In the last few years or so, there has been a tremendous demand for fair division services to provide systematic and fair ways of dividing a set of (indivisible) goods such as tasks, courses, and properties among a group of agents so that the agents do not envy each other. Such demand gave rise to, for examples, Spliddit111http://www.spliddit.org/, the University of Pennsylvania’s Course Match222https://mba-inside.wharton.upenn.edu/course-match/, and Fair Outcomes, Inc333https://www.fairoutcomes.com/ which are all based on mathematical and fairness notions. To capture the fairness of an allocation, which is arguably initiated by the work of [Foley, 1967], envy-freeness (EF) (and its relaxations, such as EF1 and EFX) is often used to ensure that each agent should not envy or prefer the allocated goods of other agents.
In this paper, we study an envy-free allocation domain where the planner of the division tasks wishes to withhold allocation information of others from the user or the user simply does not know the allocation of others in the system. There are a couple of good reasons why it is desirable for the planner to withhold such information. First, in many private fair allocations of goods such as tasks or gifts, the planner requires the system to preserve anonymity as not to give away the received bundles of other agents. Second, due to the large number of (unrelated) agents and items that could be potentially be involved in the division tasks (e.g., on the Internet such as MTurks), it is not meaningful for the planner to provide such information due to various reasons. Motivated by this domain, we focus on answering the following questions.
When indivisible goods are to be allocated among unaware agents, what is the appropriate envy-free notion and how efficiently can the allocation be found subject to the notion?
Proportionality (PROP) and maximin share (MMS) [Budish, 2011] are two widely studied and well accepted fair allocation notions, both of which are defined for unaware agents. In PROP, it is required that the value of every agent’s bundle is at least a fraction of her value for the whole goods, where is the number of agents. It is well known that such an allocation may not exist for indivisible goods, thus a weaker and more realistic notion is desired for indivisible goods. MMS is a proper relaxation of PROP, which studies an adversarial situation: when the goods are partitioned into bundles and an agent would always get the least preferred bundle of goods, what is the best way she can partition the goods. The value of such a bundle is the MMS value of the agent. In addition to its non-existence result, MMS allocation only guarantees each agent’s best minimum value, and the value of some agent’s bundle can still be the least compared with others, which may cause significant envy (demonstrated by the below example).
Suppose there are agents and goods, , and , , where is a sufficiently large number and and are valuation functions. Then each agent’s MMS value is 1 and , is an MMS allocation, but each agent envies each other. However, if we exchange their allocations, everyone’s value can be improved to .
Recently, [Aziz et al., 2018] introduced epistemic envy-free (EEF) notion to study unaware setting. With respect to EEF, each agent is satisfied if there exists one reallocation of the goods that she does not get among the other agents, such that her value for her bundle is at least as good as every bundle in this reallocation. This measure is not robust if the reallocations are restricted by agent’s reasoning (e.g., adversarial settings in MMS). Moreover, it can be shown that EEF and PROP allocations (and their relaxations) barely exist and cannot be properly approximated444Consider three agents with two goods such that every agent has value 1 for each good. Then the agent that receives no good has no bounded guarantee for both PROP and EEF (and their relaxations by removing any item from the items she has not obtained)..
In this paper, we focus on modeling the envy-freeness for indivisible goods allocation as well as deriving new algorithms to find (approximately) fair allocations subject to different fairness notions in an unaware environment. Our main contributions include the following.
First, we introduce a novel fairness notion of maximin aware (MMA), which guarantees that the agent’s bundle value is at least as much as her value for some other agent’s bundle, no matter how the remaining goods are distributed, i.e., there is always somebody who gets no more than her. MMA combines the notion of epistemic envy-freeness [Aziz et al., 2018] and MMS [Budish, 2011] where each agent may not know or care about the exact allocation of the remaining goods to other agents, but can still guarantee that the value for her own bundle is at least as much as she can obtain if she is given the chance to repartition the remaining goods and receive the least valued portion. We provide a detailed picture of the relationship between MMA and classic fairness notions for different valuations.
Then we show MMA is a strong requirement and cannot be guaranteed in general, and provide two relaxations of MMA: MMA1 and MMAX. We also prove that MMA1 (and MMAX) potentially has stronger egalitarian guarantee than EF1 and such an allocation is guaranteed to exist for the following situations: (1) there are at most three agents with submodular valuations and (2) there are any number of agents but all of them have identical submodular valuation. If the above requirement of submodularity is replaced by strictly increasing subadditivity, an MMAX allocation is guaranteed to exist. In contrast, MMS allocation may not exist even for three agents with additive valuations [Kurokawa et al., 2018] and an EFX allocation is only known to exist when there are two agents [Plaut and Roughgarden, 2018].
We present a polynomial-time algorithm that computes an allocation such that every agent is either -approximate MMA or exactly MMAX for additive valuations. For several specific classes of valuations, such as binary additive or additive with identical preference ranking, our algorithm returns an exact MMAX and EFX allocation. It is shown in [Plaut and Roughgarden, 2018] that a -approximate EFX allocation exists for general subadditive valuations, but finding it may need exponential time, which leaves an open question whether such an allocation can be found in polynomial time. We show that the allocation by our algorithm computes is also -approximate EFX when all agents have subadditive valuations, and thus answer this problem affirmatively.
To show the existence of MMA1 allocations, we first show that if all agents have identical submodular valuations, a leximin -partition of the goods is MMA1. Next, we provide a divide-and-choose style algorithm for 3 agents with submodular valuations (need not to be identical). Basically, we let the first agent make a leximin -partition, and other 2 agents select. Then no matter which subset is left for the first agent, she is satisfied with respect to MMA1. However, the difficulty comes from how the other 2 agents should select. By carefully examining all possible cases, we show that there is a way to select and redivide the three subsets such that both of them are satisfied with respect to MMA1.
To design an efficient algorithm for MMAX or EFX allocations, we construct a bipartite graph between agents and goods. We show that there is a way to use a maximum weighted matching between the unenvied agents and the remaining goods to guide our allocation. Eventually, we show that all the goods are allocated, and the final allocation is either -approximate MMA or exact MMAX for additive valuations and -approximate EFX for subadditive valuations.
Other Related Works.
For indivisible goods allocations, EF or PROP allocations may not always exist555For example, there are two agents but only one good.. [Budish, 2011] introduced the relaxed concept of envy-freeness up to one good (EF1). In an EF1 allocation, it is only required that each agent’s value for a bundle is at least as much as her value for every other agent’s bundle minus a single good (in the bundle). It is shown in [Lipton et al., 2004] that an EF1 allocation always exists, and can be found in polynomial time. In [Caragiannis et al., 2016], the authors introduced a strictly stronger fairness notation than EF1 called envy-free up to any good (EFX), where the comparison is made to “any” single good instead of “a” single good. The state-of-the-art results show that an EFX allocation exists in the following settings: (1) there are 2 agents, or (2) there are any number of agents but all of them have the identical valuation [Plaut and Roughgarden, 2018]. It is still an open question whether an EFX allocation exists in general, even for additive valuations.
Recently, we see many new fairness notions adapted for different settings, such as Nash social welfare [Caragiannis et al., 2016], epistemic envy-freeness [Aziz et al., 2018], pairwise maximin share [Caragiannis et al., 2016], and groupwise maximin share [Barman et al., 2018]. There is also a line of works making connections (or constraints) of fair allocations to graphs or networks [Abebe et al., 2017, Bei et al., 2017, Aziz et al., 2018]. More works on the fair allocation problem under different constraints can be found in [Ferraioli et al., 2014, Bouveret et al., 2017, Biswas and Barman, 2018].
We provide necessary definitions and introduce MMA in Section 2. We show its connections with classic fairness notions for different classes of valuations in Sections 3. In Sections 4 and 5, we study the two relaxations of MMA, MMA1 and MMAX, and discuss their connections with other fairness notions and their existences in different settings. Finally, in Section 6, we present a polynomial time algorithm to compute a -approximate MMAX allocation for additive valuations; and -approximate EFX allocation for subadditive valuations.
2 Preliminaries and Definitions
In the envy-free division problem on indivisible goods, there is a set of agents and a set of indivisible goods. Each agent has an valuation function that maps every subset of goods to a non-negative real number. We assume that is normalized (i.e., ) and montone (i.e., for every ) for each agent . For convenience we use to denote , for every valuation and every good . Throughout the whole paper, we call an valuation
additive if for each ;
binary additive (BA) if is additive, and for any good ;
subadditive (SA) if for any ;
submodular (SM) if for any and , .
An allocation is a partition of among agents in , i.e., and for any . In other words, is the set of goods allocated to agent . Let be the goods not assigned to agent . Without loss of generality, we assume that for every , there exists such that .
Below we state some standard fairness definitions.
Definition 2.1 (Ef)
For any , an allocation is -envy-free (-EF) if for any , . The allocation is EF when .
Definition 2.2 (Ef1)
For any , an allocation is -EF1 if for any , there exists such that . The allocation is EF1 when .
Definition 2.3 (Efx)
For any , an allocation is -EFX if for any , for any . The allocation is EFX when .
Next, we introduce another fairness notion called maximin share (MMS). Let be the set of all -partitions of a set . The maximin share of agent on among players is
Definition 2.4 (Mms)
For any , an allocation is -MMS if for any , . The allocation is MMS when .
Next we define the epistemic envy-free (EEF) [Aziz et al., 2018], which is the most relevant fairness notion to this work.
Definition 2.5 (Eef)
An allocation is EEF if for any , there exists an allocation such that and for any .
For the purpose of presenting our results, we provide the definition of proportionality fairness.
Definition 2.6 (Prop)
An allocation is PROP if for any , .
Finally, we say that if every allocation is also a allocation when agents have valuations, where and are the fairness notions (e.g., MMA1 and MMS) and is the function type (e.g. SA and SM).
3 Maximin-Aware Allocation
In this section, we introduce the fairness notion of maximin-aware (MMA), and study its connection to other existing fairness notations.
Definition 3.1 (Mma)
For any , an allocation is -MMA if for any agent , . The allocation is MMA when .
There is a modeling and conceptual advantages of MMA: an MMA allocation guarantees for every agent, no matter how the goods that she does not get are distributed among the other agents, there is always an agent who gets no more than her. Compared to EEF, where the happiness of each each agent depends on the existence of a reallocation of the other goods, MMA provides a robust way for the agents to reason about their reallocations of the remaining goods.
Observe that an MMS allocation need not be MMA. Recall, Example 1.1 where and is an MMS allocation. However, this allocation is far from being MMA since and . The only MMA allocation in this example is and .
Seemingly MMA being a stronger definition than MMS, Example 3.1 shows that this is not true.
Suppose there are four agents () and eight goods (). Agent 1 has the additive valuation shown in Table 1.666We do not explicitly describe the valuations for the other agents since it is easy to design other agents’ valuations such that is allocated to agent 1. It is easy to check that . Let . Then . That is, for agent 1, does not satisfy the condition of MMS . However, , which satisfies the condition of MMA.
However, when all agents have binary additive (BA) valuations, MMA actually implies MMS.
Fix any agent , let be the number of goods agent is interested in. Then . Fix any MMA allocation , we show that it is also MMS. Suppose otherwise, i.e., . Then we have
which contradicts the fact that is MMA. Hence is also an MMS allocation.
Indeed, similar to MMS, MMA is a more realistic and appropriate definition for indivisible goods allocation, in the sense that MMA is more approachable than EF and PROP for additive valuations. In the following, we show that PROP MMA. Combining with Theorem 4 of [Aziz et al., 2018], it implies EF EEF PROP MMA.
EF EEF PROP MMA
Fix any PROP allocation and any agent . It follows that . Since the valuations are additive,
Hence, the allocation is also MMA.
Note that the other direction is not true: in Example 3.1, agent 1 is satisfied with with respect to MMA, but the portion is only . On the other hand, MMA is still a strong requirement for indivisible goods allocation, in the sense that it cannot be satisfied even in some simple settings.
MMA allocations may not exist even when the agents have identical BA valuation.
Assume there are agents and goods, where . Each agent’s value on every good is . For any agent , if she gets goods then
Thus in any MMA allocation, we have for every agent , which is impossible with goods in total.
In the following two sections, we introduce two relaxations of MMA based on EF1 and EFX, respectively.
4 Maximin-Aware up to One Good
In this section, we relax the MMA notion to maximin-aware up to one good (MMA1) that is analogous to EF1.
Definition 4.1 (Mma1)
Fix any , an allocation is -MMA1 if for any , there exists such that
The allocation is MMA1 when .
4.1 Connection to EF1 and MMS
We first note that while MMA1 is a relaxed version of MMA, it may have better egalitarian guarantee than EF1 allocations. An MMA1 allocation guarantees that for each agent and her favorite item (assume ), is at least as large as the worst bundle in any -partition of . However, EF1 implies that there exists an -partition of , i.e., for and , such that is at least as large as the worst bundle, i.e. . We illustrate this as the following example.
Consider agents and goods, where every agent has identical additive valuation with for and for . Thus allocation for and is EF1. In such an allocation, agent only gets value 1. However, there is an allocation (for example: for and ) where everyone obtains value at least .777We note that there is no exact EF allocation for this example.
In Example 4.1, an EF1 allocation can be arbitrarily bad to agent . However, we show that any MMA1 allocation in this example guarantees that each agent’s value is at least . If everyone’s value is at least , then our claim trivially holds. Suppose for some agent , . Then among all goods in , there are goods with value and goods with value 1. Thus by removing the most favorite good, the maximin share for the remaining set is . Since is an MMA1 allocation, , which means .
In the following, we show that MMS implies MMA1 when the agents’s valuations are submodular.
Let be an MMS allocation. It follows that for every agent . Fix any agent , it suffices to show that there exists such that .
If for every , , then by the submodularity of , we have
where the inequality holds since the marginal contribution of each good does not increase. Thus is also an MMA1 allocation for agent .
Otherwise, let be such that . For the sack of contradiction, suppose there exists a partition of such that for all . Then if we include into , the resulting partition of (into parts) has minimum value strictly larger than on every agent, which contradicts the definition of .
By a similar argument, every -MMS allocation is naturally -MMA1. Recall Example 3.1 where we show an MMA allocation is not necessarily MMS (thus neither an MMA1 allocation), this implication is strict.
Next, we show that for binary additive valuations, MMA1 and MMS are actually equivalent.
Since binary additive functions are submodular, it suffices to prove the ”” direction. Fix any MMA1 allocation , we prove that for every agent , where .
Fix any agent and assume the contrary that . Let . Then for any we have
which contradicts the fact that is MMA1.
It is shown in [Kurokawa et al., 2018] that even for three agents with additive valuations, MMS allocations do not always exist. As we will show later, for a broader class of situations, an MMA1 allocation is guaranteed to exist.
4.2 The Existence of MMA1 Allocations
By Lemma 4.2, we know that an MMA1 allocation always exists when the valuations are binary additive. In this section, we explore the existence of MMA1 allocations for instances with subadditive valuations.
First, for any valuation over a set of goods, we define the leximin -partition as follows. A leximin -partition is a partition which divides into subsets and maximizes the lexicographical order when the values of the partitions are sorted in non-decreasing order. In other words, a leximin -partition maximizes the minimum value over all possible -partitions, and if there is a tie it selects the one maximizing the second minimum value, and so on.
In the following, we show that MMA1 allocations exist when all agents have identical submodular valuation, or when there are three agents with (different) submodular valuations.
Identical submodular valuation.
We first show that when all agents have identical submodular valuation, the leximin -partition of is indeed an MMA1 allocation.
When all agents have the identical submodular valuation, the leximin -partition of is MMA1.
Let be the submodular valuation and be the leximin -partition of . We show that allocating each partition to agent gives an MMA1 allocation. As before, if for all , , then we have
by submodularity of . Otherwise let be any good such that .
Assume the contrary that .
Let be the partition of such that for all . Then by including into , we obtain a partition of such that every bundle has value strictly larger than . In other words, the resulting partition is strictly better than the leximin -partition, which is a contradiction.
Three agents with different submodular valuations.
Next we show a divide-and-choose style algorithm (shown in Algorithm 1) which gives an MMA1 allocation.
When and the valuations are submodular, Algorithm 1 computes an MMA1 allocation.
First, observe that since is a leximin -partition of agent 3, and she always gets one of , and , by Theorem 4.1, she is satisfied with respect to MMA1.
Next, we consider agents 1 and 2.
If the final allocation is from Case (2), i.e., both of them get their favorite bundles, then they are satisfied with respect to MMA1, given that the valuations are submodular.
If the final allocation is from Case (3), i.e., agent 2 repartitions and , the favorite two bundles of both agents, and let agent 1 choose first, then we show that both agents are satisfied with respect to MMA1. First, Agent 1 will always be happy, since the bundle she chooses must have value at least , by submodularity of . For agent 2, by leximin partitioning into and (suppose ), we have . Thus if is left for agent 2, she will be satisfied. Otherwise agent 2 gets , which satisfies , since agent 2 is still able to leave and unchanged. Observe that there must exist some , such that , as otherwise , which is impossible. Moreover, we have , otherwise and is not a leximin partition. Thus , which implies that agent 2 is also satisfied with respect to MMA1.
If the final allocation is from Case (4a), agent 1 and agent 2 get a bundle with value at least and , respectively. Thus they are both satisfied.
If the final allocation is from Case (4b), i.e., agent 2 takes away her favorite two bundles and and leximin-partitions them, then similar to Case 3, no matter which one is left for her, she will be satisfied. Note that for agent 1, since , the bundle she gets has value at least . Thus agent 1 is also satisfied.
We note that when there are two agents and both of them have subadditive valuations (do not need to be identical), the definition of MMA1 degenerates to be EF1. Thus an MMA1 allocation always exists. Indeed, using approaches similar to Theorems 4.1 and 4.2, we can show that a leximin allocation is an MMA1 allocation if (1) all agents have identical strictly increasing subadditive valuations, or (2) there are three agents with (different) strictly increasing subadditive valuations. We say that a subadditive function is strictly increasing if for any .
5 Maximin-Aware up to Any Good
First, we define MMAX as follows.
Definition 5.1 (Mmax)
Fix any , an allocation is -MMAX if for any ,
for any . The allocation is MMAX when .
We first note that MMAX allocations give an even better egalitarian guarantee than MMA1 and EF1 allocations. Recall Example 4.1, where we show there is an EF1 allocation such that some agent only gets value 1 and any MMA1 allocation guarantees each agent’s value is at least . It is not difficult to show that any MMAX allocation guarantees each agent’s value is at least .
Obviously, an MMAX allocation is also MMA1. Together with Lemma 4.2, we have
However, beyond BA valuations, the above implication does not hold. We can see this by borrowing an example from [Kurokawa et al., 2018], for which there is no MMS allocation, but MMAX (and thus MMA1) allocations do exist.
Consider the following example with 4 agents with additive valuations on 14 goods, where agent 1 and 2 have valuation shown in matrix , while valuation of agent 3 and 4 are shown in matrix . The goods correspond to the non-zero elements in and . Here is a small number; say and is an even smaller number compared with ; say . Partitioning the goods based on the rows gives an MMAX (and MMA1) allocation . Note that in , every agent gets a value of at least and the least favorite good in values at least to her. Then by removing the least favorite good, the proportional value is at most (the total value of goods is ), which is no more than . Thus for any and .
On the other hand, it is easy to check that for all agent . However, by enumerating all possible allocations, one can check that there does not exist any allocation that guarantees a bundle of value at least for every agent.
Next we show the connection between MMAX and EFX when the valuations are binary additive.
Let be any EFX allocation. Consider any agent and let be the value of ’s bundle under the allocation . Since is EFX, then for any , one of the following statements holds:
and (i.e., agent has value 1 for every good in );
(i.e., may contain more than goods, but agent has value 1 for exact goods).
This is because if both of them do not hold, then and , which means agent has value for at least one good in . Thus agent still envies agent after removing the good with value from .
Note that as long as there exists an agent such that , then . Hence
Otherwise we have . Then by removing any good , we have
In both case, agent is satisfied with respect to MMAX.
However, when agents have general additive valuations, EFX does not imply MMAX. More specifically, for the following example with additive valuations, an EFX allocation exists but is not MMAX.
Suppose and . Agent 1 has the additive valuation shown in Table 2. It is easy to check that , , is an EFX allocation to agent 1. But this is not MMAX to her. Consider the following partition of : , . Then , which are strictly larger than .
The Existence of MMAX Allocations.
First, note that for two agents, MMAX is equivalent to EFX and thus exists for general subadditive valuations [Plaut and Roughgarden, 2018]. Indeed, following exactly the same analysis as in Theorems 4.1 and 4.2, we obtain the following result.
When there are at most three agents having strictly increasing subadditive valuations, or any number of agents having identical strictly increasing subadditive valuations, an MMAX allocation exists.
6 Computing Allocations Efficiently
In this section, we provide a polynomial-time algorithm for computing an allocation such that when all agents have additive valuations every agent is either -MMA or MMAX. Moreover, the allocation our algorithm outputs is -EFX when all agents have subadditive valuations.
Outline of Algorithm.
In each round (each while loop of Algorithm 2), we assign at most one good to each agent that is not envied by any other agent. The assignment of goods to agents in each round is given by a maximum weight matching between the unenvied agents and the unallocated goods , where the weight of every edge is given by the marginal increase in the value, after allocating the good to the corresponding agent. If all edges between and have weight , we assign goods to unenvied agents according to some maximum cardinality matching. Observe that the set of unenvied agents may change after each round, e.g., after getting one more good, an unenvied agent may become envied by some other agents. We repeat the procedure until all goods are allocated. To make sure that there is always an agent not envied by any other agent, we use the approach (procedure in the following) introduced in [Lipton et al., 2004], which is now widely used in the resource allocation literature.
Given an allocation , we define its corresponding envy-graph as follows. Every agent is represented by a node in and there is a directed edge from node to node iff , i.e., is envies . A directed cycle in is called an envy-cycle. Let be such a cycle. If we reallocate to agent for all , and reallocate to agent , the number of edges of will be strictly decreased. Thus, by repeatedly using this procedure, we eventually get another allocation whose envy-graph is acyclic. It is shown in [Lipton et al., 2004] that can be done in time .
Next, we prove the following main result of this section.
Algorithm 2 computes an allocation in polynomial time that is (1) -EFX and EF1 when all agents have subadditive valuations; (2) either -MMA or exact MMAX for each agent when the valuations are additive.
We first show that Algorithm 2 is well defined. In each round, there must be at least one agent who is not envied by anybody, since after Step 6, the envy-graph is acyclic. Thus there will be at least one agent whose in-degree is , i.e., . Moreover, since at least one good will be allocated in each round (each while loop), Algorithm 2 terminates in at most rounds. Given that each round (including procedure , the computation of maximum weight matching, and the updates of edge weights and unenvied agents ) can be done in polynomial time, Algorithm 2 runs in polynomial time.
Fix any agent
. We classify other agents into three sets:
agents not envied by : ;
agents envied by that receive only one good: ;
agents envied by that receive more than one good: .
By definition we have .
We first show that the final allocation is -EFX and EF1 when the valuations are subadditive. Note that it suffices to consider , as the other agents are either not envied by , or allocated only one good.
Fix any agent . Let be the last good allocated to . As we only allocate goods to unenvied agents, we have , which implies that agent is satisfied with respect to EF1. On the other hand, we also have , since otherwise in the first round, matching (which is not allocated yet) with (which is not envied) gives a matching with strictly larger weight. Combing the two inequalities and by subadditivity of valuation , we have
which implies that agent is satisfied with respect to -EF, and -EFX.
Finally, we show that when the valuations are additive, agent is also satisfied with respect to -MMA or MMAX.
The case when , i.e., all other agents receive one good and is envied by agent , is trivial: by removing any good from , there is always some agent with no good. Thus we have
which implies that agent is also satisfied with respect to MMAX.
Now suppose that . In other words, we have , and . We prove that in this case we have , which implies that agent is satisfied with respect to -MMA.
We first prove that is at most the average value of bundles allocated to agents in :
Let . Observe that if we have for every , then
Indeed, the assumption is w.l.o.g., since otherwise we can reset every to , which does not change . The reason is, in any partitioning of into bundles, the bundle with minimum value (whose value is at most ) is not touched.
As we have shown before (for subadditive valuations), for every , we have . Since we have for all , we conclude that
which implies that agent is satisfied with respect to -MMA.
-MMA if at least two agents receive more than one good;
-MMAX if exactly one agent receives more than one good;
Similarly, when the valuations are subadditive, we are able to show that the allocation is -EF, when at least two agents receive more than one good; -EFX, when exactly one agent receives more than one good; and EFX, otherwise.
In this paper, we introduce a novel fairness notions MMA and its two relaxations MMA1 and MMAX in an unaware environment. We study their connections with other fairness notations, and propose an efficient algorithm for computing allocations that is (approximately) MMA and MMAX.
We leave the existence of MMA1 and MMAX allocations for the broader class of valuations as a future direction. Another promising direction would be to extend our work to other preference representations, including ordinal preferences, or to chores instead of goods. Finally, it will be interesting to obtain analogues concepts for asymmetric agents, where agents may have different entitlements or shares.
The authors thank Haris Aziz for reading a draft of this paper and for helpful discussions. This work is partially supported by NSF CAREER Award No. 1553385.
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