · · · · · 8ℎ−1 8ℎ · · · · · · 8:−1 8:

>1 >2 >3 · · · · · · >ℎ−1 >ℎ >ℎ+1 · · · · · · >:

80 81 82 83 · · · · · · 8ℎ−1 8ℎ · · · · · · 8:−1 8:

>1 >2 >3 · · · · · · >ℎ−1 >ℎ >ℎ+1 · · · · · · >:

Fig. 3. Example of a swap operation. The thick lines correspond to the pairs (8C , >C+1) such that agent 8C ownsitem >C+1; the do�ed lines correspond to the pairs (>C , 8C ) such that item >C is an MBB item for 8C but is not

owned by 8C . The upper figure (respectively, the lower figure) represents the MBB network before the swap

operation (respectively, a�er the swap operation) along % = (80, >1, 81, . . . , >: , 8: ).

Now consider the three cases

(1) �′8ℎ = �8ℎ ;(2) �′8ℎ = �8ℎ ∪ {>ℎ};(3) �′8ℎ = (�8ℎ ∪ {>ℎ}) \ {>ℎ+1}.In case (1), the claim clearly holds due to the fact that 1

F8ℎ

? (�8ℎ ) ≥ wmin(�,p) and inequality

(8). In case (2), the claim holds because of the fact that 1F8ℎ

? (�′8ℎ) ≥1

F8ℎ

? (�8ℎ ) ≥ wmin(�,p) andthewhile-condition in Line 11. In case (3), we have�8ℎ \ {>ℎ+1} = �′8ℎ \ {>ℎ}. Combining this withinequality (8) yields

(1 + n)wmin(�,p) ≥ 1

F8ℎ

? (�8ℎ \ {>ℎ+1})

=1

F8ℎ

? (�′8ℎ \ {>ℎ}) ≥ min>∈�′8ℎ

1

F8ℎ

? (�′8ℎ \ {>}).

By the while-condition in Line 11, we further have 1F8ℎ

? (�′8ℎ) =1

F8ℎ

? ((�8ℎ ∪ {>ℎ}) \ {>ℎ+1}) ≥(1 + n)wmin(�,p). This proves the claim. �

As a corollary of the above lemmas, assuming that wmax−1(�,p) ≥ (1 + n)wmin(�,p) (whichalways holds before a swap phase due to thewhile-condition in Line 4), the function wmax−1 doesnot increase due to the swap operations along % ; further, the weighted spending wmin(�,p) ofthe least weighted spender does not decrease, and strictly increases only if agent 8: receives item>: .

Corollary A.4. Suppose that wmax−1(�,p) ≥ (1 + n)wmin(�,p). Then, wmax−1(�,p) ≥wmax−1(�′,p).

Proof. The spending of agents who do not appear in % does not change; so consider agent 8ℎon % . For the n-path-violator 80, Lemma A.2 implies

wmax−1(�,p) ≥ min>∈�80

1

F80

? (�80 \ {>}) ≥ min>∈�′80

1

F80

? (�′80 \ {>}).


For agent 8ℎ with ℎ > 0, by Lemma A.3, we have

wmax−1(�,p) ≥ (1 + n)wmin(�,p) ≥ min>∈�′8ℎ

1

F8ℎ

? (�′8ℎ \ {>}).

This proves the claim. �

Corollary A.5. It holds that wmin(�,p) ≤ wmin(�′,p). Further, wmin(�,p) < wmin(�′,p)only if the least weighted spender 8: receives item >: , i.e., �

′8:= �8: ∪ {>: }.

Proof. By definition, 1F80

? (�′80) ≥ wmin(�,p). Also, by Lemma A.3, 1F8ℎ

? (�′8ℎ ) ≥ wmin(�,p)for ℎ = 1, 2, . . . , : . Since no other agent changes her bundle, we have wmin(�,p) ≤ wmin(�′,p).It is also trivial to see that wmin(�,p) < wmin(�′,p) only if agent 8: receives item >: , i.e., �′8: =

�8: ∪ {>: }. �

We now show that the potential function decreases by at least 1 after the swap operationsalong % .

LemmaA.6 (Lemma 13 in the extendedversion of [Barman et al., 2018]). We haveΦ8∗ (�,p)−1 ≥ Φ8∗ (�′,p).

Proof. Suppose that the exchange terminates when giving item >ℓ to agent 8ℓ with 1 ≤ ℓ ≤ : .Applying Lemma A.1 repeatedly to the pair of agents 8ℎ and 8ℎ+1 for 0 ≤ ℎ ≤ ℓ − 1, we get thatΦ8∗ (�,p) − 1 ≥ Φ8∗ (�′,p). �

A.2.3 Price-rise phase. Next, let us consider the price-rise phase starting with Line 16 and endingwith Line 28 of Algorithm 3. Let (�,p) be the pair of allocation and price vector just before theprice-rise phase. Let 8∗ be the corresponding least weighted spender defined in Line 6, � be theset of agents who can reach 8∗ (� includes 8∗), and G,~, I be the multiplicative factors defined inLines 17, 18, and 19. Furthermore, let

W :=

G if wmin(�,p) > 0 and G = min{G,~, I};~ if wmin(�,p) = 0 or ~ = min{G,~, I};1 + n otherwise.

Note that if wmin(�,p) = 0, then we always set W = ~, so we do not have to worry about thewell-definedness of G in this case.

We assume that up until this point,

• for each item > ∈ $ , ?> = (1 + n)0 for some non-negative integer 0; and• � is not 3n-WpEF1with respect to p, i.e., wmax−1(�,p) > (1 + 3n)wmin(�,p); and• � is complete, and �8 ⊆ MBB8 (p) for each 8 ∈ # .

Let p′ be the new price vector after the price-update where ? ′> = W?> for each > ∈ MBB(p, � ) and? ′> = ?> for each > ∈ $ \MBB(p, � ). We prove the following auxiliary lemmas. First, we observethat since (�,p) admits a violator, there is an item that will not experience the price-update.

Lemma A.7. $ \MBB(p, � ) ≠ ∅. In particular, # \ � ≠ ∅, and$ \MBB8 (p) ≠ ∅ for each 8 ∈ � .Proof. Suppose towards a contradiction that $ \MBB(p, � ) = ∅, i.e., MBB(p, � ) = $ . Take an

n-violator 9 (so � 9 ≠ ∅) such that wmax−1(�,p) = min>∈� 9? (� 9 \ {>}). Then, since every item

in � 9 belongs to MBB(p, � ), 9 has a directed path to the least weighted spender 8∗ via any itemin � 9 . Thus, 9 ∈ � . This means that 9 is an n-path-violator for 8∗ as the weighted spending of 9 isstrictly higher than (1 + n)wmin(�,p) even after removing any item from � 9 . However, by theif-condition in Line 8, 9 must not be an n-path-violator for 8∗, a contradiction. Since every agentonly receives her MBB items in �, and � is a complete allocation, we have # \ � ≠ ∅. �


Another observation is that the price of each item in MBB(p, � ) increases by a power of (1 + n)when W ≠ G , and the price of each item not in MBB(p, � ) does not change after the price-rise phase.Further, the price-rise phase makes MBB edges from items in MBB(p, � ) to agents outside � vanishwhile preserving the other edge structures.

Lemma A.8. The following properties hold:

(i) ~ = (1 + n)0 for some positive integer 0.

(ii) For each 8 ∉ � and each > ∈ �8 , it holds that ?> = ? ′> .(iii) For each > ∈ MBB(p, � ) and each 8 ∈ # \ � with �8 ≠ ∅, item > does not have a directed path

to agent 8 in � (�,p ′). In particular, > ∉ MBB8 (p ′).Proof. (i) Observe that by definition of the maximum bang per buck ratio, U8 (p) > E8 (>)

?>for

any 8 ∈ # and > ∉ MBB8 (p); from Lemma A.7 we have~ > 1. Since the prices and valuationsare assumed to be integral powers of (1 + n), ~ is also an integral power of (1 + n) and thusthe statement (i) holds.

(ii) We know that ?> ≠ ? ′> only if > ∈ MBB(p, � ). If some > ∈ �8 with 8 ∉ � belongs to MBB(p, � ),namely > ∈ MBB8′ (p) for some agent 8 ′ ∈ � , then 8 has a directed path to the least weightedspender 8∗ which starts with the edges (8, >) and (>, 8 ′) followed by a directed path from 8 ′

to 8∗, a contradiction.(iii) We will first show that > ∉ MBB8 (p ′) for each > ∈ MBB(p, � ) and each 8 ∉ � with �8 ≠ ∅.

Suppose otherwise, i.e., > ∈ MBB8 (p ′) for some > ∈ MBB(p, � ) and some 8 ∉ � with �8 ≠ ∅.Let > ′ ∈ �8 . Then, since ? ′>′ = ?>′ by (ii) and > ′ ∈ MBB8 (p) by our assumption, we haveE8 (>′)?′>′

=E8 (>′)?>′

= U8 (p). However, recall that W > 1 since min{G,~, 1 + n} > 1. This implies

? ′> = W?> > ?> , and

E8 (>)? ′>

<

E8 (>)?>≤ U8 (p) =

E8 (> ′)? ′>′

≤ U8 (p ′),

a contradiction.Now, we will show that for each > ∈ MBB(p, � ) and each 8 ∉ � with �8 ≠ ∅, there is nodirected path from > to 8 in � (�,p ′). Assume towards a contradiction that there is a directedpath % from item > ∈ MBB(p, � ) to agent 8 ∉ � in � (�,p ′); let % = (>1, 81, >2, . . . , >: , 8:) where>1 = > and 8: = 8 . We will show by induction that 8C ∈ � for each C = 1, 2, . . . , : . Observefirst that by the above reasoning, item > will not be desired by the agents outside � afterthe price-rise phase, namely, > ∉ MBB8′ (p ′) for any 8 ′ ∈ # \ � . Thus, 81 ∈ � . Suppose that8C′ ∈ � for each 1 ≤ C ′ ≤ C ; we will prove that 8C+1 ∈ � . Since agent 8C has an edge towards>C+1 in � (�,p ′), we have >C+1 ∈ �8C ⊆ MBB8C (p). Thus, >C+1 ∈ MBB(p, � ); by applying thesame reasoning as the above observation about item > , this implies 8C+1 ∈ � , completing theinduction. But this means 8 = 8: ∈ � , contradicting the assumption that 8 ∉ � . �

We also observe that the identity of the least weighted spender does not changewhenW ∈ {G,~}.Lemma A.9. If min{G,~} ≤ I, then 8∗ remains a least weighted spender with respect to p

′, i.e.,1F8∗

? ′(�8∗) = wmin(�,p ′).Proof. If ? (�8∗) = 0, then the weighted spending of 8∗ remains 0 after the price-update, and

thus the claim holds. So suppose ? (�8∗) > 0. Since the prices of the items owned by agents in �

uniformly increased, ?′ (�8∗ )F8∗

≤ ?′ (�8 )F8

for any 8 ∈ � . Also, by definition of I, for any 9 ∈ # \ � wehave

? ′(�8∗)F8∗

= W · ? (�8∗)F8∗

≤ I · wmin(�,p) ≤ ? (� 9 )F 9wmin(�,p) · wmin(�,p) = ? ′(� 9 )

F 9,


which implies that 8∗ remains a least weighted spender with respect to p ′. �

We now prove that � still satisfies the MBB condition with respect to the new price vector.

Lemma A.10. For each agent 8 ∈ # , it holds that �8 ⊆ MBB8 (p ′).Proof. Take any 8 ∈ # ; by our assumption, �8 ⊆ MBB8 (p). Consider the following two cases.

First, suppose 8 ∉ � . Then, ?> = ? ′> for each > ∈ �8 by Lemma A.8(ii). Thus, the price of every> ∈ �8 remains the same while the price of every other item does not decrease, so �8 ⊆ MBB8 (p ′).Second, suppose 8 ∈ � . Then, for any > ∈ MBB8 (p) and > ′ ∈ $ \MBB(p, � ), the bang per buck ratiofor > at p ′ still remains higher or equal to that for > ′, i.e.,

E8 (>)? ′>≥ E8 (>)

?>· 1~≥ E8 (>)

?>· E8 (> ′)U8 (p)?>′

=E8 (> ′)? ′>′

,

where the first inequality follows from the fact that W ≤ ~ (recall that ~ = (1 + n)0 ≥ 1 + n from

Lemma A.8(i)) and the second inequality follows from the definition of ~ (namely, 1~≥ E8 (>′)

U8 (p)?>′ ).Further, for any > ∈ MBB8 (p) and > ′ ∈ MBB(p, � ) \MBB8 (p), the bang per buck ratio for > at p′

remains higher than that for > ′, i.e.,

E8 (>)? ′>

=E8 (>)W?>

>

E8 (> ′)W?>′

=E8 (> ′)? ′>′

.

Thus, every item in MBB8 (p) still has a higher or equal bang per buck ratio as the other items, andall such items have the same bang per buck ratio since the prices of the items in MBB8 (p) havebeen increased by the same factor W . This means that MBB8 (p) ⊆ MBB8 (p ′). Since �8 ⊆ MBB8 (p),we conclude that �8 ⊆ MBB8 (p ′). �

Finally, we show that the algorithm terminates if it raises prices by scaling with G .

Lemma A.11. If W = G and wmax−1(�,p) ≥ wmax−1(�,p′), then (�,p′) is WpEF1.Proof. Suppose that W = G and wmax−1(�,p) ≥ wmax−1(�,p ′). Then, the algorithm termi-

nates by Lemma A.9 and by definition of G . Indeed, at price vector p′, we have

wmin(�,p ′) = G ·wmin(�,p) = wmax−1(�,p) ≥ wmax−1(�,p′),and thus (�,p ′) is WpEF1. �

Lemma A.12. If wmax−1(�,p) < wmax−1(�,p ′), then (�,p′) is 3n-WpEF1.Proof. Suppose that wmax−1(�,p) < wmax−1(�,p ′), and consider an agent 9 ∈ # such that

wmax−1(�,p′) = 1F9

min>∈� 9? ′(� 9 \ {>}). If 9 ∉ � , then the prices of items owned by 9 do not

change by Lemma A.8(ii) and hence wmax−1(�,p ′) = wmax−1(�,p), a contradiction. Thus, 9 ∈ � ,meaning that 9 has a directed path % to the least weighted spender 8∗ in� (�,p). By the if-conditionin Line 8, 9 is not an n-path-violator with respect to (�,p); thus, by deleting the item > ( 9 , %) from� 9 , 9 ’s weighted spending becomes smaller than or equal to (1 + n)wmin(�,p), which implies

(1 + n)wmin(�,p) ≥ min>∈� 9

1

F 9? (� 9 \ {>}). (9)

Consider first the case where wmin(�,p ′) = Wwmin(�,p), meaning that the identity of 8∗ did notchange in going from p to p ′. Then, (�,p ′) is n-WpEF1 since

(1 + n)wmin(�,p ′) = W (1 + n)wmin(�,p)

≥ W min>∈� 9

1

F 9? (� 9 \ {>})


= wmax−1(�,p ′),where the inequality follows from (9). Next, consider the case where wmin(�,p′) ≠ Wwmin(�,p).By definition, wmin(�,p ′) < Wwmin(�,p), which means that wmin(�,p) > 0 and G takes a finitevalue. In this case, the identity of 8∗ has changed, so by LemmaA.9, min{G,~} > I. Thus,W = (1+n).Since wmin(�,p) ≤ wmin(�,p′), Inequality (9) implies

(1 + n)wmin(�,p′) ≥ min>∈� 9

1

F 9? (� 9 \ {>}).

Multiplying this with W = (1 + n), we get(1 + 3n)wmin(�,p ′) ≥ (1 + n)2wmin(�,p′)

≥ (1 + n) min>∈� 9

1

F 9? (� 9 \ {>}) = min

>∈� 9

1

F 9? ′(� 9 \ {>}) = wmax−1(�,p ′).

We conclude that (�,p ′) is 3n-WpEF1, as desired. �

The two lemmas immediately imply that scalingwith G ensures the termination of the algorithm.

Corollary A.13. If W = G , then (�,p ′) is 3n-WpEF1.Proof. SupposeW = G . If wmax−1(�,p) ≥ wmax−1(�,p ′), then (�,p ′) is WpEF1 by LemmaA.11.

If wmax−1(�,p) < wmax−1(�,p ′), then (�,p ′) is 3n-WpEF1 by Lemma A.12. This proves the claim.�

A.2.4 Convergence of Algorithm 3. Using Lemmas A.11 and A.12, we will show that during itscourse, the algorithm repeatedly increases the minimum weighted spending wmin(�,p) by afactor of (1 + n) within polynomial time, and this spending is bounded above by the functionwmax−1(�,p), which is non-increasing. We then show that the algorithm indeed converges to a3n-WpEF1 allocation while keeping the MBB condition.To begin with, we show that the MBB condition is satisfied and each price is a power of (1+n) at

any point in the execution of the algorithm. Further, we obtain the monotonicity of the functionswmax−1(�,p) and wmin(�,p) by the observations we made in the previous subsection.

Lemma A.14. During the execution of Algorithm 3, the following statements hold:

(i) � is complete and satisfies the MBB condition, i.e., �8 ⊆ MBB8 (p) for any 8 ∈ # .

(ii) If p is not the price vector at termination, then the price ?> of each item > ∈ $ is a power of

(1 + n), i.e., ?> = (1 + n): for some non-negative integer : .

(iii) Except possibly in the last iteration of the algorithm, the functionwmax−1(�,p) never increases.(iv) The function wmin(�,p) never decreases.Proof. (i) The allocation � is clearly complete because each item is assigned to some agent

at every step. Take any 8 ∈ # . We will inductively show that agents are kept assigned totheir MBB items. Recall that at the initial allocation �(0) and price vector p (0) , every agentreceives an item only if she has the highest valuation for the item, and the price for eachitem is set to be the maximum such value. This means that each agent 8 has U8 (p (0) ) at most1 and is assigned to an item > only if her bang per buck ratio for > is exactly 1. Thus, theclaim holds for the initial allocation. Moreover, each agent remains assigned to her MBBitems after the swap phase (between Line 8 and Line 14). Also, by Lemma A.10, each agentremains assigned to her MBB items after the price-rise phase (between Line 16 and Line 28).This proves the claim.


(ii) Recall that the initial price of each item is set to be the maximum value of the agents for thatitem, which is (1+n): for some : ≥ 0. The price of each item never decreases, and increasesby a power of (1 + n) at each price-rise phase by Lemma A.8(i) and Corollary A.13 exceptpossibly for the final iteration. Thus, the claim holds.

(iii) The claim immediately follows from Corollary A.4 and Lemma A.12.(iv) The function wmin(�,p) does not decrease due to Corollary A.5 and the fact that the spend-

ing of each agent does not decrease during the price-rise phase. �

Now, we prove that not only is the function wmin(�,p) weakly increasing, but it is also strictlyincreasing during the execution of the algorithm. We first consider the corner case where theweighted spending of the least weighted spender 8∗ is zero. In this case, the value wmin(�,p) maynot increase even after the price-rise phase. However, we will show that the number of agents whocan reach 8∗ strictly increases, and eventually the algorithm transfers an item of positive price froma violator to 8∗. By a time step we refer to a point in the execution of the algorithm.

Lemma A.15. Suppose that at time step C , agent 8 is chosen as the least weighted spender 8∗ inLine 6 of Algorithm 3 but ? (�8) = 0. Then, after = + 1 iterations of the while loop in Line 4, either the(unweighted) spending of 8 strictly increases by at least 1, or the algorithm terminates.

Proof. Recall that the initial prices of all items are positive, so the prices of all items are positiveat time step C . Thus, agent 8 is not assigned to any item at time step C . Further, since allocation �

is complete at any point of the algorithm, there is always an agent 9 ≠ 8 with |� 9 | ≥ 2 as long as? (�8 ) = 0 and wmin(�,p) < wmax−1(�,p).

We first show that whenever the algorithm performs the swap phase, 8 receives a new item,which means that 8’s spending strictly increases and proves the desired claim. To see this, supposethat the algorithm enters the swap phase at some time step when ? (�8) = 0. Observe that agentswith at least one item have a positive spending as all prices are positive. Hence, the swap operationalong the directed path % = (80, >1, 81, . . . , >: , 8:) ends when it gives the last item >: to 8: = 8 , sinceif some intermediate agent 8ℎ with 0 < ℎ < : owns at least two items, then she becomes a path-violator for 8 . (Recall that by definition of � (�,p), before the swaps are performed, each of theagents 80, 81, . . . , 8:−1 has at least one item.) Thus, 8’s spending becomes positive if the algorithmenters the swap phase, and the increase is at least 1 since the prices are integral powers of (1 + n).It remains to show that after at most = consecutive iterations of the price-rise phase during

which ? (�8 ) = 0, some violator becomes a path-violator for 8 and hence the algorithm enters theswap phase. To this end, it is sufficient to show that whenever the algorithm performs the price-rise phase, the number of agents who can reach 8 (namely, agents in � ) strictly increases. Supposethat the algorithm increases the price of MBB items of the set � of agents who can reach 8 . Since theset � does not include an n-path-violator for 8 , no agent in � has more than one item: Indeed, if someagent 9 ∈ � has at least two items, 9 still has a positive weighted spending even after removingany item in � 9 ; however, at least one of 9 ’s items has a directed path to 8 since 9 ∈ � , meaningthat 9 is an n-path-violator. Further, when ? (�8) = 0, the algorithm raises prices of MBB items of� by scaling with ~ defined in Line 18. By definition of ~, after the price-rise phase, some item >

not in MBB(p, � ) becomes a new MBB item for an agent in � ; hence, some agent not in � can nowreach 8 , while all agents in � can still reach 8 . It follows that the number of agents who can reach 8strictly increases after each price-rise phase. We conclude that the algorithm can perform at most= consecutive price-rise phases. �

The following lemmas further ensure that whenwmin(�,p) > 0, theminimumweighted spend-ing strictly increases within a polynomial number of time steps.


Lemma A.16. During the execution of Algorithm 3, suppose that agent 8 ceases to be the leastweighted spender 8∗ just after time step C1. Suppose further that 8 is chosen again later as 8∗, and let

C2 be the first time step when this happens. Let (�,p) (respectively, (�′,p′)) be the pair of allocationand price vector at time C1 (respectively, time C2). Then, it holds that �8 ( �′8 or

1F8? ′(�′8 ) ≥ 1+n

F8? (�8).

Proof. Since agent 8 is not the least weighted spender at time C1 + 1, the weighted spending of8 strictly increases in going from C1 to C1 + 1, either by receiving a new item in Line 12 or by theprice-rise of her items in Line 28.First, consider the case when 8 does not lose any item between C1 and C2. Then, the claim clearly

holds when 8 gains some new item at C1. Thus, suppose that the algorithm raises prices at time C1.By Corollary A.13, it raises prices by a factor of either ~ or (1 + n), both of which are powers of(1+ n). This implies that the weighted spending of 8 strictly increases at least by a factor of (1 +n)in going from C1 to C2.Now consider the case when 8 loses some item between C1 and C2, whichmeans that 8 becomes an

n-path-violator for the least weighted spender 8∗ at some time step between C1 and C2. Let C ′ be thelast time step before C2 when 8 becomes an n-path-violator and loses an item > . Then, 8’s weightedspending after giving item > to her neighbor is strictly higher than the value of (1 + n)wmin(�,p)at time step C ′, which is greater than or equal to (1 + n) times 8’s weighted spending at time C1 byLemma A.14(iv). Further, the weighted spending of 8 does not decrease between C ′ and C2. Hence,8’s weighted spending increases by a factor of at least (1 + n). �

Due to Lemma A.6, the number of consecutive swap phases is bounded by a polynomial in theinput size.

Lemma A.17. During the execution of Algorithm 3, the number of consecutive swap phases duringwhich 8∗ remains the chosen least weighted spender is at most<=2.

Proof. Let 8 be the least weighted spender 8∗ chosen in Line 6 of Algorithm 3 at some timestep C . By Lemma A.6, each swap operation between Line 11 and Line 14 causes a drop of at least 1in the value of the function Φ8 (�,p), which is non-negative and bounded above by<=2, implyingthat Algorithm 3 can perform at most<=2 iterations of the swap phase during which 8 is the leastweighted spender. �

Lemma A.18. During the execution of Algorithm 3, after poly(<,=) time, either the algorithm

terminates, or wmin(�,p) strictly increases by a factor of at least (1 + n) .Proof. Note that each swap and price-rise phase can be executed in poly(<,=) time: for the

swap phase, finding a nearest n-path-violator 9 as well as the associated path % from 9 to 8∗ can bedone in polynomial time (see Sections 4 and �.1 in the extended version of [Barman et al., 2018]for details). By Lemma A.15, the minimum weighted spending becomes strictly positive after = + 1iterations of the while loop when wmin(�(0) ,p (0) ) = 0 at the initial pair of allocation and pricevector (�(0) ,p (0) ), so suppose without loss of generality that wmin(�(0) ,p (0) ) > 0.

Consider an arbitrary time step C during the execution of the algorithm. If the algorithm per-forms the price-update without changing the identity of 8∗ chosen in Line 6 of Algorithm 3, thenthe minimum weighted spending wmin(�,p) increases by a factor of at least (1 + n) (if the algo-rithm raises prices by a factor of G , then by Corollary A.13, the algorithm terminates immediatelyafterwards). Thus, consider the case where the identity of 8∗ changes at every price-update. In thiscase, by Lemma A.17, the identity of 8∗ must change after at most<=2 consecutive swap phases.Observe that, after (< + 1)= identity changes, some agent 8 becomes 8∗ at least< + 1 times by thepigeonhole principle. By LemmaA.16 and the fact that the size of 8’s bundle can grow up to at most<, 8’s weighted spending must increase by a factor of at least (1 + n) within poly(<,=) time. �


Let E<0G := max8 ∈#,>∈$ E8 (>) and F<0G := max8 ∈# F8 . We are ready to show the convergenceof Algorithm 3.

Theorem A.19. Algorithm 3 terminates in poly(<,=, 1n, E<0G ,F<0G ) time.

Proof. By Lemma A.18, after poly(<,=) time, either the algorithm terminates, or the minimumweighted spending wmin(�,p) strictly increases by a factor of at least (1+n) (or becomes positiveif it is 0; see the proof of Lemma A.18). Observe that wmin(�,p) is at least 1

F<0Gat the initial

outcome where it is positive, and during the execution of the algorithm, wmin(�,p) is at mostwmax−1(�,p), which is bounded above by <E<0G due to Lemma A.14(iii) and the choice of theinitial price vector. Now, log(1+n) (<E<0G/F<0G ) = log(<E<0G/F<0G )/log(1 + n), and we havelog(1 + n) > n

1+n ; the latter holds since the function 5 (G) := log(1 + G) − G1+G satisfies 5 (0) = 0 and

5 ′(G) > 0 for all G > 0. The desired running time follows. �

Similarly to Barman et al. [2018], we show that the output price vector can be bounded aboveby a product of weights and valuations. We start by showing the following auxiliary lemma.

Lemma A.20. During the execution of Algorithm 3, suppose that the price of item > has been in-creased with the multiplicative factor W (line 28) just after time step C . Let (�,p) be the pair of alloca-tion and price vector at time C . Then, it holds that W · ?> ≤ <E2<0GF<0G .

Proof. Suppose that at time step C , agent 8 is chosen as the least weighted spender 8∗ in Line 6of Algorithm 3. Thus, 1

F8? (�8) = wmin(�,p). Let G and ~ be the multiplicative factors defined

in Line 17 and Line 18 at time step C , respectively. By definition of W , we have W ≤ ~. Indeed, ifW = (1 + n), then (1 + n) ≤ ~ because ~ is a power of (1 + n) by Lemma A.8(i); and if W = G , thenG ≤ ~ by definition of W .

Let � be the set of agents who can reach 8 at time step C . Since > ∈ MBB(p, � ) and� is a completeallocation, we have that > ∈ �8′ for some 8 ′ ∈ � . Now, since the algorithm did not terminate attime step C , there is a violator 9 at time step C such that even after removing any item > ′ ∈ � 9

from 9 ’s bundle, the weighted spending of 9 does not go below the least weighted spending, i.e.,1F9? (� 9 \ {> ′}) > wmin(�,p) ≥ 0. In particular, this means that � 9 contains at least two items.

Thus, there is an item >∗ ∈ � 9 such that ?>∗ ≤ ? (� 9 \ {> ′}) ≤ F 9wmax−1(�,p), which is boundedabove by<E<0GF<0G due to Lemma A.14(iii) and the choice of the initial price vector. We furtherobserve that� 9∩MBB(p, � ) = ∅ since otherwise 9 would becomea path-violator for 8 , contradictingthe fact that the algorithm performs the price-rise phase just after time step C . Thus, by definitionof ~ and the fact that � 9 ∩MBB(p, � ) = ∅, we have that

~ ≤ U8′ (p)?>∗

E8′ (>∗)=E8′ (>)?>

?>∗

E8′ (>∗). (10)

By the fact that W ≤ ~ and (10), we get

W · ?> ≤ ~ · ?> ≤E8′ (>)E8′ (>∗)

?>∗ ≤ E8′ (>)?>∗ ,

which is bounded above by<E2<0GF<0G since E8′ (>) ≤ E<0G and ?>∗ ≤ <E<0GF<0G . �

We are now ready to upper-bound the final price of each item by applying the above lemma.

Lemma A.21. Suppose that (�,p) is the output of Algorithm 3. Then, ?> ≤ <E2<0GF<0G for each> ∈ $ .

Proof. If item > never experiences the price-rise phase, then its final price ?> is the same as theinitial price, so ?> ≤ E<0G by the definition of the initial price vector. Thus, suppose otherwise,


namely, the price of > increased at some time step. Let C be the last such step. Then by LemmaA.20, the updated price of item > is at most<E2<0GF<0G , which proves the claim. �

A.3 Approximate Instance

As in Barman et al. [2018], given an arbitrary instance, we will consider an approximate versionwhere the valuations as well as weights are integral powers of (1+n) > 1. Formally, we define then-rounded instance I ′ = (#,$, {E ′8 }8 ∈# , {F ′8 }8 ∈# ) of a given instance I = (#,$, {E8 }8 ∈# , {F8}8 ∈# )as follows: For each 8 ∈ # and > ∈ $ , the value E ′8 (>) is given by

E ′8 (>) :={(1 + n) ⌈log1+n E8 (>) ⌉ if E8 (>) > 0;

0 otherwise.

For each 8 ∈ # , the weightF ′8 is given by

F ′8 := (1 + n) ⌊log1+n F8 ⌋

Note that for each 8 ∈ # and > ∈ $ , we haveE8 (>) ≤ E ′8 (>) ≤ (1 + n)E8 (>) (11)

and 1(1+n)F8 ≤ F ′8 ≤ F8 , where the latter implies

1

F8≤ 1

F ′8≤ (1 + n)

F8. (12)

We will show below that the WEF1 property for the approximate instance translates into the WEF1property for the original instance. We write the minimum positive difference of the weightedvalues as follows:

n8 9 := min

{E8 (- )F 9

− E8 (. )F8

��-,. ⊆ $ ∧ E8 (- )F 9

− E8 (. )F8

> 0

},

andn<8= := min

8, 9 ∈#n8 9 . (13)

For the n-rounded instance I′ = (#,$, {E ′8 }8 ∈# , {F ′8 }8 ∈# ) and a price vector p, we denote the

maximum bang per buck ratio for agent 8 by U ′8 (p) = max>∈$E′8 (>)?>

and the MBB set by

MBB′8 (p) ={> ∈ $

��E ′8 (>)?>

= U ′8 (p)}.

Lemma A.22. Let I ′ = (#,$, {E ′8 }8 ∈# , {F ′8 }8 ∈# ) be the n-rounded instance of a given instance

I = (#,$, {E8 }8 ∈# , {F8 }8 ∈# ). Suppose that 0 < n < min{n<8=

632 , 1}, where 2 := max8 ∈#

E8 ($)F8

. Let

(�,p) be a pair of a complete allocation and a price vector such that �8 ⊆ MBB′8 (p) for each 8 ∈ # ,

and � is 3n-WpEF1 for I′ with respect to p. Then � is also WEF1 for I.Proof. Take any pair of agents 8, 9 ∈ # . Agent 8 does not envy 9 when � 9 = ∅; so suppose

� 9 ≠ ∅. Since � is 3n-WpEF1 for I ′, there exists an item > ∈ � 9 such that

1 + 3nF ′8

? (�8 ) ≥1

F ′9? (� 9 \ {>}).

Multiplying both sides of this inequality with U ′8 (p), we get1 + 3nF ′8

E ′8 (�8 ) ≥1

F ′9

∑

>′∈� 9 \{> }?>′U

′8 (p) ≥

1

F ′9E ′8 (� 9 \ {>}).


Thus, we get

1 + 3nF ′8

E ′8 (�8 ) ≥1

F ′9E ′8 (� 9 \ {>})

⇒ (1 + 3n)3

F8E8 (�8 ) ≥

1

F 9E8 (� 9 \ {>})

⇒ 63n

F8E8 (�8 ) ≥

1

F 9E8 (� 9 \ {>}) −

1

F8E8 (�8 )

⇒ n<8= >

63n

F8E8 (�8 ) ≥

1

F 9E8 (� 9 \ {>}) −

1

F8E8 (�8 ),

where the second inequality follows from (11) and (12), and for the third inequality we use (1 +3n)3 ≤ 1 + 63n , which follows from expansion and n < 1. By definition of n<8=, we conclude that1F9E8 (� 9 \ {>}) − 1

F8E8 (�8 ) ≤ 0, as desired. �

For small enough n , the MBB condition for I′ translates to the MBB condition for the originalinstance.

Lemma A.23 (Proof of Lemma 5 in the extended version of Barman et al. [2018]). Let p bea price vector such that 1 ≤ ?> ≤ \ for each > ∈ $ , for some \ > 1. Let I′ = (#,$, {E ′8 }8 ∈# , {F ′8 }8 ∈# )be the n-rounded instance of the given instance where 0 < n <

1\<E<0G

. If �8 ⊆ MBB′8 (p) for each8 ∈ # , then the allocation � is Pareto-optimal for the original instance.

A.4 Pu�ing Things Together: Proof of Theorem 4.6

We assumewithout loss of generality that the valuations as well as weights are at least 1; otherwise,one can scale up these values by multiplying with the same factor " and obtain an equivalentinstance. Now let n be such that

n =1

2min

{1

<2E3<0GF<0G

,n<8=

632, 1

},

where 2 := max8 ∈#E8 ($)F8

and n<8= is defined as in (13). Consider the n-rounded instance I′ ofthe given instance I. By Lemma A.14(i) and Theorem A.19, Algorithm 3 computes a pair (�,p)such that � is a complete allocation that satisfies 3n-WpEF1 and each agent 8 ∈ # receives items inMBB′8 (p) for I ′. By Lemmas A.21, A.22 and A.23, � is WEF1 and Pareto optimal for I. The boundon the running time follows from Theorem A.19. �

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