When dividing discrete objects, one often strives for a fairness notion called envy-freeness (Foley, 1967), under which no agent prefers the allocation of another agent to its own. Such outcomes might not exist in general (even with only two agents and a single indivisible good), motivating the need for approximations. Among the many approximations of envy-freeness proposed in the literature (Lipton et al., 2004; Budish, 2011; Nguyen and Rothe, 2014; Caragiannis et al., 2016), the one that has found impressive practical appeal is envy-freeness up to one good (EF). In an EF allocation, agent can envy agent as long as there is some good in ’s bundle whose removal makes the envy go away. It is known that an EF allocation always exists and can be computed in polynomial time (Lipton et al., 2004).
A closer scrutiny, however, reveals that EF is not as strong as one might think: In the worst case, an EF allocation could entail the (hypothetical) removal of every good. To see this, consider an instance with six goods and three agents whose (additive) valuations are as follows:
Observe that the allocation shown via circled goods is EF, since any pairwise envy can be addressed by removing an underlined good. However, each pair of agents requires the removal of a different good (e.g., ’s envy towards is addressed by removing whereas ’s envy towards is addressed by removing , and so on), resulting in a weak approximation in aggregate (over all pairs of agents).
The above example shows that EF, on its own, is too coarse to distinguish between allocations that remove a large number of goods (such as the one with circled entries) and those that remove only a few (such as the one with underlined entries, which, in fact, is envy-free). This limitation highlights the need for a fairness notion that (a) can distinguish between allocations in terms of their aggregate approximation, and (b) retains the “up to one good” style approximation of EF that has proven to be practically useful (Goldman and Procaccia, 2014). Our work aims to fill this important gap.
We propose a new fairness notion called envy-freeness up to hidden goods (HEF-), defined as follows: Say there are agents, goods, and an allocation . Suppose there is a set of goods (called the hidden set) such that each agent withholds the goods in (i.e., the hidden goods owned by ) and only discloses the goods in to the other agents. Any other agent only observes the goods disclosed by (i.e., those in ), and its valuation for ’s bundle is therefore instead of . Additionally, agent ’s valuation for its own bundle is (and not ) because it can observe its own hidden goods. If, under the disclosed allocation, no agent prefers the bundle of any other agent (i.e., if for every pair of agents ), then we say that is envy-free up to hidden goods (HEF-). In other words, by withholding the information about , allocation can be made free of envy.
Notice how HEF- addresses the previous concerns: Like EF, HEF- is a relaxation of envy-freeness that is defined in terms of the number of goods. However, unlike EF, HEF- offers a precise quantification of the extent of information that must be withheld in order to achieve envy-freeness.
Of course, any allocation can be made envy-free by hiding all the goods (i.e., if ). The real strength of HEF- lies in being small; indeed, an HEF- allocation is envy-free. As we will demonstrate below, there are natural settings that admit HEF- allocations with a small (i.e., hide only a small number of goods) even when (exact) envy-freeness is unlikely.
Information Withholding is Meaningful in Practice
To understand the usefulness of HEF-, we generated a synthetic dataset where we varied the number of agents from to , and the number of goods from to (we ignore the cases where ). For every fixed and , we generated instances with binary valuations. Specifically, for every agent and every good , the valuation is drawn i.i.d. from . Figure 0(a) shows the heatmap of the number of instances out of that do not admit envy-free outcomes. Figure 0(b) shows the heatmap of the number of goods that must be hidden in the worst-case. That is, the color of each cell denotes the smallest such that each of the corresponding instances admits some HEF- allocation.
It is evident from Figure 1 that even in the regime where envy-free outcomes are unlikely (in particular, the red-colored cells in Figure 0(a)), there exist HEF- allocations with (the light blue-colored cells in Figure 0(b)). This observation, along with the foregoing discussion, shows that fairness through information withholding is a well-motivated approach towards approximate envy-freeness that yields promising existence results in practice.
We make contributions on three fronts.
On the conceptual side, we propose a novel fairness notion called HEF- as a fine-grained generalization of envy-freeness in terms of aggregate approximation.
Our theoretical results (Section 4) show that computing HEF- allocations is computationally hard even for highly restricted classes of valuations (Corollaries 1 and 1). We show a similar result when HEF- is coupled with Pareto optimality (Theorem 2). We also show that finding an envy-free allocation is NP-complete even for binary valuations (Lemma 1). Surprisingly, this fundamental problem was open prior to our work.
Our experiments show that HEF- allocations with a small often exist, even when envy-free allocations do not (Figure 1). We also compare several known algorithms for computing EF allocations on synthetic and real-world preference data, and find that the round-robin algorithm and a recent algorithm of Barman et al. (2018) withhold close-to-optimal amount of information, often hiding no more than three goods (Section 5).
2 Related Work
An emerging line of work in the fair division literature considers relaxations of envy-freeness by limiting the information available to the agents. Notably, Aziz et al. (2018) consider a setting where each agent is aware only of its own bundle and has no knowledge about the allocations of the other agents. They propose the notion of epistemic envy-freeness (EEF) under which each agent believes that an envy-free allocation of the remaining goods among the other agents is possible. Note that in EEF, each agent might consider a different hypothetical assignment of the remaining goods, and each of these could be significantly different from the actual underlying allocation. By contrast, under HEF-, each agent evaluates its valuation with respect to the same (underlying) allocation. Chen and Shah (2017) study a related model where agents have probabilistic beliefs about the allocations of the other agents, and envy is defined in expectation. Chan et al. (2019) study a setting similar to Aziz et al. (2018) where each agent is unaware of the allocations of the other agents, but is guaranteed that its bundle is not the worst.
Another related line of work considers settings where the agents constitute a social network and can only observe the allocations of their neighbors (Abebe et al., 2017; Bei et al., 2017; Chevaleyre et al., 2017; Aziz et al., 2018; Beynier et al., 2018; Bredereck et al., 2018). These works place an informational constraint on the set of agents, whereas our model restricts the set of revealed goods per agent.
An instance of the fair division problem is defined by a set of agents , a set of goods , and a valuation profile that specifies the preferences of every agent over each subset of the goods in via a valuation function . We will assume that the valuation functions are additive, i.e., for any and , , where . We will write instead of for a singleton good . We say that an instance has binary valuations if for every and every , .
An allocation refers to an -partition of the set of goods , where is the bundle allocated to agent . Given an allocation , the utility of agent for the bundle is .
Definition 1 (Envy-freeness).
An allocation is envy-free (EF) if for every pair of agents , . An allocation is envy-free up to one good (EF) if for every pair of agents such that , there exists some good such that . An allocation is strongly envy-free up to one good (sEF) if for every agent such that , there exists a good such that for all , . The notions of EF, EF, and sEF are due to Foley (1967), Budish (2011), and Conitzer et al. (2019) respectively.111A slightly weaker notion than EF was previously studied by Lipton et al. (2004). However, their algorithm can be shown to compute an EF allocation.
Definition 2 (Envy-freeness with hidden goods).
An allocation is said to be envy-free up to hidden goods (HEF-) if there exists a set of at most goods such that for every pair of agents , we have . An allocation is envy-free up to uniformly hidden goods (uHEF-) if there exists a set of at most goods satisfying for every such that for every pair of agents , we have . We say that allocation hides the goods in and reveals the remaining goods. Notice that a uHEF- allocation is also HEF- but the converse is not necessarily true. Indeed, in Proposition 2, we will present an instance that, for some , admits an HEF- allocation but no uHEF- allocation.
It follows from the definitions that an allocation is EF if and only if it is HEF-. It is also easy to verify that an allocation is sEF if and only if it is uHEF-. This is because the unique hidden good for every agent is also the one that is (hypothetically) removed under sEF.
We say that allocation is HEF with respect to set if becomes envy-free after hiding the goods in , i.e., for every pair of agents , we have . We say that goods must be hidden under if is HEF with respect to some set such that , and there is no set with such that is HEF with respect to .
Definition 3 (Pareto optimality).
An allocation is Pareto dominated by another allocation if for every agent with at least one of the inequalities being strict. A Pareto optimal (PO) allocation is one that is not Pareto dominated by any other allocation.
Definition 4 (Ef algorithms).
We will now describe four known algorithms for finding EF allocations that are especially relevant to our work.
Round-robin algorithm (RoundRobin):
Fix a permutation of the agents. The RoundRobin algorithm cycles through the agents according to . In each round, an agent gets its favorite good from the pool of remaining goods.
Envy-graph algorithm (EnvyGraph):
This algorithm was proposed by Lipton et al. (2004) and works as follows: In each round, one of the remaining goods is assigned to an agent that is not envied by any other agent. The existence of such an agent is guaranteed by resolving cyclic envy relations in a combinatorial structure called the envy-graph of an allocation.
Fisher market-based algorithm (Alg-EF1+PO):
This algorithm, due to Barman et al. (2018), uses local search and price-rise subroutines in a Fisher market associated with the fair division instance, and returns an EF and PO allocation. The bound on running time of this algorithm is pseudopolynomial (i.e., is a polynomial in instead of ).
Maximum Nash Welfare solution (Mnw):
The Nash social welfare of an allocation is defined as . The MNW algorithm computes an allocation with the highest Nash social welfare (called a Nash optimal allocation). Caragiannis et al. (2016) showed that a Nash optimal allocation is both EF and PO.
Conitzer et al. (2019) observed that RoundRobin, Alg-EF1+PO, and MNW algorithms all satisfy sEF. It is easy to see that EnvyGraph algorithm is also sEF. However, note that among the above algorithms, only MNW and Alg-EF1+PO are known to also satisfy PO.222It is also known that RoundRobin and EnvyGraph fail to satisfy PO; see, e.g., (Conitzer et al., 2017). The allocations computed by all four algorithms have the property that there exists some agent that is not envied by any other agent. Indeed, MNW and Alg-EF1+PO are both PO and therefore cannot have cyclic envy relations, and RoundRobin and EnvyGraph algorithms have this property by design. For such an agent (not necessarily the same agent for all algorithms), no good needs to be removed under sEF. Therefore, from Remark 1, all these algorithms are also envy-free up to uniformly hidden goods, or uHEF-.
Given an instance with additive valuations, a uHEF- allocation always exists and can be computed in polynomial time, and a allocation always exists and can be computed in pseudopolynomial time.
Note that for any , an HEF- allocation might fail to exist. Indeed, with agents that have identical and positive valuations for goods, some agent will surely miss out and force the allocation to hide all (i.e., or more) goods. Therefore, the bound in Proposition 1 for uHEF- (and hence, for HEF-) is tight in terms of .
3.1 Relevant Computational Problems
Definition 5 formalizes the decision problem of whether a given instance admits an HEF- allocation.
Definition 5 (Hef--Existence).
Given an instance , does there exist an allocation and a set of at most goods such that is HEF with respect to ?
Notice that a certificate for HEF--Existence consists of an allocation as well as a set of at most hidden goods.
Another relevant computational question involves checking whether a given allocation is HEF with respect to some set of at most goods.
Definition 6 (Hef--Verification).
Given an instance and an allocation , does there exist a set of goods such that is HEF with respect to ?
For additive valuations, both HEF--Existence and HEF--Verification are in NP. The next problem pertains to the existence of envy-free allocations.
Definition 7 (Ef-Existence).
Given an instance , does there exist an envy-free allocation for ?
4 Theoretical Results
We will now present our theoretical results concerning the existence and computation of HEF- and uHEF- allocations. Our first result (Proposition 2) shows that uHEF- is a strictly more demanding notion than HEF-.
There exists an instance that, for some , admits an HEF- allocation but no uHEF- allocation.
Consider the fair division instance with five agents and six goods shown in Table 1. Observe that the allocation with , , , , satisfies HEF- with respect to the set .
We will show that does not admit a uHEF- allocation. Suppose, for contradiction, that there exists an allocation satisfying uHEF-. Then, must hide and (otherwise, at least one of , or will envy the owner(s) of these goods). Thus, in particular, the good must be revealed by . Assume, without loss of generality, that is not assigned to in (otherwise, a similar argument can be carried out for ). Then, must assign both and to (so that does not envy the owner of ). However, this violates the one-hidden-good-per-agent property of uHEF-—a contradiction. ∎
Recall from Section 3.1 that HEF--Existence is NP-complete when . This still leaves open the question whether HEF--Existence is NP-complete for any fixed . Our next result (Theorem 1) shows that this is indeed the case, even under the restricted setting of identical valuations (i.e., for every , for every ).
Theorem 1 (Hardness of HEF--Existence).
For any fixed , HEF--Existence is NP-complete even for identical valuations.
We will show a reduction from Partition, which is known to be NP-complete (Garey and Johnson, 1979). An instance of Partition consists of a multiset with for all . The goal is to determine whether there exists such that , where .
We will construct a fair division instance with agents and
goods. The goods are classified intomain goods and dummy goods . The (identical) valuations are defined as follows: Every agent values the goods at respectively; the good at , and each dummy good at .
() Suppose is a solution of Partition. Then, an HEF- allocation can be constructed as follows: Assign the main goods corresponding to the set to agent and those corresponding to to agent . The good is assigned to agent . Each of the remaining agents is assigned a unique dummy good. Note that every agent in the set envies every agent in the set , and these are the only pairs of agents with non-zero envy. Therefore, the allocation can be made envy-free by hiding the dummy goods, i.e., the allocation is HEF with respect to the set .
() Now suppose there exists an HEF- allocation . Since there are dummy goods and agents, there must exist at least three agents that do not receive any dummy good in . Without loss of generality, let these agents be , and (otherwise, we can reindex). We claim that all dummy goods must be hidden under . Indeed, agent does not receive any dummy good, and therefore its maximum possible valuation can be for any dummy good . If some dummy good is not hidden, then will envy the owner of , contradicting HEF-. Therefore, all dummy goods must be hidden, and since there are such goods, these are the only ones that can be hidden.
The above observation implies that the good must be revealed by . Furthermore, must be assigned to one of , or (otherwise, by pigeonhole principle, one of these agents will have valuation at most and will envy the owner of ). If is assigned to , then the remaining main goods must be divided between and such that and . This gives a partition of . ∎
Another commonly used preference restriction is that of binary valuations (i.e., for every and , ). We show that even under this restriction, HEF--Existence remains NP-complete when (Corollary 1). This follows from Lemma 1, which shows that for binary valuations, determining the existence of an envy-free allocation is NP-complete. Somewhat surprisingly, the computational complexity of this fundamental problem was not addressed prior to our work,333We remark that our contribution is to show that EF-Existence remains NP-complete even under binary valuations; without this restriction, NP-completeness was already known (Lipton et al., 2004). and might therefore be of independent interest.444A closely related problem of determining the existence of an envy-free (EF) and Pareto optimal (PO) allocation was shown to be NP-complete for binary valuations by Bouveret and Lang (2008). It is easy to check that our proof of Lemma 1 actually constructs a PO allocation (while assuming only EF), and therefore implies the result of Bouveret and Lang (2008) as a corollary. The proof of Lemma 1 appears in Section 6.1 in the appendix.
EF-Existence is NP-complete even for binary valuations.
For , HEF--Existence is NP-complete even for binary valuations.
Theorem 2 (Hardness of HEF-+Po).
Given any instance with binary valuations and any fixed , it is NP-hard to determine if admits an allocation that is envy-free up to hidden goods and Pareto optimal .
(Sketch) Starting from any instance of EF-Existence with binary valuations (Lemma 1), we add to it new goods and new agents such that all new goods are approved by all new agents (and no one else). Also, the new agents have zero value for the existing goods. In the forward direction, an arbitrary allocation of new goods among the new agents works. In the reverse direction, PO forces each new (respectively, existing) good to be assigned among new (respectively, existing) agents only. The imbalance between new agents and new goods means that all (and only) the new goods must be hidden. Then, the restriction of the HEF- allocation to the existing agents/goods gives the desired EF allocation. ∎
We will now proceed to analyzing the computational complexity of HEF--Verification. Here, we show a hardness-of-approximation result (Theorem 3). The inapproximability factor is stated in terms of the aggregate envy, defined as follows: Given any allocation , the aggregate envy in is the sum of all pairwise envy values, i.e.,
Theorem 3 (Hef--Verification inapproximability).
Given any , it is NP-hard to approximate HEF--Verification to within even for binary valuations, where is the aggregate envy in the given allocation.
We will show a reduction from Hitting Set. An instance of Hitting Set consists of a finite set , a collection of subsets of , and some . The goal is to determine whether there exists , that intersects every member of (i.e., for every , ). It is known that given any , it is NP-hard to approximate Hitting Set to within a factor (Dinur and Steurer, 2014).
We will construct a fair division instance with agents and goods. The agents are classified into dummy agents and one main agent . The goods are classified into main goods and distinct sets of dummy goods, where the set consists of the goods .
The valuations are as follows: The main agent approves all the main goods, i.e., for all , . Each dummy agent approves the dummy goods in the set as well as those main goods that intersect with , i.e., for every , for all , and whenever . All other valuations are set to .
The input allocation is defined as follows: The main agent is assigned all the main goods, i.e., . For every , the dummy agent is assigned the dummy goods in the set, i.e., . Note that in the allocation , each dummy agent envies the main agent by one approved good, and these are the only pairs of agents with envy.
() Suppose , is solution of the Hitting Set instance. We claim that the allocation is HEF with respect to the set with . Indeed, since is induced by a hitting set, each dummy agent approves at least one good in . Therefore, by hiding the goods in , the envy from the dummy agents can be eliminated.
() Now suppose there exists , such that is HEF with respect to . Then, for every , the set must contain at least one good that is approved by the dummy agent (otherwise will not be envy-free after hiding the goods in ). It is easy to see that the set constitutes the desired hitting set of cardinality at most .
Finally, to show the hardness-of-approximation, notice that the aggregate envy in is because each dummy agent envies the main agent by one unit of utility. The claim now follows by substituting in the inapproximability result of Hitting Set stated above. ∎
Our next result (Theorem 4) provides an approximation algorithm that (nearly) matches the hardness-of-approximation result in Theorem 3. We remark that while Theorem 3 holds even for binary valuations, the algorithm works for any instance with additive valuations.
Theorem 4 (Approximation algorithm).
There is a polynomial-time algorithm that, given as input any instance of HEF--Verification, finds a set with such that the given allocation is HEF with respect to . Here, and denote the aggregate envy and the number of goods that must be hidden under the given allocation respectively.
The proof of Theorem 4 is deferred to Section 6.2 in the appendix, but a brief idea is as follows: For any set , define the residual envy function so that is the aggregate envy in allocation after hiding the goods in . That is,
The relevant observation is that is supermodular. Given this observation, the approximation guarantee in Theorem 4 can be obtained by the standard greedy algorithm for submodular maximization, or, equivalently, supermodular minimization (Nemhauser et al., 1978); see Algorithm 1 in Section 6.2.
5 Experimental Results
|Normalized average-case regret|
|Number of goods that must be hidden on average (averaged over non-EF instances only)|
We have seen that the worst-case computational results for HEF-, even in highly restricted settings, are mostly negative (Section 4). In this section, we will examine whether the known algorithms for computing approximately envy-free allocations—in particular, the four EF algorithms described in Definition 4 in Section 3—can provide meaningful approximations to HEF- in practice. Recall from Footnote 2 that all four discussed algorithms—RoundRobin, MNW, Alg-EF1+PO, and EnvyGraph—satisfy uHEF-.
We evaluate each algorithm in terms of (a) its regret (defined below), and (b) the number of goods that the algorithm must hide. Given an instance and an allocation , let denote the number of goods that must be hidden under . The regret of allocation is the number of extra goods that must be hidden under compared to the optimal. That is, . Similarly, given an algorithm Alg, the regret of Alg is given by , where is the allocation returned by Alg for the input instance . Note that the regret can be large due to the suboptimality of an algorithm, but also due to the size of the instance. To negate the effect of the latter, we normalize the regret value by , which, as discussed above, is the worst-case upper bound on the number of hidden goods for all four algorithms of interest.
5.1 Experiments on Synthetic Data
The setup for synthetic experiments is similar to that used in Figure 1. Specifically, the number of agents, , is varied from to , and the number of goods, , is varied from to (we ignore the cases where ). For every fixed and , we generated instances with binary
valuations drawn i.i.d. from Bernoulli distribution with parameter(i.e., ). Table 2 shows the heatmaps of the normalized regret (averaged over instances) and the number of goods that must be hidden (averaged over non-EF instances, i.e., whenever ) for all four algorithms.555The appendix provides additional results for in Table 4, and for in Table 5.
It is clear that Alg-EF1+PO and RoundRobin algorithms have a superior performance than MNW and EnvyGraph. In particular, both Alg-EF1+PO and RoundRobin have small normalized regret, suggesting that they hide close-to-optimal number of goods. Additionally, the number of hidden goods itself is small for these algorithms (in most cases, no more than three goods need to be hidden), suggesting that the worst-case bound of is unlikely to arise in practice. Overall, our experiments suggest that Alg-EF1+PO and RoundRobin can achieve useful approximations to HEF- in practice, especially in comparison to MNW and EnvyGraph.666In Section 6.3 in the appendix, we present two families of instances where the normalized worst-case regret of MNW is large.
5.2 Experiments on Real-World Data
For experiments with real-world data, we use the data from the popular fair division website Spliddit (Goldman and Procaccia, 2014). The Spliddit data has instances in total, where the number of agents varies between and , and the number of goods varies between and . Unlike the synthetic data, the distribution of instances here is rather uneven (see Figure 3 in the appendix); in fact, of the instances have agents and
goods. Therefore, instead of using heatmaps, we compare the algorithms in terms of their normalized regret (averaged over the entire dataset) and the cumulative distribution function of the hidden goods (seeFigure 2).
Figure 2 presents an interesting twist: MNW is now the best performing algorithm, closely followed by RoundRobin and Alg-EF1+PO. For any fixed , the fraction of instances for which these three algorithms compute an HEF- allocation is also nearly identical. As can be observed, these algorithms almost never need to hide more than three goods. By contrast, EnvyGraph has the largest regret and significantly worse cumulative performance. Therefore, once again, Alg-EF1+PO and RoundRobin algorithms perform competitively with the optimal solution, making them attractive options for achieving fair outcomes without withholding too much information.
We are grateful to Ariel Procaccia and Nisarg Shah for sharing with us the data from Spliddit, and to Rupert Freeman and Neeldhara Misra for very helpful discussions. LX acknowledges NSF #1453542 and #1716333 for support.
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6.1 Proof of Lemma 1
Our proof uses a reduction from Equitable Coloring, which is defined below.
Definition 8 (Equitable Coloring).
Given a graph and a number , does there exist a proper coloring of such that all color classes are of equal size?
The standard definition of Equitable Coloring requires the color classes to differ in size by at most one. We overload the term to refer to the version where all color classes are of the same size. Equitable Coloring (in Definition 8) can be shown to be NP-complete by a straightforward reduction from Graph -Colorability (Garey and Johnson, 1979). In addition, we can assume without loss of generality.
(of Lemma 1) We will show a reduction from Equitable Coloring. Recall from Definition 8 that an instance of Equitable Coloring consists of a graph and a number . The goal is to determine a proper coloring of where the color classes are of the same size. For simplicity, we will write and .777Not be confused with the number of agents, , and the number of goods, , as defined in Section 3. Note that we can assume, without loss of generality, that is connected. Since a connected graph with vertices has at least edges, we have that
In addition, we will also assume that each vertex in has degree at least two. Indeed, for any degree one vertex that is adjacent to some edge , we can add new vertices to create the following cycle: . Call the new graph . It is easy to see that has an equitable -coloring if and only if does.
We will construct a fair division instance with goods and agents. The agents are classified into edge agents and dummy agents . The goods are classified into vertex goods and edge goods . Note that we use the same notation for the vertices (edges) and the corresponding vertex (edge) goods.
The preferences of the agents are defined as follows: For every edge , an edge agent approves all the edge goods and exactly two vertex goods and . Each dummy agent approves all the vertex goods and has zero value for the edge goods.
() Suppose admits an equitable coloring with each color class of size . Then, an envy-free allocation can be constructed as follows: Assign each edge good to the edge agent and each vertex good to the dummy agent if vertex has color . Notice that all goods are allocated under . Also note that no two edge agents envy each other since each of them gets exactly one edge good. Furthermore, due to the proper coloring condition, for any edge in , the corresponding vertex goods and are assigned to distinct dummy agents. Hence, no edge agent envies a dummy agent. The dummy agents have zero value for the edge goods and therefore do not envy the edge agents. Finally, since all color classes are of the same size, each dummy agent gets exactly approved goods, and therefore does not envy any other dummy agent. Overall, the allocation is envy-free.
No edge agent can get two or more edge goods under .
(of Property 1) Suppose, for contradiction, that some edge agent gets two or more edge goods. Then, any other edge agent has a utility of at least for the bundle of . For to be envy-free, must have a utility of at least for its own bundle. For binary valuations, this means that must be assigned two or more goods that it approves. Therefore, we need at least goods to satisfy the edge agents. The total number of available goods is , which, using Equation 1, evaluates to at most . This leaves at most one good to be allocated among dummy agents. Since (Definition 8), some dummy agent is bound to be envious, contradicting the envy-freeness of . ∎
Every dummy agent gets at least one vertex good under .
(of Property 2) Fix a vertex good and a dummy agent . Then, either is assigned to , or gets some other (approved) good to prevent it from envying the owner of . Since the only goods approved by the dummy agents are the vertex goods, the claim follows. ∎
No dummy agent can get an edge good under .
(of Property 3) Suppose, for contradiction, that a dummy agent gets an edge good under . From Property 2, we know that also gets some vertex good, say . By assumption, the graph has minimum degree two, so there must exist some edge adjacent to the vertex . Notice that the edge agent has a utility of (at least) for the bundle of . Therefore, for to be envy-free, must get at least two goods that it approves. Property 1 limits the number of edge goods assigned to any edge agent to at most one. Therefore, in addition to some edge good, must also get the vertex good . Once again using the bound on minimum degree of , we get that there must exist some edge adjacent to the vertex . A similar argument shows that the vertex good must be assigned to the edge agent . Continuing in this manner, we will eventually encounter an edge such that is already assigned to and is already assigned to either or some other edge agent. This would imply that is envious of some other agent under —a contradiction. ∎
Every edge agent gets exactly one edge good under .
No edge agent can get a vertex good under .
(of Property 5) The argument is similar to that of Property 3. Suppose, for contradiction, that some edge agent is assigned a vertex good . Let be an edge incident to the vertex in (such an edge must exist due to the bound on minimum degree). From Property 4, we know that each edge agent gets exactly one edge good. Thus, the edge agent has a utility of (at least) for the bundle of the agent . For to be envy-free, must receive two or more goods that it approves, only one of which can be an edge good. Therefore, agent must also receive the vertex good . Now let be an edge incident to the vertex in . A similar argument implies that the vertex good must be assigned to the edge agent . Continuing in this manner, we will encounter an edge such that the vertex good is assigned to the edge agent and the vertex good is assigned to some other edge agent. Thus, the edge agent has a utility of for the bundle of some other edge agent, even though it values its own bundle at . This contradicts the envy-freeness of . ∎
For any edge , no dummy agent is assigned both vertex goods and under .
It follows from Property 5 that all vertex goods must be allocated among the dummy agents. Now consider the following coloring of the graph : For each vertex , the color of is the index of the dummy agent that gets the vertex good . Property 6 implies that the coloring is proper. Furthermore, due to envy-freeness of , agents with identical valuations must have equal utilities. Therefore, each dummy agent gets the same number of vertex goods, implying that the coloring is equitable. This completes the proof of Lemma 1. ∎
6.2 Proof of Theorem 4
Recall the statement of Theorem 4.
Recall from Section 4 that given any allocation , the residual envy function is defined as follows:
Here, is the aggregate envy in after hiding the goods in . We will show in Lemma 2 that is supermodular, i.e., for any pair of sets such that and any good , . The proof of Theorem 4 will then follow from the standard greedy algorithm for submodular maximization, or, equivalently, supermodular minimization (Nemhauser et al., 1978).
The residual envy function is supermodular.
We will start with the necessary notation. For any agent and any other agent , define as the envy of towards after hiding the goods in . Also, let denote the total (aggregate) envy towards . We therefore have .
Notice that is a monotone non-increasing set function, i.e., for any , we have . Also notice that for any and any , we have that .
For any set of goods and any agent , define as the set of agents that envy agent even after the goods in are hidden. Notice that if , then . Thus, if for some agent we have that , then , and therefore .
Define as the set of agents that have a strictly positive valuation for the good .
We will now prove that is supermodular, i.e., for any and any good , . Let be the owner of good under , i.e., . Notice that if (i.e., ), then additivity of valuations implies . Thus,
where the first equality uses the fact that for any , we have , the third equality uses the fact that if , then , and the fourth equality uses the fact that whenever . By a similar reasoning for the set , we get that
Recall that . Therefore, Equation 2 can be rewritten as
where the inequality follows from the use of the monotonicity of for all .
Suppose, for contradiction, that for some , we have . Then, we must have , since otherwise we get and therefore . This would imply that , which contradicts the monotonicity of . Hence, for any , we also have that .
Notice that for any , we have by the additivity of valuations. However, this would require that , which is a contradiction. Therefore, the function must be supermodular. ∎
We are now ready to prove Theorem 4.
(of Theorem 4) Note that allocation is HEF with respect to a set if and only if . For integral valuations, if and only if . Therefore, it suffices to compute a set in polynomial time such that and .
Consider the greedy algorithm described in Algorithm 1.
At each step, the algorithm adds to the current set the good that provides the largest reduction in the residual envy. This process is continued as long as . Since there are goods, it is clear that the algorithm terminates in at most steps. Furthermore, from the above observation, it follows that the allocation is HEF with respect to the set returned by the algorithm. Therefore, all that remains to be shown is a bound on .
Observe that . Recall from the proof of Lemma 2 that is a sum of monotone non-increasing set functions, and is therefore itself monotone non-increasing. Define another set function as follows:
Notice that is a non-negative, monotone non-decreasing, and integer-valued submodular function with . Therefore, our goal is to find a set such that .
We will now use the result of Nemhauser et al. (1978) for submodular maximization stated below as Proposition 3. In particular, let be the size of the optimal hidden set (i.e., the number of goods that must be hidden under ). Then,
From the bound in Proposition 3, we have that