1 Introduction
A significant body of recent work, in algorithmic game theory, has been directed towards the study of fair and efficient allocation of indivisible goods among agents; see, e.g.,
[End17] and [BCE16]. This thread of research has led to the development of multiple algorithms and platforms (e.g., Spliddit [GP15]) which, in particular, address settings wherein discrete resources (that cannot be fractionally allocated) need to be partitioned among multiple agents. Contributing to this line of work, the current paper studies discrete fair division from a welfarist perspective.We specifically address the problem of finding allocations (of indivisible goods) that (approximately) maximize the generalized means of the agents’ valuations. Formally, for exponent parameter , the th generalized mean, of nonnegative values , is defined as . Parameterized by , this family of functions includes wellstudied fairness and efficiency objectives, such as average social welfare (), Nash social welfare (), and egalitarian welfare (). In fact, generalized means—with the exponent parameter in the range —admit a fundamental axiomatic characterization: up to monotonic transformations, generalized means (with ) exactly constitute the family of welfare functions that satisfy the PigouDalton transfer principle and a few other key axioms [Mou04].^{1}^{1}1Note that generalized means are ordinally equivalent to CES (constant elasticity of substitution) functions. Hence, by way of developing a single approximation algorithm for maximizing generalized means, the current work provides a unified treatment of multiple fairness and efficiency measures.
With generalized mean as our objective, we focus on fairdivision instances in which the agents have a common subadditive (i.e., complement free) valuation. Formally, a set function , defined over a set of indivisible goods , is a said to be subadditive iff, for all subsets and of , we have . This class of functions includes many other wellstudied valuation families, namely XOS, submodular, and additive valuations.^{2}^{2}2Recall that a submodular function is defined by a diminishing returns property: , for all subsets and . These function classes have been used extensively in computer science and mathematical economics to represent agents’ valuations. Of particular relevance here are results that (in the context of combinatorial auctions) address the problem of maximizing social welfare under submodular, XOS, and, more generally, subadditive valuations [NRTV07].
The focus on a common valuation function across the agents provides a technically interesting and applicable subclass of fairdivision problems–as a stylized application, consider a setting in which the agents’ values represent money, i.e., for every agent, the value of each subset (of the goods) is equal to the subset’s monetary worth. Here, one encounters subadditivity when considering goods that are substitutes of each other. Also, from a technical standpoint, we note that the problem of maximizing social welfare is APXhard even under identical submodular [KLMM08] and subadditive valuations [DNS05]. Appendix B extends this hardness result to all .
Our Results: Addressing fairdivision instances with identical subadditive valuations, we develop an efficient constantfactor approximation algorithm for the generalizedmean objective (Theorem 1). Specifically, our algorithm computes an allocation (of the indivisible goods among the agents), , with the property that its generalizedmean welfare, , is at least times the optimal mean welfare, for all . This result in fact implies an interesting existential guarantee as well: if in a fairdivision instance the agents’ valuations are identical and subadditive, then there exists a single allocation that uniformly approximates the optimal mean welfare for all .
The tradeoff between fairness and economic efficiency is an important consideration in fair division literature.^{3}^{3}3For example, consider the work on price of fairness [BFT11; BLMS19] The relevance of the abovementioned existential guarantee is substantiated by the fact that this result reasonably mitigates the fairnessefficiency tradeoff in the current context; it shows that for identical subadditive valuations there exists a single allocation which is near optimal with respect to efficiency objectives (in particular, social welfare) as well as fairness measures (e.g., egalitarian welfare). Note that such an allocation cannot be simply obtained by selecting an arbitrary partition that (approximately) maximizes social welfare: under identical additive valuations, all the allocations have the same social welfare, even the ones with egalitarian welfare equal to zero. One can also construct instances, with identical subadditive valuations, wherein particular allocations have optimal egalitarian welfare, but subpar social welfare.
Even specific instantiations of our algorithmic guarantee provide novel results: while the problem of maximizing Nash social welfare, among agents, admits an approximation under nonidentical submodular valuations [GKK20], the current work provides a novel (constantfactor) approximation guarantee for maximizing Nash social welfare when the agents share a common subadditive (and, hence, submodular) valuation.^{4}^{4}4Under nonidentical additive valuations, there exists a polynomialtime approximation algorithm for maximizing Nash social welfare [BKV18a]. Furthermore, under identical additive valuations, maximizing Nash social welfare admits a polynomialtime approximation scheme [NR14; BKV18b]. Analogously, the instantiation of our result for egalitarian welfare is interesting in and of itself.
Given that the valuations considered in this work express combinatorial preferences, a naive representation of such set functions would require exponential (in the number of goods) values, one for each subset of the goods. Hence, to primarily focus on the underlying computational aspects and not on the representation details, much of prior work assumes that the valuations are provided via oracles that can only answer particular type of queries. The most basic oracle considered in literature answers value queries: given a subset of the indivisible goods, the value oracle returns the value of this subset. In this value oracle model, the work of Vondrák [Von08] considers submodular valuations and provides an efficient approximation algorithm for maximizing social welfare. Using this method as a subroutine and, hence, completely in the value oracle model, our algorithm achieves the abovementioned approximation guarantee for identical submodular valuations.
Another wellstudied oracle addresses demand queries. Specifically, such an oracle, when queried with an assignment of prices to the goods, returns , for the underlying valuation function .^{5}^{5}5Observe that a value query can be simulated via polynomially many demand queries. Though, the converse is not true [NRTV07]. Demand oracles have been often utilized in prior work for addressing social welfare maximization in the context of subadditive and XOS valuations [NRTV07]. In particular, the work of Fiege [Fei09] shows that, under subadditive valuations and assuming oracle access to demand queries,^{6}^{6}6This result holds even if the agents have distinct, but subadditive, valuations. the social welfare maximization problem admits an efficient approximation algorithm. Demand queries are unavoidable in the subadditive case: one can directly extend the result of Dobzinski et al. [DNS10] to show that, even under identical (subadditive) valuations, any sublinear (in ) approximation of the optimal social welfare requires exponentially many value queries. At the same time, we note that our algorithm requires demand oracle access only to implement the approximation algorithm of Fiege [Fei09] as a subroutine. Beyond this, we can work with the value oracle.
Related Work: Multiple algorithmic and hardness results have been developed to address welfare maximization in the context of indivisible goods/discrete resources. Though, in contrast to the present paper, prior work in this direction has primarily addressed one welfare function at a time.
As mentioned previously, maximizing social welfare and Nash social welfare (see, e.g., [CG18] and references therein) has been actively studied in algorithmic game theory. Egalitarian welfare has also been addressed in prior work–this welfare maximization problem is also referred to as the maxmin allocation problem (or the Santa Claus problem); see, e.g., [AKS14]. Specifically, for maximizing egalitarian welfare under additive and nonidentical valuations, the result of Chakrabarty et al. [CCK09] provides an approximation algorithm that runs in time ; here denotes the number of agents and . Furthermore, under nonidentical submodular valuations, the problem of maximizing egalitarian welfare is known to admit a polynomialtime approximation algorithm [GHIM09]; here is the number of goods. In contrast to these sublinear approximations, this paper shows that, if the agents’ valuations are identical, then even under subadditive valuations the problem of maximizing egalitarian welfare admits a constantfactor approximation guarantee.
2 Notation and Preliminaries
An instance of a fairdivision problem corresponds to a tuple , where denotes the set of indivisible goods that have to be allocated (partitioned) among the set of agents, . Here, represents the (identical) valuation function of the agents;^{7}^{7}7Recall that this work addresses fairdivision instances in which all the agents have a common valuation function. specifically, is the value that each agent has for a subset of goods .
We will assume throughout that the valuation function is (i) normalized: , (ii) monotone: for all , and (iii) subadditive: for all subsets .
Write to denote the collection of all partitions of the indivisible goods . We use the term allocation to refer to an partition of the goods. Here, denotes the subset of goods allocated to agent and will be referred to as a bundle.
Generalized (Hölder) means, , constitute a family of functions that capture multiple fairness and efficiency measures. Formally, for an exponent parameter , the th generalized mean of nonnegative numbers is defined as .
Note that, when , reduces to the arithmetic mean. Also, as tends to zero,
, in the limit, is equal to the geometric mean and
. Hence, following standard convention, we will write and .Considering generalized means as a parameterized collection of welfare objectives, we define the mean welfare, , of an allocation as
(1) 
Here, is the (common) valuation function of the agents. Indeed, with equal to one, zero, and , the mean welfare, respectively, corresponds to (average) social welfare, Nash social welfare, and egalitarian welfare.
Given a fairdivision instance and , ideally, we would like to find an allocation with as large an value as possible, i.e., maximize the mean welfare. An allocation that achieves this goal will be referred to as a optimal allocation and denoted by .
We note that, under identical, subadditive valuations, finding a optimal allocation is APXhard, for any (Appendix B). Hence, the current work considers approximation guarantees. In particular, for fairdivision instances in which the agents have a common subadditive valuation, we develop a polynomialtime algorithm that computes a single allocation with the property that for all . That is, the developed algorithm achieves an approximation ratio of uniformly for all .
The work of Fiege [Fei09] shows that, for subadditive valuations, the socialwelfare maximization problem (equivalently, the problem of maximizing ) admits an efficient approximation algorithm, assuming oracle access to demand queries. In particular, such an oracle, when queried with an assignment of prices to the goods, returns . Our algorithm requires demand oracle access only to implement the approximation algorithm of Fiege [Fei09] as a subroutine. Beyond this, we can work with the basic value oracle, which when queried with a subset of goods , returns .
In fact, if the underlying valuation is submodular, then one can invoke the result of Vondrák [Von08] (instead of using the approximation algorithm by Feige [Fei09]) and efficiently obtain a approximation for the socialwelfare maximization problem in the value oracle model. Hence, under a submodular valuation, our algorithm can be implemented entirely in the standard value oracle model.
For a fairdivision instance , write to denote the mean welfare (i.e., the average social welfare) of the allocation computed by the approximation algorithm of Feige [Fei09]. The approximation guarantee established in [Fei09] ensures that—for any instance with a subadditive valuation—we have . Here, denotes a optimal allocation, i.e., it maximizes the (average) social welfare in .
3 Maximizing Mean Welfare
Addressing fairdivision instances with identical subadditive valuations, this section presents an efficient algorithm for computing a constantfactor approximation to the mean welfare objective, uniformly for all .
The algorithm consists of two phases, Algorithm 1 (Alg) and Algorithm 2 (AlgLow). In the first phase, “highvalue” goods are assigned as singletons–we use the approximation algorithm of Feige [Fei09]
to obtain an estimate of the optimal
mean welfare and deem a good to be of high value if its valuation is at least a constant (specifically, ) times this estimate. Intuitively, the estimate provides a useful benchmark, since the optimal mean welfare upper bounds the optimal mean welfare for all (Proposition 1); this bound essentially follows from the generalized mean inequality [BMV88] which asserts that, for all , the mean welfare of any allocation is at most its mean welfare, .Therefore, in phase one of the algorithm, we sort the goods in nonincreasing order by value and iteratively select goods, which by themselves provide a value comparable to that of the optimal mean welfare. In each iteration, the selected good is assigned as a singleton to an agent and this agentgood pair is removed from consideration. Note that such an update leads to a new fairdivision instance with one less good and one less agent, as well as a potentially different optimal mean welfare. The key technical issue here is that the change in the optimal mean welfare (and, hence, its estimate obtained via Feige’s algorithm) can be nonmonotonic. Nonetheless, via an inductive argument, we show that the welfare contribution of the goods assigned (as singletons) in the first phase is sufficiently large (Lemma 3).
The first phase terminates when we obtain an instance wherein each good is of value no more than a constant times its optimal mean welfare. The second phase (AlgLow) is designed to address such a fairdivision instance. In particular, we show that, in the absence of highvalue goods, we can efficiently find an allocation such that each bundle is of value at least constant times the optimal mean welfare of . To obtain the allocation , we first compute (via Feige’s approximation algorithm) an allocation that provides a approximation to the optimal mean welfare of . Subsequently, we show that the subsets s, that have appropriately high value, can be partitioned to form the desired bundles s, which constitute the allocation .
Multiple technical lemmas (in Sections 5 and 6) are required to show that the two phases in combination lead to the desired welfare bound. It is also relevant to note that, while the abovementioned ideas hold at a high level, the formal guarantees are obtained by separately analyzing different ranges of the exponent parameter .
The following theorem constitutes the main result of the current work. It asserts that Algorithm 1 (Alg) achieves a constantfactor approximation ratio for the mean welfare maximization problem.
Theorem 1 (Main Result).
Let be a fairdivision instance wherein all the agents have an identical, subadditive valuation function . Given demand oracle access to , Alg computes in polynomial time an allocation that, for all , provides a approximation to the optimal mean welfare, i.e., , for all ; here, is the optimal allocation in .
We first consider an instance wherein all the goods are of value a constant times less than and prove that, for such an instance, AlgLow finds an allocation in which the value of every bundle is comparable to the optimal average social welfare of . In Section 7, we use this fact and supporting lemmas from Sections 5 and 6 to prove Theorem 1 for . Finally, in Section 8, we prove the main result for
We start with the following observation to upper bound the optimal mean welfare in terms of the optimal mean welfare.
Proposition 1.
Let be a fairdivision instance in which all the agents have an identical, subadditive valuation . Then, for each the optimal mean welfare is at least as large as the optimal mean welfare:
Proof The generalized mean inequality (see, e.g., [BMV88]) applied to the allocation , gives us , for all . By definition, the allocation maximizes the mean welfare, , and, hence, the claim follows .
4 Approximation Guarantee for AlgLow
This section addresses the second phase of the algorithm (AlgLow) that—by the processing performed in the whileloop of Alg—solely needs to consider fairdivision instances wherein all the goods satisfy , i.e., the goods are of “low value.” The following lemma establishes that, for such instances, AlgLow finds bundles each with value comparable to the optimal mean welfare (and, hence, comparable to the optimal mean welfare) of .
Recall that here is a subset of the original set of goods , is a subset of the agents, and denotes the mean welfare (average social welfare) of the allocation computed by Feige’s approximation algorithm for instance .
Lemma 1.
Let be a fairdivision instance in which all the agents have an identical subadditive valuation function , and every good satisfies . Then, in the demand oracle model, the algorithm AlgLow efficiently computes an allocation with the property that, for all ,
Proof For input instance , Feige’s algorithm returns an allocation with nearoptimal average social welfare:
Given allocation , we show that one can partition s to form bundles such that the value of each bundle is at least . Note that, by the assumption in the lemma, for each we have
(2) 
Let denote the subsets in with value at least . These are the subsets we split to form bundles that form the output allocation .
Since is subadditive, we have the following lower bound on the cumulative value of the subsets in
(3) 
Claim 1.
Let be the allocation computed by Feige’s algorithm for input instance Then, every , with the property that , can be partitioned into at least subsets such that for each .
Proof Initialize to be the empty set. Then, we keep transferring goods—in any order and one at a time—from to till the value of goes over Returning the last such good back into , the populated set satisfies
Note that, by construction, . Therefore, using the subadditivity of , we get .
We can repeat the above process to obtain subsets and stop when Given that, for each , we remove a subset of value atmost from , the subadditivity of gives us . Hence, the following lower bound holds . In other words, we can extract at least bundles, each of value no less than , from .
We now apply the same procedure to every subset in to obtain bundles, each of value at least . Using this equation and inequality (3), we get
(Since )  
In conclusion, one can construct at least bundles of value at least . Note that this observation implies that AlgLow successfully finds bundles each of value at least .
Proposition 1 gives us and, hence, the stated claim follows.
With an approximation guarantee for AlgLow in hand, we next analyze the first phase of the algorithm–specifically, analyze the whileloop in Alg. Note that in each iteration of this whileloop (Steps (4) to (7)) a good of value at least constant times the optimal mean welfare (of the current instance) is allocated as a singleton to an agent. We will show that these singleton assignments complement the subsequent use of AlgLow, in the sense that the two phases together retain welfare guarantees. To formally prove Theorem 1, we first state and prove two useful results, Lemma 2 (in Section 5) and Lemma 3 (in Section 6).
5 Structural Lemma
The following lemma provides a structural property of optimal allocations , for . It states that the only way allocation has a bundle of notably high value is through a single good that by itself has high value.
Lemma 2.
Let be a fairdivision instance wherein all the agents have an identical, subadditive valuation over the set of goods. In addition, let be a mean optimal allocation in , for any .
If for any bundle , with , we have , then there exists a good with the property that that
The proof of the above lemma is divided into three parts (Sections 5.1, 5.2, and 5.3) depending on the range of the exponent parameter
5.1 Proof of Lemma 2 for
Assume, towards a contradiction, that , for some , and for all Recall that the mean welfare of the allocation returned by Feige’s algorithm satisfies . Pick a bundle (in ) with the property that . Such a bundle exists, since .
Define a partition— and —of as follows:
(i) Initialize to be the empty set. Then, we keep transferring goods from to (one at a time and in an arbitrary order) and stop as soon as the value of exceeds .
(ii) Denote the remaining set of goods as . Note that, by construction, and
. The last inequality follows from the fact that is subadditive and the assumption that goods in are of value at most .
(iii) Write to denote the allocation obtained by replacing the bundles and in by and , respectively: and along with for all .
We will show that has mean welfare strictly greater than that of the optimal allocation . Hence, by way of contradiction, the desired result follows.
Recall that the current case addresses exponent parameters that are negative, . Hence, the following inequality implies that the mean welfare of is strictly greater than that of :
(4) 
However, for negative , the lower bounds on the values of and gives us
In addition, using the bounds , we get
5.2 Proof of Lemma 2 for
Assume, towards a contradiction, that , for some , and for all Recall that the mean welfare of the allocation returned by Feige’s algorithm satisfies . Pick a bundle (in ) with the property that . Such a bundle exists, since .
Define a partition— and —of as follows:
(i) Initialize to be the empty set. Then, we keep transferring goods from to (one at a time and in an arbitrary order) and stop as soon as the value of exceeds .
(ii) Denote the remaining set of goods as . Note that, by construction, and
. The last inequality follows from the fact that is subadditive and the assumption that goods in are of value at most .
(iii) Write to denote the allocation obtained by replacing the bundles and in by and , respectively: and along with for all .
We will show that has mean welfare strictly greater than that of the optimal allocation . Hence, by way of contradiction, the desired result follows.
Notice that the construction of , and hold for as well. We will use these sets in the next section to prove an analogous result for Nash social welfare.
The current case addresses exponent parameters that are positive . Hence, the following inequality implies that the mean welfare of is strictly greater than that of
(5) 
However, for positive , the lower bounds on the values of and gives us
In addition, using the bounds , we get
Therefore, the desired equation (5) follows from the following numeric inequality, which is established in Appendix A.
This establishes Lemma 2 for .
5.3 Proof of Lemma 2 for Nash Social Welfare ()
Recall the sets , , and the allocation defined in Section 5.2. We will show that has Nash welfare strictly greater than that of , and hence, by way of contradiction, establish the desired result.
In order to obtain , it suffices to prove that The lower bounds we obtained on the values of and gives us
In addition, we have
Therefore, the lemma holds for as well.
6 Combination Lemma
The following lemma shows that the goods assigned as singletons in the whileloop of Alg (Algorithm 1), along with a optimal allocation of instance that remains at the termination of the loop, lead to a mean welfare that is comparable to the optimal, for all .
As shown previously in Lemma 1, AlgLow—with instance as input—achieves a constantfactor approximation for the mean welfare objective. Hence, the lemma established in this section will enable us to combine the welfare guarantees of the goods assigned in the whileloop of Alg and the allocation computed by AlgLow to obtain the desired approximation result for .
Specifically, given a fairdivision instance as input, let denote the set of goods that get assigned as singletons in the whileloop of Alg (Algorithm 1). Furthermore, for , let denote the instance obtained at the end of the th iteration of this whileloop. Since in the first iterations Alg assigns goods to the first agents as singletons, we have . In particular, is the instance that remains after the termination of the whileloop in Alg and this instance is passed on to AlgLow as input.
Instance consists of agents and, hence, in , any optimal allocation contains bundles. For notational convenience, we will index these bundles from to , i.e., .
Lemma 3.
Given a fairdivision instance with an identical subadditive valuation , let denote the set of goods that get assigned as singletons in the whileloop of Alg and let be the instance that remains after the termination of this loop. In addition, let and denote optimal allocations of instances and , respectively. Then, with constant ,

For , we have .

For , we have .

For , we have
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