# Equitable Allocations of Indivisible Goods

In fair division, equitability dictates that each participant receives the same level of utility. In this work, we study equitable allocations of indivisible goods among agents with additive valuations. While prior work has studied (approximate) equitability in isolation, we consider equitability in conjunction with other well-studied notions of fairness and economic efficiency. We show that the Leximin algorithm produces an allocation that satisfies equitability up to any good and Pareto optimality. We also give a novel algorithm that guarantees Pareto optimality and equitability up to one good in pseudopolynomial time. Our experiments on real-world preference data reveal that approximate envy-freeness, approximate equitability, and Pareto optimality can often be achieved simultaneously.

• 11 publications
• 14 publications
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01/24/2021

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## 1 Introduction

We consider fair division problems that require a central planner to divide a set of goods among a group of agents—each with their own individual preferences over the goods—such that the resulting allocation is fair. How exactly one can certify that an allocation is “fair” remains a subject of debate, but the literature suggests two distinct viewpoints. In the first viewpoint, an agent should prefer her bundle of goods to some comparison bundle. The gold standard of fairness here is envy-freeness, which says that each agent should prefer her bundle of goods to any other agents’ bundle.

In this work, we consider the second viewpoint, in which agents compare their happiness levels, or utilities. Here, an allocation is considered fair if the planner is able to make all agents equally well-off. A central fairness notion in this context is equitability: An equitable allocation is one where agents derive equal utilities from their assigned shares. Stated differently, an equitable allocation seeks to minimize the disparity between the best-off and the worst-off agents.

Both perspectives have merit, but the practical importance of equitability as a fairness criterion has been highlighted in an experimental study conducted by Herreiner and Puppe (2009). They asked human subjects to deliberate over an assignment of indivisible goods subject to a time limit. It was found that the chosen outcomes were equitable (and Pareto optimal) far more often than they were envy-free. They concluded that equitability is a significant predictor of the perceived fairness of an allocation, often more so than envy-freeness.

Like many other fairness notions, equitability has been traditionally studied for divisible goods (also called cake-cutting). In this setting, it is known that an equitable allocation always exists (Dubins and Spanier, 1961; Alon, 1987). On the computability side, it is known that no finite procedure can find an (exact) equitable division (Procaccia and Wang, 2017), though an -equitable division can be computed in a finite number of steps (Cechlárová and Pillárová, 2012a, b).

For indivisible goods, an equitable (EQ) allocation might fail to exist even with two agents and a single good, motivating the need for approximations. To this end, Gourvès et al. (2014) proposed the notion of near jealousy-freeness, under which for any pair of agents, the disparity can be reversed by removing any good from the bundle of the agent with higher utility. We refer to this notion as equitability up to any good (EQx) in keeping with the nomenclature for a similar relaxation of envy-freeness (Caragiannis et al., 2016). We also study equitability up to one good (EQ1), requiring only that inequity can be eliminated by removing some good from the higher-utility-agent’s bundle. Gourvès et al. (2014) showed that for additive valuations, an EQx (hence, EQ1) allocation always exists and can be computed in polynomial time. However, they did not study Pareto optimality (PO), a fundamental and often desirable notion of economic efficiency that may still be violated by an (approximately) equitable allocation.

Our work takes a deeper dive into the study of (approximately) equitable allocations of indivisible goods—in conjunction with Pareto optimality as well as other well-studied notions of fairness (envy-freeness and its relaxations)—and considers a host of existence and computational questions. Table 1 provides a comprehensive summary of our results. Some of the highlights are:

• We strengthen the aforementioned result of Gourvès et al. (2014) to show that an EQx and PO allocation always exists for strictly positive valuations (creftypecap 3). Without the positivity assumption, even an EQ1+PO allocation might fail to exist (creftypecap 1), and finding an EQ+PO/EQx+PO/EQ1+PO allocation becomes strongly NP-hard (Remarks 1 and 1).

• As a step towards making the above existence result constructive, we design a pseudopolynomial-time algorithm that always returns an EQ1+PO allocation for strictly positive valuations (Theorem 3).

• We construct an instance in which no allocation can be EQ1+EF1+PO (creftypecap 4).111EF1 stands for envy-freeness up to one good, which is a (necessary) relaxation of envy-freeness defined for indivisible goods; see Section 2 for the relevant definitions. We show that determining whether such an allocation exists is, in general, strongly NP-hard (Corollary 1), but the special case of binary valuations is efficiently solvable (Theorem 4).

• We validate our theoretical results via experiments on the data from the popular fair division website Spliddit as well as on synthetically generated instances (Section 4).

##### Related Work

For divisible goods (i.e., cake-cutting), Dubins and Spanier (1961) showed that an equitable division always exists (without providing a bound on the number of cuts). Subsequent work has established the existence of equitable divisions where each agent gets a contiguous piece (Cechlárová et al., 2013; Aumann and Dombb, 2015; Chèze, 2017).

Equitability has also been studied in combination with other fairness and efficiency notions. It is known that there always exists a cake division that is simultaneously equitable and envy-free (Alon, 1987). However, existence might fail if, in addition, one also requires Pareto optimality (Brams et al., 2013) or contiguous pieces (Brams et al., 2006). Connections between Pareto optimality and social welfare maximizing equitable divisions have also been studied (Brams et al., 2012).

For indivisible goods, in addition to the work of Gourvès et al. (2014) discussed above, Suksompong (2019) studies equitable and connected allocations of indivisible goods (i.e., when the goods constitute the vertices of a graph and a feasible allocation assigns every agent a connected subgraph).

## 2 Preliminaries

##### Problem instance

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 .333The assumption about integrality of valuations is required only for Theorem 3. All other positive results (i.e., existence and algorithmic results) hold even in the absence of this assumption. Similarly, all negative results (i.e., non-existence and hardness results) hold even if the valuations are restricted to be integral. We will assume that the valuation functions are additive, i.e., for any agent and any set of goods , , where . For a singleton good , we will write instead of .

##### Allocation

An allocation is an -partition of the set of goods , where is the bundle allocated to the agent ( is allowed to be an empty set). Given an allocation , the utility of agent for the bundle is .

##### Equitable allocations

An allocation is said to be equitable (EQ) if for every pair of agents , we have . An allocation is equitable up to one good (EQ1) if for every pair of agents such that , there exists some good such that . An allocation is equitable up to any good (EQx) if for every pair of agents such that and for every good such that , we have .444Our results hold analogously for the following variant of EQx due to Gourvès et al. (2014): For every pair of agents such that , for every good .

##### Envy-free allocations

An allocation is envy-free (EF) if for every pair of agents , we have . An allocation is envy-free up to one good (EF1) if for every pair of agents such that , there exists some good such that . An allocation is envy-free up to any good (EFx) if for every pair of agents such that and for every good such that , we have . The notions of EF, EF1, and EFx are due to Foley (1967), Budish (2011),555 Lipton et al. (2004) previously defined a slightly weaker notion than EF1, but their algorithm can, in fact, compute an EF1 allocation. and Caragiannis et al. (2016), respectively.

##### 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.

##### Nash social welfare

Given an instance , the Nash social welfare of an allocation is defined as . An allocation is called Nash optimal or MNW (Maximum Nash Welfare) if it maximizes the Nash social welfare among all allocations.666Caragiannis et al. (2016)

define a Nash optimal allocation as one that provides positive utility to the largest set of agents, and subject to that, maximizes the geometric mean of valuations. Our results hold even under this extended definition.

##### Leximin-optimal allocations

A Leximin-optimal allocation (Dubins and Spanier, 1961) is one that maximizes the minimum utility that any agent achieves, subject to which the second-minimum utility is maximized, and so on. The utilities induced by a Leximin-optimal allocation are unique, although there may exist more than one such allocation.

## 3 Results

This section presents our theoretical results, summarized in Table 1. We first consider equitability and its relaxations, then consider them in conjunction with Pareto optimality, before finally adding envy-freeness (and its relaxations) to the mix.

### 3.1 Existence and Computation of EQ, EQ1, EQx

We will start by observing that envy-freeness and equitability (and their corresponding relaxations) become equivalent when the valuations are identical (i.e., when, for every good , for all ).

###### Proposition 1.

For identical valuations, an allocation is EF/EF1/EFx if and only if it is EQ/EQ1/EQx.

It is known that determining whether a given instance has an envy-free (EF) allocation is NP-complete even for identical valuations (via a straightforward reduction from Partition) (Lipton et al., 2004).777In fact, the problem is strongly NP-complete due to a similar reduction from 3-Partition (Garey and Johnson, 1979). creftypecap 1 implies that the same holds for equitable (EQ) allocations. By contrast, an EQx (and therefore EQ1) allocation always exists and can be efficiently computed (creftypecap 2) even for non-identical valuations. This result is due to Gourvès et al. (2014), who showed the existence of EQx allocations under the more general setting of matroids.

###### Proposition 2 (Gourvès et al., 2014).

An EQx allocation always exists and can be computed in polynomial time.

Briefly, Gourvès et al. (2014) prove creftypecap 2 using a greedy algorithm. In each round, the algorithm assigns a least-happy agent its favorite good from among the remaining goods. Thus, at any stage, the most recent good assigned to an agent is also its least-favorite good in its own bundle. Since each new good is assigned to an agent with the least utility, an allocation that is EQx prior to the assignment continues to be so after it (up to the removal of the most recently assigned good). The claim now follows by induction over the rounds.

creftypecap 2 presents an interesting contrast between the notions of EQx and EFx: An EQx allocation is guaranteed to exist and can be efficiently computed, whereas for EFx, even the question of guaranteed existence is an open problem.

### 3.2 Equitability and Pareto Optimality

We now turn our attention to computing an allocation that is both equitable up to one good and Pareto optimal (we use the shorthand EQ1+PO for such allocations). Unfortunately, such allocations might fail to exist when the valuations are allowed to be zero-valued (creftypecap 1). This provides an interesting contrast with the analogous relaxation of envy-freeness; it is known that an allocation satisfying EF1 and PO always exists (Caragiannis et al., 2016; Barman et al., 2018a).

###### Example 1 (Non-existence of EQ1+PO).

Consider an instance with three agents and six goods . The goods are valued at by and at by and . The goods are valued at by and and at by . Any PO allocation must assign to (giving it a utility of ) and allocate between and . Either or receives at most one good, creating an EQ1 violation with . Thus, an EQ1 and PO allocation might fail to exist even under binary valuations. ∎

Worse still, when the valuations can be zero-valued, determining whether there exists an EQ1+PO allocation is strongly NP-hard. Similar hardness results hold for EQx+PO and EQ+PO allocations as well (Remark 1).

###### Theorem 1 (Hardness of EQ1 + PO).

Given any fair division instance with additive valuations, determining whether there exists an allocation that is equitable up to one good and Pareto optimal is strongly NP-hard.

###### Proof.

We will show a reduction from 3-Partition, which is known to be strongly NP-hard (Garey and Johnson, 1979). An instance of 3-Partition consists of a set of numbers where , and the goal is to find a partition of into subsets such that the sum of numbers in each subset is , where .888Note that we do not require to be of size three each; 3-Partition remains strongly NP-hard even without this constraint.

We will construct a fair division instance as follows: There are agents and goods . For every and , agent values the good at . The agents all value the goods and at . Finally, the agent values and at each, and all other goods at .

Suppose is a solution of 3-Partition. Then, an EQ1 and PO allocation can be constructed as follows: For every , , and . Notice that is EQ1 because each of the agents has utility , and the utility of the agent exceeds only by a single good . Furthermore, is PO because each good is assigned to an agent with the highest valuation for it.

Now suppose that is an EQ1 and PO allocation. Since is PO, it must assign and to . Furthermore, since is EQ1, each of the agents should have a utility of at least under , i.e., for every , . This induces a solution of the 3-Partition instance. ∎

###### Remark 1 (Hardness of EQx+PO/EQ+PO).

The reduction in Theorem 1 can also be used to prove strong NP-hardness of finding an EQx+PO allocation (same construction works) or an EQ+PO allocation (if values at ).

Our next result shows that for the special case of binary valuations (i.e., for all , ), an EQ+PO allocation, if it exists, can be computed in polynomial time. Later, we will show similar tractability results for EQ1+PO and EQx+PO allocations (Theorem 4).

###### Theorem 2 (Algorithm for EQ+PO for binary valuations).

There is a polynomial-time algorithm that given as input any fair division instance with additive and binary valuations, returns an allocation that is equitable and Pareto optimal whenever such an allocation exists.

###### Proof.

We will use a maximum flow algorithm. For binary valuations, an allocation is PO if and only if it assigns each good to an agent that approves it. For an EQ allocation , we have (say) for every . Consider a bipartite graph over the set of agents and goods with an edge for every and such that . For any fixed , construct a flow network where the source node is connected to each agent node in with an edge of capacity . Each node corresponding to a good in is connected to the sink node with an edge of capacity . The edges between agents and goods are of capacity . It is straightforward to check that there exists an EQ+PO allocation in the fair division instance (with common utility ) if and only if the above network admits a feasible flow of value . The desired algorithm simply iterates over all integral values of between and . ∎

On the other hand, when all valuations are strictly positive (i.e., for all ), there always exists an allocation that is both equitable up to any good and Pareto optimal.

###### Proposition 3 (Existence of EQx+PO for positive valuations).

Given any fair division instance with additive and strictly positive valuations, an allocation that is equitable up to any good and Pareto optimal always exists.

###### Proof.

(Sketch.) We will show that any Leximin-optimal allocation, say , satisfies EQx (Pareto optimality is easy to verify). Suppose, for contradiction, that there exist agents and some good such that . Let be an allocation derived from by transferring the good from agent to agent . Notice that under , both agents and have strictly greater utility than , while all other agents have exactly the same utility as under . Thus, is a ‘Leximin improvement’ over , which contradicts that is Leximin-optimal. ∎

Although creftypecap 3 offers a strong existence result, it does not automatically provide a constructive procedure for finding such allocations. Indeed, computing a Leximin-optimal allocation is known to be intractable (Bezáková and Dani, 2005; Plaut and Roughgarden, 2018). Our next result (Theorem 3) addresses this gap by providing a pseudopolynomial-time algorithm for finding an EQ1 and PO allocation when the valuations are strictly positive.

###### Theorem 3 (Algorithm for EQ1+PO for positive valuations).

Given any fair division instance with additive and strictly positive valuations, an allocation that is equitable up to one good and Pareto optimal always exists and can be computed in time, where .

In particular, when the valuations are polynomially bounded (i.e., for every and , ), our algorithm runs in polynomial time. In contrast, computing a Leximin-optimal allocation remains NP-hard even under this restriction (Bezáková and Dani, 2005).

One might expect to prove Theorem 3 via a standard relax-and-round approach: Start with a fractional maximin allocation (i.e., a fractional allocation that maximizes the minimum utility) followed by a rounding step. However, in creftypecap 2, we provide an instance where every rounding of the fractional maximin solution fails to satisfy EQ1. Therefore, the relax-and-round approach might be inadequate for finding EQ1+PO allocations.

Our proof of Theorem 3 is deferred to Section 6.1 but a brief idea is as follows: Our algorithm (Algorithm 1) uses the framework of Fisher markets (Brainard and Scarf, 2000), which are well-studied models of a set of buyers spending their budgets of virtual money on utility-maximizing bundles of goods. Standard welfare theorems in economics guarantee that equilibrium (i.e., market clearing) outcomes in these markets are economically efficient. However, such outcomes could, in general, lead to fractional allocations and be highly inequitable. Our algorithm addresses the first challenge by starting with (and always maintaining) an integral equilibrium of some Fisher market. To meet the second challenge, our algorithm uses a combination of local search and price-rise routines to gradually move towards an approximately equitable equilibrium. The analysis for achieving the desired running time and correctness guarantees is intricate, and involves a number of structural observations and potential function arguments.

Our techniques are inspired from a similar recent algorithm of Barman et al. (2018a) for finding allocations that are envy-free up to one good (EF1) and Pareto optimal (PO). A key difference between the two algorithms lies in the way a local improvement is defined: For Barman et al. (2018a), a local improvement is defined in terms of equalizing the agents’ spendings, whereas for us, it pertains to equalizing the agents’ utilities. We believe that the latter approach is more direct, and leads to a simpler algorithm and analysis. This distinction is also necessary, because as we will show in creftypecap 4, an EQ1+EF1+PO allocation might fail to exist even with strictly positive valuations. Therefore, any algorithm that is tailored to return an EF1 outcome—including the algorithm of Barman et al. (2018a)—will invariably fail to find the desired EQ1+PO allocation, motivating the need for an alternative approach.

Given the success of market-based algorithms in finding EQ1+PO allocations, it is natural to ask whether these techniques can be extended to find an EQx+PO allocation. Unfortunately, this is where these techniques hit a roadblock. The problem stems from the fact that the market-based algorithm always outputs a fractionally Pareto optimal (fPO) allocation (refer to Section 6.1 for the definition), but there exist instances where no EQx allocation satisfies fPO (Section 6.6). Whether an EQx+PO allocation can be computed in (pseudo-)polynomial time with strictly positive valuations is an intriguing question for future research.

### 3.3 Equitability, Envy-Freeness and Pareto Optimality

We will now consider all three notions—equitability, envy-freeness, and Pareto optimality—together. Recall from creftypecap 3 that for strictly positive valuations, an EQ1+PO (in fact, an EQx+PO) allocation is guaranteed to exist. It is also known that an EF1+PO allocation always exists. One might therefore ask whether an EQ1+EF1+PO allocation also always exists. Our next result (creftypecap 4) dismisses that possibility.

###### Proposition 4 (Non-existence of EQ1+EF1+PO).

There exists an instance with strictly positive valuations in which no allocation is simultaneously equitable up to one good , envy-free up to one good and Pareto optimal .

###### Proof.

Fix some and . Consider an instance with agents and goods . Each of values each of at and each of at . Agent values every good at . By the pigeonhole principle for the goods , some agent among must have utility at most . This means that can be assigned at most one good (otherwise EQ1 is violated). Therefore, if all the goods are allocated (which is a necessary condition for a PO allocation), at least goods must be assigned among . This means that one of these agents gets at least three goods, creating an EF1 violation with . ∎

###### Remark 2.

creftypecap 4 has several interesting implications. First, it shows that a Nash optimal allocation—which is guaranteed to be EF1 and PO (Caragiannis et al., 2016)—need not satisfy EQ1. Similarly, the algorithm of Barman et al. (2018a) for computing an EF1 and PO allocation could also fail to return an EQ1 allocation. By contrast, our algorithm in Theorem 3 is guaranteed to find an EQ1 and PO allocation. Finally, it shows that the Leximin-optimal allocation—which is guaranteed to be EQx and PO for strictly positive valuations (creftypecap 3)—need not be EF1.

##### Comparison with cake-cutting

It is worth comparing creftypecap 4 with the corresponding results for divisible goods (i.e., cake-cutting). Brams et al. (2013) have shown that there might not exist a division of the cake that simultaneously satisfies EQ, EF, and PO. Our result in creftypecap 4 shows an analogous impossibility for indivisible goods. Interestingly, the impossibility for cake-cutting goes away when PO is relaxed to completeness (i.e., only requiring that the entire cake is allocated). Under this relaxation, it is known that a perfect allocation of the cake exists (Alon, 1987).999An allocation is perfect if for every , . By contrast, for indivisible goods, the impossibility remains even when PO is relaxed to completeness and EF1 is relaxed to proportionality up to one good (Prop1).101010An allocation is proportional if for every , we have . An allocation is proportional up to one good (Conitzer et al., 2017) if for every , there exists a good such that . Indeed, the proof of creftypecap 4 works even under these relaxations. Moreover, the proof can be easily extended to show the non-existence of EQk, Prop and complete allocations for any constants .

We now turn to the computational aspects of allocations with all three properties. Note that the allocation constructed in the proof of Theorem 1 is envy-free. Therefore, from Remarks 1 and 1, we obtain strong NP-hardness of all combinations of the three properties.

###### Corollary 1 (Hardness of EF+EQ+PO).

Let , , and . Then, determining whether a given instance admits an allocation that is simultaneously , , and is strongly NP-hard.

The intractability in Corollary 1 can, in certain cases, be alleviated when the valuations are restricted to be binary. We will start with an observation concerning EQ and PO allocations under this restriction.

###### Proposition 5.

For binary valuations, an allocation that is equitable and Pareto optimal is also envy-free .

###### Proof.

Suppose each agent gets a utility under the said EQ allocation. For binary valuations, PO implies that each agent approves all the goods in its bundle. Furthermore, any other agent gets at most goods approved by (simply because agent gets exactly goods). Hence, the allocation is EF. ∎

###### Remark 3.

creftypecap 5 shows that for binary valuations, an EQ+PO allocation (if it exists) is, in fact, EQ+PO+EF (hence also EQ+PO+EFx/EQ+PO+EF1). From Theorem 2, we know that there is a polynomial-time algorithm for determining whether an instance with binary valuations admits an EQ+PO allocation. A similar implication therefore also holds for EQ+PO+EF/EQ+PO+EF1/EQ+PO+EFx allocations.

Theorem 4 shows that binary valuations are also useful when one considers the combination of EQ1, EF1, and PO.

###### Theorem 4 (Algorithm for EQ1+EF1+PO for binary valuations).

There is a polynomial-time algorithm that given as input any fair division instance with additive and binary valuations, returns an allocation that is equitable up to one good , envy-free up to one good , and Pareto optimal , whenever such an allocation exists.

The proof of Theorem 4 is provided in Section 6.7. The idea is to show that any EQ1+PO allocation, if it exists, is also Nash optimal. For binary valuations, all Nash optimal allocations induce identical utility profiles (up to renaming of agents). As a result, every Nash optimal allocation satisfies EQ1. It is known that every Nash optimal allocation satisfies EF1 and PO (Caragiannis et al., 2016). Moreover, for binary valuations, a Nash optimal allocation can be computed in polynomial time (Darmann and Schauer, 2015; Barman et al., 2018b). Therefore, determining the existence of an EQ1+EF1+PO allocation reduces to checking whether an arbitrary Nash optimal allocation satisfies EQ1, which can be done in polynomial time.

Notice that for binary valuations, a Pareto optimal allocation is EF1 if and only if it is EFx, and is EQ1 if and only if it is EQx. Therefore, when the valuations are binary, the above algorithm works for all combinations of + + PO, where and .

We conclude this section by observing that some of the problems discussed in Corollary 1 continue to be intractable even for binary valuations. This follows from a result of Bouveret and Lang (2008), who showed that finding an envy-free (EF) and Pareto optimal (PO) allocation under binary valuations is NP-complete (refer to Proposition 21 in their paper).

###### Proposition 6 (Bouveret and Lang, 2008).

Given any fair division instance with additive and binary valuations, determining whether there exists an envy-free and Pareto optimal allocation is NP-complete.

###### Remark 4.

It is easy to verify that the allocation constructed in the reduction of Bouveret and Lang (2008) is, without loss of generality, equitable up to one good (EQ1). Therefore, for binary valuations, determining whether there exists an allocation that is EF + EQ1+ PO/EF + EQx+ PO is NP-complete.

## 4 Experiments

In this section, we compare the proposed and existing algorithms (in particular, Alg-eq1+po, MNW, and Leximin) in terms of how frequently they satisfy various fairness and efficiency properties in the real-world and synthetic datasets.

For real-world preferences, we used the data obtained from the popular fair division website Spliddit (Goldman and Procaccia, 2014). Out of the instances in the Spliddit data, we used the instances that had strictly positive valuations and . The instances have between and agents, and between and goods.111111More than of the instances have three agents and six goods. Users are restricted to normalized, integral valuations. For synthetic data, we generated instances with , , and (strictly positive) valuations drawn i.i.d. from Dirichlet distribution. The concentration parameter for each item is set to to generate normalized valuations.121212We normalize the valuations in the synthetic data to allow for a fair comparison with the Spliddit data, which has normalized valuations by design. We remark that all algorithms studied in this paper work even in the absence of this assumption.

We consider the following combinations of fairness and efficiency properties: EQ+PO, EQ1+PO, EQx+PO, EQ1+EF1+PO, and EQx+EFx+PO. For each instance of the Spliddit and synthetic datasets, we check whether the property is satisfied by the output of Alg-eq1+po, MNW, and Leximin. Figure 1 presents the relevant histograms.131313All codes and synthetic data generation files are available at https://github.com/sujoyksikdar/fairdivision. Note that each of the algorithms we consider is Pareto optimal, so the histograms would be unaltered even if we did not assess PO.

Not surprisingly, we see that very few instances permit a solution that is Pareto optimal and exactly equitable. Whenever such a solution exists, it is provably achieved by Leximin, but this happens in only 1% of Spliddit instances and none of the synthetic instances. For the EQ1 relaxation, we see that not only do Leximin and Alg-eq1+po satisfy both EQ1 and PO, but so does MNW on over of Spliddit instances (and over of synthetic instances). However, this trend changes when we consider EQx. Alg-eq1+po, despite being guaranteed to satisfy EQ1, only satisfies EQx on of Spliddit instances (and of synthetic instances). A similar drop off is observed with MNW. Thus, for the purpose of achieving (approximately) equitable and Pareto optimal allocations, Leximin is a clear winner.

We observe little change when, in addition to approximate equitability and Pareto optimality, we also require approximate envy-freeness. Indeed, in most cases, an allocation that is EQ1+PO/EQx+PO is also EF1/EFx. It is interesting to note that while MNW—which is appealing from the perspective of achieving relaxed envy-freeness—quite often fails to satisfy EQx, Leximin provably satisfies relaxed equitability while also achieving EFx on a large fraction of instances.

## 5 Discussion

We studied equitable allocations of indivisible goods in conjunction with other well-known notions of fairness (envy-freeness) and economic efficiency (Pareto optimality), and provided a number of existential and computational results. In the appendix, we provide simulation results comparing the algorithms considered in Section 4 with respect to relaxations of envy-freeness (Section 6.8). We also analyze EQ1 and EQx allocations from the perspective of approximating the optimal solutions to Max-Min Fairness, otherwise known as the Santa Claus problem (Section 6.10).

Our work reveals some intriguing similarities and differences between equitability and envy-freeness. In many places, our work parallels the existing literature on envy-freeness: We present Leximin as a canonical algorithm for EQ1+PO, just like MNW achieves EF1+PO. Also, our pseudopolynomial-time algorithm for EQ1+PO uses similar techniques to that of Barman et al. (2018a) for EF1+PO. However, in other places, the differences are more pronounced. Most notably, EQx comes with a universal existence guarantee (often in conjunction with PO), while the existence of EFx allocations remains an open problem. Finally, exact equitability is a knife-edge property often hard to achieve in practice, unlike envy-freeness which is often satisfiable (Dickerson et al., 2014).

Going forward, it would be very interesting to extend our results to the public decisions model of Conitzer et al. (2017). Extensions to models with additional feasibility constraints on the allocations (Bouveret et al., 2017), or settings with both goods and chores (Aziz et al., 2018) will also be interesting.

## Acknowledgments

We are grateful to the anonymous IJCAI-19 reviewers for their helpful comments, and to Ariel Procaccia and Nisarg Shah for sharing with us the data from Spliddit. LX acknowledges NSF #1453542 and #1716333 for support.

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## 6 Appendix

### 6.1 Proof of Theorem 3

Recall the statement of Theorem 3.

See 3

The proof of Theorem 3 relies on the algorithm Alg-eq1+po (presented in Algorithm 1), and spans Sections 6.5, 6.4, 6.3, 6.2 and 6.1. We will start with some necessary definitions that will help us state Theorem 5, of which Theorem 3 is a special case.

##### Fractional allocations

A fractional allocation refers to a fractional assignment of the goods to the agents such that no more than one unit of any good is allocated, i.e., for every good , . We will use the term allocation to refer to a discrete allocation and explicitly write fractional allocation otherwise.

##### ε-Pareto optimality

Given any , is -Pareto optimal (-PO) if there does not exist an allocation such that for every agent with one of the inequalities being strict.

##### Fractional Pareto optimality

An allocation is fractionally Pareto optimal (fPO) if it not Pareto dominated by any fractional allocation. Thus, a fractionally Pareto optimal allocation is also Pareto optimal, but the converse is not necessarily true (creftypecap 8).

##### ε-EQ1 allocation

Given any , an allocation is -equitable up to one good (-EQ1) if for every pair of agents such that , there exists some good such that .

###### Theorem 5.

Given any fair division instance with additive and strictly positive valuations and any , an allocation that is -equitable up to one good and -Pareto optimal always exists and can be computed in time, where .

When , we recover Theorem 3 as a special case of Theorem 5 (see Lemmas 18 and 16).

The remainder of this section develops the necessary preliminaries that will enable us to present our algorithm (Algorithm 1) and the analysis of its running time (Lemma 1) and correctness (Lemma 2). The detailed proofs of these results are presented subsequently in Sections 6.5, 6.4, 6.3 and 6.2.

#### Market Preliminaries

##### Fisher market

A Fisher market is an economic model that consists of a set of divisible goods and a set of agents (or buyers), each of whom is given a budget (or endowment) of virtual money (Brainard and Scarf, 2000). The agents can use the virtual money to purchase a utility-maximizing subset of the goods but do not derive any utility from the money itself. Formally, a Fisher market is given by a tuple consisting of a set of agents , a set of divisible goods , a valuation profile

and a vector of

endowments or budgets .

A market outcome refers to a pair , where is a fractional allocation of the goods, and is a price vector that associates a price with every good . The spending of agent under the market outcome is given by . The utility derived by the agent under depends linearly on the valuations as .

##### Induced fair division instance

A Fisher market naturally defines a fair division instance , which we will refer to as the induced fair division instance. This correspondence between Fisher markets and the fair division problem allows us to extend the fairness and efficiency notions defined in Section 2 to Fisher markets. Thus, we will say that an allocation is equitable/envy-free/Pareto optimal for a market if it is equitable/envy-free/Pareto optimal for the induced fair division instance .

##### MBB ratio and MBB set

Given a price vector , define the bang-per-buck ratio of agent for good as .141414If and , then . The maximum bang-per-buck ratio (or MBB ratio) of agent is . The maximum bang-per-buck set (or MBB set) of agent is the set of all goods that maximize the bang-per-buck ratio for agent at the price vector , i.e., .

A market outcome constitutes an equilibrium if it satisfies the following conditions:

• Market clearing: Each good is either priced at zero or is completely allocated. That is, for every good , either or .

• Budget exhaustion: Agents spend their budgets completely, i.e., for all .

• MBB consistency: Each agent’s allocation is a subset of its MBB set. That is, for every agent and every good , . Note that MBB consistency implies that every agent maximizes its utility at the given prices under the budget constraints.

creftypecap 7 presents the well-known first welfare theorem for Fisher markets (Mas-Colell et al., 1995, Chapter 16).

###### Proposition 7.

For a Fisher market with linear utilities, any equilibrium outcome is fractionally Pareto optimal .

##### MBB-allocation graph and alternating paths

Let be Fisher market, and let and denote an integral allocation and a price vector for , respectively. An MBB-allocation graph is an undirected bipartite graph with vertex set and an edge between agent and good if either (called an allocation edge) or (called an MBB edge). Notice that if is MBB-consistent (i.e., ), then the allocation edges are a subset of MBB edges.

For an MBB-allocation graph, define an alternating path from agent to agent (and involving the agents and the goods ) as a series of alternating MBB and allocation edges such that , ,, . If such a path exists, we say that agent is reachable from agent via an alternating path.151515Note that no agent or good can repeat in an alternating path. In this case, the length of path is since it consists of MBB edges and allocation edges.

##### Reachability set

Let denote the MBB-allocation graph of a Fisher market for the outcome . Fix a source agent in . Define the level of an agent as half the length of the shortest alternating path from to if one exists (i.e., if is reachable from ), otherwise set the level of to be . The level of the source agent is defined to be . The reachability set of agent is defined as a level-wise collection of all agents that are reachable from , i.e., , where denotes the set of agents that are at level with respect to agent . Note that given an MBB-allocation graph, a reachability set can be constructed in polynomial time via breadth-first search.

Given a reachability set , we can redefine an alternating path as a set of alternating MBB and allocation edges connecting agents at a lower level to those at a higher level. Formally, we will call a path alternating if (1) , ,, , and (2) . Thus, an alternating path cannot have edges between agents at the same level.

##### Violators and path-violators

Given a Fisher market and an allocation , an agent with the least utility among all the agents is called the reference agent, i.e., .161616Ties are broken lexicographically. An agent is said to be a violator if and for every good , we have that , where is the reference agent. Notice that the allocation is EQ1 if and only if there is no violator.

Given any , an agent is an -violator if and for every good , we have . Thus, an agent can be a violator without being an -violator. An allocation is -EQ1 if and only if there is no -violator.

A closely related notion is that of a path-violator. Let and denote the reference agent and its reachability set, respectively. An agent is a path-violator with respect to the alternating path if . Note that a path-violator (along a path ) need not be a violator as there might exist some good not on the path such that . Finally, given any , an agent is an -path-violator with respect to the alternating path if .

##### ε-rounded instance

Given any , an -rounded instance refers to a fair division instance in which the valuations are either zero or a non-negative integral power of . That is, for every agent and every good , we have for some .

Given any instance , the -rounded version of is an instance obtained by rounding up the valuations in to the nearest integral power of . That is, the -rounded version of instance is an -rounded instance constructed as follows: For every agent and every good , if , and otherwise. Notice that for every agent and every good . We will assume that the rounded valuations are also additive, i.e., for any set of goods , .

#### Description of the Algorithm

Given an input instance , we first construct its -rounded version , which is then provided as an input to Alg-eq1+po (Algorithm 1).

The algorithm consists of three phases. In Phase 1, each good is assigned to an agent with the highest valuation for it (Line 1). This ensures that the initial allocation is integral as well as fractionally Pareto optimal (fPO).171717Indeed, the said allocation is MBB-consistent with respect to the prices in Line 1, and is therefore an equilibrium outcome of a Fisher market in which each agent is provided a budget equal to its spending under the allocation. From creftypecap 7, the allocation is fPO. (These two properties are always maintained by the algorithm.) If the allocation at the end of Phase 1 is -EQ1 with respect to the rounded instance , then the algorithm terminates and returns this allocation as the output (Line 1). Otherwise, it proceeds to Phase 2.

The allocation at the start of Phase 2 is not -EQ1, so there must exist an -violator. Starting from the level (Line 1), the algorithm now performs a level-by-level search for an -violator in the reachability set of the reference agent (Line 1). As soon as an -violator, say , is found (along some alternating path ), the algorithm performs a pairwise swap between and the agent that precedes it along (Line 1). Since the swapped good is in the MBB sets of both agents, the allocation continues to be MBB-consistent after the swap. If, at any stage, the reference agent ceases to be the least-utility agent, Phase 2 restarts with the new reference agent (Line 1).

The above process continues until either the current allocation becomes -EQ1 for the rounded instance (in which case the algorithm terminates and returns the current allocation as the output in Line 1), or if no -violator is reachable from the reference agent (Line 1). In the latter case, the algorithm proceeds to Phase 3.

Phase 3 involves uniformly raising the prices of all the reachable goods, i.e., the set of all goods that are collectively owned by all agents that are reachable from the reference agent (Line 1). The prices are raised until a previously non-reachable agent becomes reachable due to the appearance of a new MBB edge (Line 1). The algorithm now switches back to Phase 2 to start a fresh search for an -violator in the updated reachability set (Line 1).

##### Comparison with the algorithm of Barman et al. (2018a)

As mentioned previously in Section 3, our algorithm is inspired from the algorithm of Barman et al. (2018a) for achieving envy-freeness up to one good (EF1) together with Pareto optimality (PO). At a high-level, both algorithms involve searching for a reachable violator (along an alternating path). If such an agent exists, then it loses a good through a pairwise swap. Otherwise, both algorithms use price-rise in order to discover a new MBB edge to a previously unreachable agent. The main difference between the two algorithms is that Barman et al. (2018a) define a violator in terms of excess spending, whereas we define a violator in terms of excess utility.181818Formally, in the framework of Barman et al. (2018a), an agent is an -violator if for every good , we have that ; here is the sum of prices of all the goods in the bundle , and is the least spender. In other words, their algorithm performs local search in the space of spendings, whereas our algorithm does so in the space of utilities. As a result of this small but subtle difference, Barman et al. (2018a) achieve an approximate equitability condition in terms of spendings (which they call price envy-freeness up to one good), whereas we are able to guarantee a similar property in terms of the utilities, which is precisely the desired EQ1 condition.

##### Analysis of the algorithm

The running time and correctness of our algorithm are established by Lemma 1 and Lemma 2, respectively, as stated below.

###### Lemma 1 (Running time).

Given as input any -rounded instance with strictly positive valuations, Alg-eq1+po terminates in time steps, where .

The proof of Lemma 1 appears in Section 6.2.

###### Lemma 2 (Correctness).

Let be any fair division instance with strictly positive valuations and be its -rounded version for any given . Then, the allocation returned by Alg-eq1+po for the input is -EQ1 and -PO for . In addition, if , then is EQ1 and PO for .

The proof of Lemma 2 appears in Section 6.5.

Notice that the running time guarantee in Lemma 1 is stated in terms of time steps. A time step refers to a single iteration of Phase 1, Phase 2, or Phase 3. Since each individual iteration requires polynomial time, it suffices to analyze the running time of the algorithm in terms of the number of iterations of the three phases.191919Indeed, an iteration of Phase 1 involves assigning each good to the agent with the highest valuation and setting its price. An iteration of Phase 2 involves the construction of the reachability set (say via breadth-first or depth-first search), followed by performing a level-wise search for an -path-violator, followed by performing a swap operation. An iteration of Phase 3 involves scanning the set of reachable goods and setting an appropriate value of the price-rise factor . All of these operations can be carried out in time. We will use the terms step, time step, and iteration interchangeably.

We are now ready to prove Theorem 3.

See 3

###### Proof.

Fix . The -rounded version can be constructed in time. We run the algorithm Alg-eq1+po on the input . From Lemma 1, we know that the algorithm terminates in time. Lemma 2 implies that is EQ1 and PO for . ∎

### 6.2 Proof of Lemma 1

Recall the statement of Lemma 1. See 1

###### Proof.

The proof of Lemma 1 follows immediately from Lemmas 4 and 3, which are stated below. ∎

###### Lemma 3.

There can be at most consecutive iterations of Phase 2 before a Phase 3 step occurs.

###### Lemma 4.

There can be at most Phase 3 steps during any execution of Alg-eq1+po.

The proofs of Lemmas 4 and 3 are provided in Sections 6.4 and 6.3, respectively.

### 6.3 Proof of Lemma 3

The proof of Lemma 3 relies on several intermediate results (Lemmas 8, 7, 6 and 5) that are stated below.

###### Lemma 5.

There can be at most consecutive swap operations in Phase 2 before either the identity of the reference agent changes or a Phase 3 step occurs.

The proof of Lemma 5 is identical to (Barman et al., 2018a, Lemma 13) and is therefore omitted.

Throughout, we will use the phrase at time step to refer to the state of the algorithm at the beginning of the time step . In addition, we will use and to denote the reference agent and the allocation maintained by the algorithm at the beginning of time step , respectively. Thus, for instance, the utility of the reference agent at time step is .

###### Lemma 6.

The utility of the reference agent cannot decrease with time. That is, for any time step ,

 wit(Atit)≤wit+1(At+1it+1).
###### Proof.

The only way in which the utility of a reference agent can change is via a swap operation in Phase 2. By construction, a reference agent can never lose a good during a swap operation (though it can possibly receive a good). Therefore, the utility of a reference agent cannot decrease. ∎

###### Lemma 7.

Let be a fixed agent. Consider any set of consecutive Phase 2 steps during the execution of Alg-eq1+po. Suppose that turns from a reference to a non-reference agent during time step . Let be the first time step after at which once again becomes a reference agent. Then, either is a strict subset of or .

###### Proof.

In order for a reference agent to turn into a non-reference agent, it must receive a good during a swap operation. That is, agent must receive a good at time and hence is a strict subset of . If agent does not lose any good between and , then the claim follows. Therefore, for the rest of the proof, we will assume that agent loses at least one good between and .

Among all the time steps between and at which agent loses a good, let be the last one. Let be the reference agent at time step . Since the utility of the reference agent is non-decreasing with time (Lemma 6), we have that

 wiτ(Aτiτ)≥wi(Ati). (1)

Let denote the good lost by agent at time step . An agent that loses a good must be an -path violator (with respect to an alternating path involving that good). Therefore,

 wi(Aτi∖{g})>(1+ε)wiτ(Aτiτ). (2)

Since does not lose any good between and , we have

 wi(At′i)≥wi(Aτ+1i)=wi(Aτi∖{g}). (3)

Combining Equations 3, 2 and 1 gives

 wi(At′i)>(1+ε)wi(Ati),

as desired. ∎

###### Lemma 8.

There can be at most changes in the identity of the reference agent before a Phase 3 step occurs.

###### Proof.

From Lemma 7, we know that each time the algorithm cycles back to a some agent as the reference agent, either the allocation of agent grows strictly by at least one good, or its utility increases by at least a multiplicative factor of . By pigeonhole principle, after every consecutive changes in the identity of the reference agent, the algorithm must cycle back to some agent as the reference. Along with the fact that the utility of the reference agent is non-decreasing with time (Lemma 6), we get that after every consecutive identity changes, the utility of the reference agent must grow multiplicatively by a factor of . Since the utility of any agent can be at most (where ), there can be at most changes in the identity of the reference agent during the execution of the algorithm. Furthermore, for -rounded valuations, we have that . The stated bound now follows by observing that for every . ∎

We are now ready to prove Lemma 3. See 3

###### Proof.

From Lemma 8, we know that there can be at most changes in the identity of the reference agent (in Phase 2) before a Phase 3 step occurs. Furthermore, Lemma 5 implies that there can be at most swap operations between two consecutive identity changes or an identity change and a Phase 3 step. Combining these implications gives the desired bound. ∎

### 6.4 Proof of Lemma 4

The proof of Lemma 4 relies on several intermediate results (Corollaries 2, 12, 11, 10 and 9) that are stated and proved below. It will be useful to define the set of all -violators at time step . That is,

 Et\coloneqq{k∈[n]:wk(Atk∖{j})>(1+ε)wit(Atit)∀j∈Atk},

where is the reference agent at time step .

Some of our proofs will require the following assumption:

###### Assumption 1.

At the end of Phase 1 of Alg-eq1+po, every agent is assigned at least one good.

This assumption can be ensured via efficient preprocessing techniques similar to those used by Barman et al. (2018a). We refer the reader to Section B.1 of their paper for details.

###### Lemma 9.

Let and be two Phase 3 time steps such that . Then, .

###### Proof.

It suffices to consider consecutive Phase 3 steps and such that all intermediate time steps occur in Phase 2. Suppose, for contradiction, that there exists some agent . Observe that a non--violator cannot turn into an -violator in Phase 3 as the allocation of the goods remains fixed during price-rise. Therefore, the only way in which can turn into an -violator is via a swap operation in Phase 2. In the rest of the proof, we will argue that if there is a swap operation at time step (where ) that turns into an -violator, then there is a subsequent swap operation at time step that turns it back into a non--violator. This will provide the desired contradiction.

Suppose that agent is at level in the reachability set when it receives a good that turns it into an -violator. Recall that a swap operation involves transferring a good from an agent at a higher level