1 Introduction
In recent years, the topic of fair division of indivisible goods has received significant attention in the computer science, mathematics, and economics communities, see, for instance, Chapter 12 in [Brandt et al., 2016]. A central motivation behind this thread of research is the fact that classical notions of fairness—such as envy freeness ()^{1}^{1}1An allocation is called envy free if every agent values her bundle at least as much she values any other agent’s bundle [Foley, 1967, Varian, 1974, Stromquist, 1980]. and proportionality^{2}^{2}2An allocation, among agents, is called proportionally fair if every agent’s value for her share is at least times the total value of all the goods [Steinhaus, 1948].—which were developed for divisible goods (that can be fractionally allocated), do not translate well to the indivisible case. For instance, if there is a single indivisible good and two agents, then no allocation can guarantee or proportionality. But, given that a number of realworld settings (such as budgeted course allocations [Budish, 2011], division of inheritance, and partitioning resources in a cloud computing environment) entail allocation of discrete/indivisible goods, it is essential to define and study solution concepts which are applicable for a fair division of indivisible goods.
Classically, the applicability of solution concepts is studied via existence results. Understanding if and when a solution concept is guaranteed to exist is of fundamental importance in microeconomics and other related fields. Such existence results have been notably complemented by research—in algorithmic game theory and artificial intelligence—that has focused on computational issues surrounding the underlying solution concepts. Broadly, our results contribute to these key themes by establishing both existential and algorithmic results for a new notion of fairness.
In the context of fair division, the focus on developing efficient algorithms is motivated, in part, by websites such as Spliddit^{3}^{3}3www.spliddit.org [Goldman and Procaccia, 2015] and Adjusted Winner^{4}^{4}4http://www.nyu.edu/projects/adjustedwinner/ [Brams and King, 2005], which offer fair solutions for dividing goods. Spliddit has attracted more than fifty thousand users and, among other services, it computes allocations which are fair with respect to the standard notions of fairness. One of the solution concepts considered by Spliddit is the maximin share guarantee ().
The solution concept was defined in the notable work of [Budish, 2011], and it deems an allocation to be fair if each agent gets a bundle of value greater than or equal to an agentspecific threshold, called the maximin share of the agent. Specifically, the maximin share of an agent corresponds to the maximum value that the agent can attain for herself if she is hypothetically asked to partition the set of goods into bundles and, then, the remaining () agents pick their bundles adversarially. Hence, a riskaverse agent would partition the goods (into bundles) such that value of the least desirable bundle (according to her) in the partition is maximized. The value of the least desirable bundle in such a partition is called the maximin share of agent . This definition can be interpreted as a natural generalization of the classical cutandchoose protocol.
Although maximin shares provide a natural benchmark to define fairness, this solution concept has its own drawbacks. In particular, is not sufficient to rule out unsatisfactory allocations; see Section 3 for an example. Moreover, different allocations can be drastically different in terms of, say, the social welfare of the agents.
Motivated by these considerations, we define a strictly stronger notion of fairness, called groupwise maximin share guarantee (). Intuitively, provides an expost fairness guarantee: it ensures that, even after the allocation has been made, the maximin share guarantee is achieved not just with respect to the grand bundle of goods, but also among all the subgroups of agents . Specifically, we say that an allocation is if, for all groups and agents , the value of ’s bundle in the allocation is no less than the maximin share of restricted to . That is, if the agent were to repeat the thought experiment (of dividing all the goods allocated to the agents in group , so that the other agents pick their bundles adversarially) to calculate her maximin share restricted to , then the value of her bundle is at least this threshold. This definition directly ensures that groupwise maximin share guarantee is a stronger solution concept: implies . In Section 3, we show that can, in fact, be arbitrarily better than an allocation that just satisfies .
also strictly generalizes pairwise maximin share guarantee (), a notion defined by [Caragiannis et al., 2016]. In , the maximin share guarantee is required only for pairs of agents, but not necessarily for the grand bundle. Section 2.1, provides an example which establishes that is a strict generalization of and .
The relevance of is also substantiated by the fact that it implies other complementary notions of fairness, which do not follow from alone. In the context of indivisible goods, relaxations of envy freeness, such as ^{5}^{5}5An allocation is said to be envyfree up to one good () if no agent envies any other after removing at most one good from the other agent’s bundle; see Definition 4. [Budish, 2011] and ^{6}^{6}6An allocation is said to be envyfree up to the least valued good () if no agent envies any other agent after removing any positively valued good from the other agent’s bundle; see Definition 5. [Caragiannis et al., 2016] have also been studied. In Section 4 we show that (unlike ) fits into this scale of fairness and, in particular, a allocation is guaranteed to be (and hence ). These implications essentially follow from the observation that, by definition, a allocation is as well. [Caragiannis et al., 2016] have shown that implies , and hence we obtain the desired implications.
Throughout the paper, we focus on additive valuations, and a highlevel contribution of our work is to show that under additive valuations, a number of useful (existence, algorithmic, and approximation) results which have been established for continue to hold for as well.
1.1 Our Contributions
In addition to proving a scale of fairness for , we establish the following results:

Approximate groupwise maximin share allocations always exist under additive valuations. Prior work has shown that there are instances wherein no allocation is [Procaccia and Wang, 2014, Kurokawa et al., 2016]. These nonexistence results have motivated a detailed study of approximate allocations, i.e., allocations in which each agent gets a bundle of value (multiplicatively) close to her maximin share. Along these lines, we consider approximate (see Definition 3), and show that, under additive valuations, a  allocation always exists. In addition, such an allocation can be found in polynomial time.

allocations are guaranteed to exist when the valuations of the agents are either binary or identical.

Analogous to the experimental results for [Bouveret and Lemaître, 2014], allocations exist empirically. These simulation results indicate that we do not fall short on such generic existence results by strengthening the maximin solution concept.
1.2 Related Work
As mentioned earlier, [Budish, 2011] introduced the notion of maximin share guarantee (), and it has been extensively studied since then. In particular, [Bouveret and Lemaître, 2014] showed that if the agents’ valuations are additive, then an envy free (or proportional) allocation will be as well. They also established that exists under binary, additive valuations. Their experiments, using different distributions over the valuations, did not yield a single example wherein an allocation did not exist. [Kurokawa et al., 2016] showed that
allocations exist with high probability when valuations are drawn randomly.
However, [Procaccia and Wang, 2014] provided intricate counterexamples to refute the universal existence of allocations, even under additive valuations. This motivated the study of approximate maximin share allocations, , where each agent obtains a bundle of value at least times her maximin share. [Procaccia and Wang, 2014] established the existence of , and developed a polynomialtime algorithm to obtain such an allocation when the number of agents is a fixed constant. Later, [Amanatidis et al., 2015] showed that a  can be computed in polynomial (in the number of players) time.
Approximate maximin share allocations have also been studied for general valuations. [Barman and Krishnamurthy, 2017] have developed an efficient algorithm which finds a  allocation under submodular valuations. More recently, [Ghodsi et al., 2017] have improved the approximation guarantee for additive valuations to . They have also developed constantfactor approximation guarantees for submodular and XOS valuations, along with a logarithmic approximation for subadditive valuations.
[Aziz et al., 2017] studied the fair division of indivisible chores (negatively valued goods) and have developed an efficient algorithm which finds a  allocation.
[Caragiannis et al., 2016] defined another important fairness notion called pairwise maximin share guarantee (). As mentioned previously, under , the maximin share guarantee is required only for pairs of agents, and not even for the grand bundle. They also established that and are incomparable: neither one of these solution concepts implies the other.
2 Preliminaries and Notation
We consider the problem of finding a fair allocation of a set of indivisible goods , among a set of agents . For a subset of goods and integer , let denote the set of all partitions of . An allocation is defined as an partition , where is the set of goods allocated to agent .
The preference of the agents over the goods is specified via valuations. Specifically, we denote the valuation of an agent for a subset of goods by . In this work, we assume the valuations to be nonnegative and additive, i.e., for all and . For ease of presentation, we will use for agent ’s valuation of good , i.e., for .
As mentioned previously, the fairness notions considered in this work are defined using thresholds called maximin shares. Formally, given an agent , parameter , and subset of goods , maximin share of restricted to is defined as . Throughout, will be used to denote the maximin share of an agent with respect to the grand bundle, . We will now formally define maximin share allocation ( allocation).
Definition 1 (Maximin Share Allocation).
An allocation is said to be a maximin share allocation () iff for all agents we have .
A different solution concept defined by [Caragiannis et al., 2016] requires the maximin share guarantee to hold only for every pair of agents, i.e., an allocation is said to achieve pairwise maximin share guarantee () iff for all , we have and .
In this paper, we strengthen and , and define a stronger threshold for each agent , namely groupwise maximin share (). Formally,
Now we define groupwise maximin share allocation ( allocation).
Definition 2 (Groupwise Maximin Share Allocation).
An allocation is said to be a groupwise maximin share allocation () iff for all agents we have .
Note that an allocation is iff for all such that . Also, the threshold is a function of the underlying allocation and, in contrast, depends only the valuation of agent for the goods and the number of agents .
The fact that there are fair division instances which do not admit an allocation directly implies that allocation are not guaranteed to exist either. Therefore, we consider approximate allocations.
Definition 3 (Approximate Groupwise Maximin Share Allocation).
An allocation is said to be an approximate groupwise maximin share allocation () iff for all agents we have .
A approximate groupwise maximin share allocation is a allocation.
2.1 Strictly Generalizes and
This section shows that is a distinct solution concept which strictly generalizes both and . In fact, the instance constructed in this section shows that there exists allocations which are and, furthermore, satisfy the maximin share guarantee for all subgroups of size at most (say), but do not satisfy the criteria.
We now formally define the notion of maximin share guarantee for a subgroup of size . For ease of presentation, we call such allocations wise fair. An allocation is said to be wise fair iff for all agents and for all size subsets such that , the following holds: .
In the following example we identify an allocation which is wise fair—for each —but not wise fair; here, . Let us consider goods and agents, with . Since the number of agents is greater than the number of goods, for all and, hence, all the allocations are . Consider an agent, say , and let her valuation for the goods be

for the first goods (“large” goods),

for the next goods (“mediumsized” goods), and

for the remaining goods (“small” goods).
The valuation of other agents can be set to ensure fairness for them. Let allocation be such that , , , , and for all , i.e., no goods are allocated to the last agents. We first show that the bundle allocated to agent is at least her pairwise maximin share with any other agent . Agent does not envy agent since (since ). Moreover, for the other agents , we have . Thus, ensures . Below, we present a case analysis which shows that is also wise fair, for all .

Consider any subgroup of the first agents, containing agent . Let . In this case, exactly goods are allocated to the group of agents . In particular, this set of goods consists of “large” goods and “mediumsized” goods. Hence, any partition of these goods will have at least one bundle with at most one good. Therefore, any partition which maximizes the least valued (with respect to the valuation of agent ) bundle will have one bundle containing a “large” good, and each of the remaining bundles containing one “large” and one “mediumsized” good. Therefore, the allocation ensures maximin share guarantee for agent for all subgroups , which contain agent .

Next, consider any subgroup of size which includes agent as well as agent . Here, the set of goods allocated to agents in consists of “large” goods, “mediumsized” goods and “small” goods. To simplify the analysis, we assume that and use an averaging argument to obtain
Hence, for all such subgroups of agents of size , the bundle satisfies the maximin share guarantee of agent . A similar, but slightly involved argument establishes the result for subgroups of size .

Finally, consider a size subgroup such that and contains at least one agent , i.e., includes an agent with an empty bundle . Write and . The arguments used in the first two cases show that the maximin guarantee for agent holds for , i.e., . Using the fact that and , we get . These observations establish the required maximin share guarantee, , for this case as well.
Overall, we get that the allocation satisfies wise fairness, for any . We will complete the analysis by showing that is not wise fair.
In particular, let . Note that, here, the set consists of “large” goods, “mediumsized” goods and “small” goods. In addition, the maximin share value with respect to this subgroup is : partition the goods to obtain one bundle containing all “mediumsized” goods (each of value ) and, additionally, bundles each containing one “large” good and one “small” good. Therefore, .
This example illustrates that, in particular, is a stronger solution concept than and .
3 can be arbitrarily better than
In this section, we provide a class of examples where an allocation is not necessarily satisfactory in terms of agents’ valuations. In particular, we show that imposing leads to allocations which Pareto dominate an allocation which only satisfies .
Consider agents and a set of goods (where ), along with parameter and a sufficiently small ; . The valuations are assumed to be additive and are as follows:
,
Note that in this fair division instance the maximin share of the first agents, for all . In addition, the maximin share of the last agent, , is .
Now, consider the allocation wherein , , , and . Allocation is since for each and . In this allocation the valuation of agent is unsatisfactorily low. Another relevant observation is that this allocation is not . Below, we show that in this instance any allocation allocates a bundle of value to every agent, including agent .
The fact that is not follows by considering the goods allocated to agents and , i.e., let . Now, , but . Furthermore, it can be observed that it is necessary to allocate the agent at least one of her high valued goods to satisfy . Thus, allocation would always ensure that a bundle of at least is allocated to all the agents. That is, in this instance, unsatisfactorily low valuations can be avoided by imposing the groupwise maximin share guarantee.
4 Scale of Fairness
As mentioned earlier, envy freeness () is a wellstudied solution concept in the context of fair division of divisible items. However, for indivisible goods, a simple example with one positively valued good and two agents shows that envyfree allocations do not always exist. Hence, for indivisible goods, natural relaxations of envy freeness—in particular, and —have been considered in the literature. We now provide formal definitions of these relaxations.
Definition 4 (Envyfree up to one good [Budish, 2011]).
An allocation is said to be envyfree up to one good () iff for every pair of agents there exists a good such that .
Definition 5 (Envyfree up to the least positively valued good [Caragiannis et al., 2016]).
An allocation is said to be envyfree up to the least valued good () iff for every pair of agents and for all goods (i.e., for all goods in which are positively valued by agent ) we have .
Note that the above mentioned definitions imply that, for additive valuations, an allocation is necessarily as well. Next we show that, interestingly, these relaxed versions of envy freeness are implied by , but not by .
Proposition 1 (Scale of Fairness).
In any fair division instance with additive valuations

If an allocation is envy free () then it achieves groupwise maximin share guarantee () as well.

If an allocation is then it is (and, hence, ) as well.
Proof.
First we will show that implies : Assume that the allocation is envy free, that is, for all we have . Therefore, for any agent and any group of agents we have . Since the valuation is additive, this inequality leads to the following bound ; here . Now, an averaging argument establishes the following inequality: for any partition of if then . Hence, , and we get that is .
Next, we argue that implies : [Caragiannis et al., 2016] have shown that that implies . By definition, implies . Hence, a allocation is guaranteed to be . ∎
Note that [Caragiannis et al., 2016] also provided an example to show that maximin share guarantee by itself does not imply . Hence, an allocation is not necessarily . Consider a fair division instance with three agents, five goods, and each agent values each good at . The maximin share of all the agents is . Thus, the allocation and satisfies , but not . This is because agents and continue to envy agent even if a single good is removed from .
This, overall, shows that while is not enough to guarantee weaker notions of envy freeness, ensures fairness in terms of such notions, and secures a place in the scale of fairness.
Next we will consider the complementary direction of going from bounded envy to groupwise maximin fairness. In particular, we will establish existence and algorithmic results for approximate by considering a solution concept which is stronger than , but weaker than . Specifically, we will define allocations which are envyfree up to one lesspreferred good ()—see Definition 6 below—and show that such allocations are guaranteed to exist, when the valuations are additive. Note that, in contrast, the generic existence of allocations remains an interesting open question. Furthermore, we will prove that allocations can be computed in polynomial time and, under additive valuations, such allocations imply .
Definition 6.
An allocation is said to be envyfree up to one lesspreferred good () if for every pair of agents at least one of the following conditions hold:

contains at most one good which is positively valued by ;

There exists a good such that and .
The fact that an allocation is follows directly from the definitions of these solution concepts. Also, note that if an allocation is then for any pair of agents with the second condition in the definition of holds. In particular, write and consider two distinct goods . Since the allocation is , we have and . Note that , and, hence, . This implies that good satisfies the second condition in Definition 6. Hence, any allocation is as well.
With this new fairness notion, we have the following chain of implications: .
5 An Approximation Algorithm for
Our main result in this section shows that  allocations always exist under additive valuation, and such allocations can be found efficiently.
Theorem 1.
Every fair division instance with additive valuations admits a approximate groupwise maximin share allocation. Furthermore, such an allocation can be found in polynomial time.
ProofSketch The proof proceeds in two steps. First, we provide a constructive proof for the existence of allocations, under additive valuations (Section 5.1). Next, we complete the proof by showing that implies  (in Section 5.2) ∎
5.1 Existence of Allocations
This section shows that allocations are guaranteed to exist when the valuations are additive. Specifically, we develop an algorithm that always finds such an allocation.
Lemma 1.
Given any fair division instance with additive valuations, Algorithm 1 finds an allocation in polynomial time.
Algorithm 1 iteratively allocates the goods and maintains a partial allocation, , of the goods assigned so far. In each iteration, the algorithm selects an agent who is not currently envied by any other agent, and allocates an unassigned good of highest value (under ).
Throughout the execution of the algorithm, the existence of an unenvied agent is ensured by maintaining a directed graph, , that captures the envy between agents. The nodes in this envy graph represent the agents and it contains a directed edge from to iff envies , i.e., . Lemma 2, established in [Lipton et al., 2004], shows that if any iteration leads to a cycle in the envy graph , then we can always resolve it to obtain an acyclic envy graph without decreasing the valuation of any agents; for completeness, we provide a proof of Lemma 2. It is relevant to that, since is acyclic for a partial allocation , it necessarily contains a source node, i.e., an agent who is not envied by other agents.
Although, Algorithm 1 is similar to the algorithm developed in [Lipton et al., 2004]—which efficiently finds allocations— here, instead of assigning goods in an arbitrary order, we always allocate to an unenvied agent the available good she values the most. This is crucial for obtaining an allocation.
Lemma 2.
[Lipton et al., 2004]
Given a partial allocation of a subset of goods , we can find another partial allocation of in polynomial time such that
(i) The valuations of agents for their bundles do not decrease, that is, for all .
(ii) The envy graph is acyclic.
Proof.
If the envy graph of is acyclic then the claim holds trivially. Otherwise, let us denote a cycle in the graph by . Now, we can reallocate the bundles as follows: for all agents not in , i.e., set , and for all the agents in the cycle set to be the bundle of their successor in , i.e., set for along with .
Note that after this reallocation we have for all . Furthermore, the number of edges in strictly decreases: The edges in do not appear in the envy graph of and if an agent starts envying an agent in the cycle, say agent , then must have been envious of in . Edges between agents and which are not in remain unchanged, and edges going out of an agent in the cycle can only get removed, since valuation for the bundle assigned to her bundle increases. Therefore, we can repeatedly remove cycles and keep reducing the number of edges in the envy graph to eventually a find a partial allocation that satisfies the stated claim. ∎
Proof of Lemma 1.
Write to denote the allocation returned by Algorithm 1. First, we note that an inductive argument proves that is . In fact, we will show that the condition holds for with respect to the last (in terms of the algorithm’s allocation order) good assigned to each bundle . Write to denote the good allocated in the th iteration of Algorithm 1; hence, the goods are allocated in the following order .
The initial partial allocation is (in fact, it is envy free). Now, say that in the th iteration the algorithm allocates good to agent . Write and , respectively, to denote the partial allocations before and after the th iteration. The induction hypothesis implies that is with respect to the last assigned good. Therefore, for every pair of agents , we have , where is the last good assigned to the bundle (i.e., for any other good , we have ).
Since the good is allocated to agent , it must be the case that is a source vertex in , i.e., no agent envies under . This implies that, for all . Note that at this point of time, is the last good assigned to the bundle . In addition, from the proof of Lemma 2, we know that is a permutation of the allocation , and for all . Hence, for every pair of agents there exists a good such that . In addition, is the last good assigned to the bundle . That is, the stated property holds for as well.
Now, we will use this observation to prove that is . Specifically, we show the conditions hold for, say, agent (analogous arguments establish the claim for the other agents). Suppose that, during the execution of the algorithm, agent receives its first good, , in the th iteration. Note that the partial allocation before the th iteration, say , satisfies for all . This bound follows from the observation that during any previous iteration , the selected source vertex (i.e., the agent that gets a new good during the th iteration) does not contain any good which is positively valued by agent ; otherwise, would have been envied by , contradicting the fact that it is a source vertex. Hence, each bundle in contains at most one good from the set .
Let us now consider the final allocation and any agent . If , then the first condition in the definition of holds and we are done, else if , then bundle must have received a good after the th iteration. This is consequence of the abovementioned property of the partial allocation . Write to denote the last good allocated to the bundle . We have , since was assigned to after the th iteration. Also, in the th iteration good was selected instead of , hence it must be the case that .
Note that, as mentioned before, the condition holds for with respect to , i.e., . In addition, Lemma 2 implies that . Therefore, if for any , then there exists a good such and . Hence, is an allocation. ∎
5.2 implies Approximate
Lemma 3.
In any fair division instance with additive valuations, if an allocation is , then it is  allocation as well.
Proof.
Fix an agent and a set of agents, which contains , i.e., and . Also, let denote the set of all the goods allocated to the agents in . We will show that agent if is , then is at least times the maximin share of restricted to , i.e., . This establishes the stated claim.
Write to denote the set of goods which are positively valued by , . Now, among the set of agents consider the ones who are allocated at most one good from ; specifically, let . Write , , and . Note that the agent belongs to the group , and for all we have . Therefore, the fact that is implies that, for all , there exists a good such that and . In other words, for the additive valuation , we have .
We will now establish the multiplicative bound with respect to (the maximin share of restricted to ) and prove that . This will complete the proof. Since is additive, an averaging argument gives us
Here, the last inequality uses the bound for all . Therefore, . To complete the proof, we need to show that . Note that—while considering the maximin shares of agent —we can restrict our attention to (the set of goods which are positively valued by ). In particular, the equalities and imply that, without loss of generality, we can work under the assumptions that and . Effectively, for all , we can slightly abuse notation and denote by . Now, consider the allocation .
We have, . Also, note that for each agent . Therefore, there are at most bundles in with items from . We choose bundles from which do not contain any item from . Let us call these new bundles . By the definition of , the value of each such bundle for agent is greater than or equal to , that is, for all . Since we have assumed that all items have nonnegative values, adding more items from the remaining bundles to any of the bundles in can only increase the value of the partitions. Thus,
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