Fair Allocation of Indivisible Public Goods

05/08/2018 ∙ by Brandon Fain, et al. ∙ UNIVERSITY OF TORONTO Duke University 0

We consider the problem of fairly allocating indivisible public goods. We model the public goods as elements with feasibility constraints on what subsets of elements can be chosen, and assume that agents have additive utilities across elements. Our model generalizes existing frameworks such as fair public decision making and participatory budgeting. We study a groupwise fairness notion called the core, which generalizes well-studied notions of proportionality and Pareto efficiency, and requires that each subset of agents must receive an outcome that is fair relative to its size. In contrast to the case of divisible public goods (where fractional allocations are permitted), the core is not guaranteed to exist when allocating indivisible public goods. Our primary contributions are the notion of an additive approximation to the core (with a tiny multiplicative loss), and polynomial time algorithms that achieve a small additive approximation, where the additive factor is relative to the largest utility of an agent for an element. If the feasibility constraints define a matroid, we show an additive approximation of 2. A similar approach yields a constant additive bound when the feasibility constraints define a matching. More generally, if the feasibility constraints define an arbitrary packing polytope with mild restrictions, we show an additive guarantee that is logarithmic in the width of the polytope. Our algorithms are based on variants of the convex program for maximizing the Nash social welfare, but differ significantly from previous work in how it is used. Our guarantees are meaningful even when there are fewer elements than the number of agents. As far as we are aware, our work is the first to approximate the core in indivisible settings.

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1 Introduction

In fair resource allocation, most work considers private goods; each good must be assigned to a particular agent (and no other). However, not all goods are private. Public goods are those which can be enjoyed by multiple agents simultaneously, like a public road. Allocation of public goods generalizes the problem of allocation of private goods, and, as we will see, can provide new difficulties from both a normative and an algorithmic perspective.

Consider an example to highlight what a public resource allocation problem might look like, and why fairness might be a concern. Suppose that the next time you vote, you see that there are four referendums for your consideration on the ballot, all of which concern the allocation of various public goods in your city: A = a new school, B = enlarging the public library, C = renovating the community college, and D = improving a museum. In 2016, residents of Durham, North Carolina faced precisely these options [25]. Suppose the government has resources to fund only two of the four projects, and the (hypothetical) results were as follows: a little more than half of the population voted for , a little less than half voted for , and every other combination received a small number of votes. Which projects should be funded?

If we naïvely tally the votes, we would fund A and B, and ignore the preferences of a very large minority. In contrast, funding A and C seems like a reasonable compromise. Of course, it is impossible to satisfy all voters, but given a wide enough range of possible outcomes, perhaps we can find one that fairly reflects the preferences of large subsets of the population. This idea is not captured by fairness axioms like proportionality or their approximations [10], which view fairness from the perspectives of individual agents. Indeed, in the aforementioned example, every allocation gives zero utility to some agent, and would be deemed equally good according to such fairness criteria.

1.1 Public Goods Model

We consider a fairly broad model for public goods allocation that generalizes much of previous work [24, 10, 11, 3, 12, 9, 30]. There is a set of voters (or agents) . Public goods are modeled as elements of a ground set . We denote . An outcome is a subset of . Let denote the set of feasible outcomes.

The utility of agent for element is denoted . We assume that agents have additive utilities, i.e., the utility of agent under outcome is . Since we are interested in scale-invariant guarantees, we assume without loss of generality that for each agent , so that for all . Crucially, this does not restrict the utility of an agent for an outcome to be : can be as large as . Specifically, let , and . Our results differ by the feasibility constraints imposed on the outcome. We consider three types of constraints, special cases of which have been studied previously in literature.

Matroid Constraints.

In this setting, we are given a matroid over the ground set , and the feasibility constraint is that the chosen elements must form a basis of (see [22] for a formal introduction to matroids).

This generalizes the public decision making setting introduced by [10]. In this setting, there is a set of issues , and each issue has an associated set of alternatives , exactly one of which must be chosen. Agent has utility if alternative is chosen for issue , and utilities are additive across issues. An outcome chooses one alternative for every issue. It is easy to see that if the ground set is , the feasibility constraints correspond to a partition matroid. We note that public decision making in turn generalizes the classical setting of private goods allocation [24, 9, 30] in which private goods must be divided among agents with additive utilities, with each good allocated to exactly one agent.

Matroid constraints also capture multi-winner elections in the voting literature (see, e.g. [3]), in which voters have additive utilities over candidates, and a committee of at most candidates must be chosen. This is captured by a uniform matroid over the set of candidates.

Matching Constraints.

In this setting, the elements are edges of an undirected graph , and the feasibility constraint is that the subset of edges chosen must form a matching. Matchings constraints in a bipartite graph can be seen as the intersection of two matroid constraints. Matching constraints are a special case of the more general packing constraints we consider below.

Packing Constraints.

In this setting, we impose a set of packing constraints , where is the indicator denoting whether element is chosen in the outcome. Suppose is a matrix, so that there are packing constraints. By scaling, we can assume for all . Note that even for one agent, packing constraints encode independent set. Thus, to make the problem tractable, we assume is sufficiently large, in particular, for all . This is in contrast to matroid and matching constraints, for which single-agent problems are polynomial time solvable. A classic measure of how easy it is to satisfy the packing constraints is the width  [31]:

(1)

Packing constraints capture the general Knapsack setting, in which there is a set of items, each item has an associated size , and a set of items of total size at most must be selected. This setting is motivated by participatory budgeting applications [29, 18, 17, 11, 15, 12, 6], in which the items are public projects, and the sizes represents the costs of the projects. Knapsack uses a single packing constraint. Multiple packing constraints can arise if the projects consume several types of resources, and there is a budget constraint for each resource type. For example, consider a statewide participatory budgeting scenario where each county has a budget than can only be spent on projects affecting that county, the state has some budget that can be spent in any county, and projects might affect multiple counties. In such settings, it is natural to assume a small width, i.e., that the budget for each resource is such that a large fraction (but not all) of the projects can be funded. We note that the aforementioned multi-winner election problem is a special case of the Knapsack problem with unit sizes.

1.2 Prior Work: Fairness Properties

We define fairness desiderata for the public goods setting by generalizing appropriate desiderata from the private goods setting such as Pareto optimality, which is a weak notion of efficiency, and proportionality, which is a per-agent fair share guarantee.111Those familiar with the literature on fair division of private goods will note the conspicuous absence of the envy freeness property: that no agent should (strongly) prefer the allocation of another agent. Because we are considering public goods, envy freeness is only vacuously defined: the outcome in our setting is common to all agents.

Definition 1.

An outcome satisfies Pareto optimality if there is no other outcome such that for all agents , and at least one inequality is strict.

Recall that is the maximum possible utility agent can derive from a feasible outcome.

Definition 2.

The proportional share of an agent is . For , we say that an outcome satisfies -proportionality if for all agents . If , we simply say that satisfies proportionality.

The difficulty in our setting stems from requiring integral outcomes, and not allowing randomization. In the absence of randomization, it is reasonably straightforward to show that we cannot guarantee -proportionality for any . Consider a problem instance with two agents and two feasible outcomes, where each outcome gives a positive utility to a unique agent. In any feasible outcome, one agent has zero utility, which violates -proportionality for every .

To address this issue, [10] introduced the novel relaxation of proportionality up to one issue in their public decision making framework, inspired by a similar relaxation called envy-freeness up to one good in the private goods setting [24, 9]. They say that an outcome of a public decision making problem satisfies proportionality up to one issue if for all agents , there exists an outcome that differs from only on a single issue and . Proportionality up to one issue is a reasonable fairness guarantees only when the number of issues is larger than the number of agents; otherwise, it is vacuous and is satisfied by all outcomes. Thus, it is perfectly reasonable for some applications (e.g., three friends choosing a movie list to watch together over the course of a year), but not for others (e.g., when thousands of residents choose a handful of public projects to finance). In fact, it may produce an outcome that may be construed as unfair if it does not reflect the wishes of large groups of voters. Thus, in this work, we address the following question posed by [10]:

Is there a stronger fairness notion than proportionality in the public decision making framework…? Although such a notion would not be satisfiable by deterministic mechanisms, it may be satisfied by randomized mechanisms, or it could have novel relaxations that may be of independent interest.

1.3 Summary of Contributions

Our primary contributions are twofold.

  • We define a fairness notion for public goods allocation that is stronger than proportionality, ensures fair representation of groups of agents, and in particular, provides a meaningful fairness guarantee even when there are fewer goods than agents.

  • We provide polynomial time algorithms for computing integer allocations that approximately satisfy this fairness guarantee for a variety of settings generalizing the public decision making framework and participatory budgeting.

1.4 Core and Approximate Core Outcomes

Below, we define the notion of core outcomes, which has been extensively studied (in similar forms) as a notion of stability in economics [13, 34, 26] and computer science [23, 11] in the context of randomized or fractional allocations. Our main contribution is to study it in the context of integer allocations.

Definition 3.

Given an outcome , we say that a set of agents form a blocking coalition if there exists an outcome such that for all and at least one inequality is strict. We say that an outcome is a core outcome if it admits no blocking coalitions.

Note that non-existence of blocking coalitions of size is equivalent to proportionality, and non-existence of blocking coalitions of size is equivalent to Pareto optimality. Hence, a core outcome is both proportional and Pareto optimal. However, the core satisfies a stronger property of being, in a sense, Pareto optimal for coalitions of any size, provided we scale utilities based on the size of the coalition. Another way of thinking about the core is to view it as a fairness property that enforces a proportionality-like guarantee for coalitions: e.g., if half of all agents have identical preferences, they should be able to get at least half of their maximum possible utility. It is important to note that the core provides a guarantee for every possible coalition. Hence, in satisfying the guarantee for a coalition , a solution cannot simply make a single member happy and ignore the rest as this would likely violate the guarantee for the coalition .

Approximate Core.

Since a proportional outcome is not guaranteed to exist (even allowing for multiplicative approximations), the same is true for the core. However, an additive approximation to the core still provides a meaningful guarantee, even when there are fewer elements than agents because it provides a non-trivial guarantee to large coalitions of like-minded agents.

Definition 4.

For , an outcome is a -core outcome if there exists no set of agents and outcome such that

for all , and at least one inequality is strict.

A -core outcome is simply a core outcome. A -core outcome satisfies -proportionality. Similarly, a -core outcome satisfies the following relaxation of proportionality that is slightly weaker than proportionality up to one issue: for every agent , . We note that this definition, and by extension, our algorithms satisfy scale invariance, i.e., they are invariant to scaling the utilities of any individual agent. Because we normalize utilities of the agents, the true additive guarantee is times the maximum utility an agent can derive from a single element. Since an outcome can have many elements, an approximation with small remains meaningful.

The advantage of an approximate core outcome is that it fairly reflects the will of a like-minded subpopulation relative to its size. An outcome satisfying approximate proportionality only looks at what individual agents prefer, and may or may not respect the collective preferences of sub-populations. We present such an instance in Example 1 (Section 2.1), in effect showing that an approximate core outcome is arguably more fair.

In our results, we will assume to be a small constant, and focus on making as small as possible. In particular, we desire guarantees on that exhibit sub-linear or no dependence on , , or any other parameters. Deriving such bounds is the main technical focus of our work.

1.5 Our Results

We present algorithms to find approximate core outcomes under matroid, matching, and general packing constraints. Our first result (Section 3) is the following:

Theorem 1.

If feasible outcomes are constrained to be bases of a matroid, then a -core outcome is guaranteed to exist, and for any , a -core outcome can be computed in time polynomial in and .

In particular, for the public decision making framework, the private goods setting, and multi-winner elections (a.k.a. Knapsack with unit sizes), there is an outcome whose guarantee for every coalition is close to the guarantee that Conitzer et al. provide to individual agents [10].

In Section 4, we consider matching constraints. Our result now involves a tradeoff between the multiplicative and additive guarantees.

Theorem 2.

If feasible outcomes are constrained to be matchings in an undirected graph, then for constant , a -core outcome can be computed in time polynomial in and .

Our results in Section 5 are for general packing constraints. Here, our guarantee depends on the width from Equation (1), which captures the difficulty of satisfying the constraints. In particular, the guarantee improves if the constraints are easier to satisfy. This is the most technical result of the paper, and involves different techniques than those used in proving Theorems 1 and 2; we present an outline of the techniques in Section 5.2.

Theorem 3.

For constant , given packing constraints with width and for all , there exists a polynomial time computable -core solution, where

Here, is the iterated logarithm, which is the number of times the logarithm function must be iteratively applied before the result becomes less than or equal to . Recall that is the maximum utility an agent can have for an outcome (thus ); our additive error bound is a vanishing fraction of this quantity. Our bound is also small if the number of agents is small. Finally, the guarantee improves for small , i.e., as the packing constraints become easier to satisfy. For instance, in participatory budgeting, if the total cost of all projects is only a constant times more than the budget, then our additive guarantee is close to a constant.

Note that (which is bounded by ), , and are all unrelated quantities — either could be large with the other two being small. In fact, in Section 5, we state the bound more generally in terms of what we call the maximally proportionally fair value , which informally captures the (existential) difficulty of finding a proportionally fair allocation. The quantity stems from three different bounds on the value of .

In Example 2 (Appendix A), we show that the lower bound on in the above theorem is necessary: if , then no non-trivial approximation to the core can be guaranteed, even when is a constant.

Finally, in Appendix C, we consider a different (and more classical) version of the core for general packing constraints, in which a deviating coalition gets a proportional share of resources rather than a proportional share of utility. We show that our techniques provide a similar approximation to this version of the core, although we do not provide an efficient algorithm in this model.

1.6 Related Work

Core for Public Goods.

The notion of core is borrowed from cooperative game theory and was first phrased in game theoretic terms by 

[34]. It has been extensively studied in public goods settings [13, 26, 11]. Most literature so far has considered the core with fractional allocations. Our definition of core (Definition 3) assumes the utility of a deviating coalition is scaled by the size of the coalition. For fractional allocations, one such core allocation coincides with the well-known notion of proportional fairness, the extension of the Nash bargaining solution [27]. This solution maximizes the product of the utilities of the agents, and we present the folklore proof in Section 2.2. Our main focus is on finding integer allocations that approximate the core, and to the best of our knowledge, this has not been studied previously.

A simpler property than the core is proportionality, which like the core, is impossible to satisfy to any multiplicative approximation using integral allocations. To address this problem, [10] defined proportionality up to one issue in the public decision making framework, inspired by related notions for private goods. This guarantee is satisfied by the integral outcome maximizing the Nash welfare objective, which is the geometric mean of the utilities to the agents. For public goods, this objective is not only NP-Hard to approximate to any multiplicative factor, but approximations to the objective also do not retain the individual fairness guarantees.

We extend the notion of additive approximate proportionality to additive approximate core outcomes, which provides meaningful guarantees even when there are fewer goods than agents. Unlike proportionality, we show in Section 2.1 that the approach of computing the optimal integral solution to the Nash welfare objective fails to provide a reasonable approximation to the core. Therefore, for our results about matroid constraints (Theorem 1) and matching constraints (Theorem 2), we slightly modify the integer Nash welfare objective and add a suitable constant term to the utility of each agent. We show that maximizing this smooth objective function achieves a good approximation to the core. However, maximizing this objective is still NP-hard [12], so we devise local search procedures that run in polynomial time and still give good approximations of the core. In effect, we make a novel connection between appropriate local optima of smooth Nash Welfare objectives and the core.

Fairness on Endowments.

Classically, the core is defined in terms of agent endowments, not scaled utilities. In more detail, in Definition 3, we assumed that when a subset of agents deviates, they can choose any feasible outcome; however, their utility is reduced by a factor that depends on . A different notion of core is based on endowments [34, 13] and has been considered in the context of participatory budgeting [11] and in proportional representation of voters in multi-winner elections with approval voting. In this notion, a deviating coalition gets a proportional share of resources rather than a proportional share of utility. For example, if the elements have different sizes, and we need to select a subset of them with total size at most , then a deviating coalition would get to choose an outcome with total size at most instead of , but would not have its utility scaled down. This notion builds on the seminal work of Foley on the Lindahl equilibrium [13], from which it can be shown that such a core outcome always exists when fractional allocations are allowed. However, it is not known how to compute such a core outcome efficiently, and further, it is difficult to define such a notion of endowments in settings such as matroid or matching constraints. In the context of integer allocations with packing constraints, we extend our techniques to provide approximations to the notion of core with endowments in Appendix C, though this is not the main focus of our paper.

The notion of core with endowments logically implies a number of fairness notions considered in multi-winner election literature, such as justified representation, extended justified representation [3], and proportional justified representation [33]. Approval-based multi-winner elections are a special case of packing constraints, in which voters (agents) have binary utilities over a pool of candidates (elements), and we must select a set of at most candidates. The idea behind proportional representation is to define a notion of large cohesive groups of agents with similar preferences, and ensure that such coalitions are proportionally represented. The core on endowments represents a more general condition that holds for all coalitions of agents, not just those that are large and cohesive. Nevertheless, our local search algorithms for Theorems 1 and 2 are similar to local search algorithms for proportional approval voting (PAV) [36, 4] that achieve proportional representation. It would be interesting to explore the connection between these various notions in greater depth.

Private Goods and Envy-freeness.

Private goods are a special case of public goods with matroid constraints. Fair allocation of private goods is a widely studied topic [37, 16, 28, 21]. A common fairness criterion for private goods is envy-freeness: that no agent should (strongly) prefer the allocation of another agent. For fractional allocations, the classic context for envy-free allocation is cake cutting [32, 5]. For integral allocations, envy-free allocations or multiplicative approximations thereof may not exist in general. Recent work has introduced envy-freeness up to one good [24, 8, 9, 30]

, an additive approximation of envy-freeness. The notion of envy does not extend as is to public goods, and the core can be thought of as enforcing envy-freeness across demographics. We note that in addition to resource allocation, group based fairness is also appearing as a desideratum in machine learning. Specifically, related notions may provide a tool against gerrymandered classifiers that appear fair on small samples, but not on structured subsets 

[20].

Strategyproofness.

In this work, we will not consider game-theoretic incentives for manipulation for two reasons. First, even for the restricted case of private goods allocation, preventing manipulation leads to severely restricted mechanisms. For instance, [35] shows that the only strategyproof and Pareto efficient mechanisms are dictatorial, and thus highly unfair, even when there are only two agents with additive utilities over divisible goods. Second, our work is motivated by public goods settings with a large number of agents, such as participatory budgeting, wherein individual agents often have limited influence over the final outcome. It would be interesting to establish this formally, using notions like strategyproofness in the large [2].

2 Prelude: Nash Social Welfare

Our approach to computing approximate core solutions revolves around the Nash social welfare, which is the product (or equivalently, the sum of logarithms) of agent utilities. This objective is commonly considered to be a natural tradeoff between the fairness-blind utilitarian social welfare objective (maximizing the sum of agent utilities) and the efficiency-blind egalitarian social welfare objective (maximizing the minimum agent utility). This function also has the advantage of being scale invariant with respect to the utility function of each agent, and in general, preferring more equal distributions of utility.

2.1 Integer Nash Welfare and Smooth Variants

The integer Max Nash Welfare (MNW) solution [9, 10] is an outcome that maximizes . More technically, if every integer allocation gives zero utility to at least one agent, the MNW solution first chooses a largest set of agents that can be given non-zero utility simultaneously, and maximizes the product of utilities to agents in .

[10] argue that this allocation is reasonable by showing that it satisfies proportionality up to one issue for public decision making. A natural question is whether it also provides an approximation of the core. We answer this question in the negative. The example below shows that even for public decision making (a special case of matroid constraints), the integer MNW solution may fail to return a -core outcome, for any and .

Example 1.

Consider an instance of public decision making [10] with issues and two alternatives per issue. Specifically, each issue has two alternatives , and exactly one of them needs to be chosen. There are two sets of agents and . Every agent has , and utility 0 for all other alternatives. Every agent has and for all issues . Visually, this is represented as follows.

1 0 0 0 0 0
0 0 1 0 0 0
0 0 0 0 1 0
1 1 1

The integer MNW outcome is because any other outcome gives zero utility to at least one agent. However, coalition can deviate, choose outcome , and achieve utility for each agent in . For to be a -core outcome, we need

Hence, is not a -core outcome for any and . In contrast, it is not hard to see that is a -core outcome because each agent in gets utility at most one in any outcome.

Further, note that outcome gives every agent utility . Since for each agent , satisfies proportionality, and yet fails to provide a reasonable approximation to the core. One may argue that , which is a -core outcome, is indeed fairer because it respects the utility-maximizing choice of half of the population; the other half of the population cannot agree on what they want, so respecting their top choice is arguably a less fair outcome. Hence, the example also shows that outcomes satisfying proportionality (or proportionality up to one issue) can be very different from and less fair than approximate core outcomes.

Smooth Nash Welfare.

One issue with the Nash welfare objective is that it is sensitive to agents receiving zero utility. We therefore consider the following smooth Nash welfare objective:

(2)

where is a parameter. Note that coincides with the Nash welfare objective. The case of was considered by [12], who showed it is NP-Hard to optimize. Recall that we normalized agent utilities so that each agent has a maximum utility of for any element, so when we add to the utility of agent , it is equivalent to adding to the utility of agent when utilities are not normalized.

We show that local search procedures for the smooth Nash welfare objective, for appropriate choices of , yield a -core outcome for matroid constraints (Section 3) and a -core outcome for matching constraints (Section 4). In contrast, in Example 3 (Appendix A) we show that optimizing any fixed smooth Nash welfare objective cannot guarantee a good approximation to the core, even with a single packing constraint, motivating the need for a different algorithm.

2.2 Fractional Max Nash Welfare Solution

For general packing constraints, we use a fractional relaxation of the Nash welfare program. A fractional outcome

consists of a vector

such that measures the fraction of element chosen. The utility of agent under this outcome is . The fractional Max Nash Welfare (MNW) solution is a fractional allocation that maximizes the Nash welfare objective (without any smoothing). Define the packing polytope as:

Then the fractional MNW solution is .

It is easy to show that the fractional MNW allocation lies in the core. Let denote the optimal fractional allocation to the MNW program. By first order optimality, for any other allocation ,

(3)

Suppose for contradiction that is not a core outcome. Then there exists a set of agents and an outcome such that , and at least one inequality is tight. This implies . However, this contradicts Equation (3). Thus , the optimal fractional solution to the MNW program, is a core solution.

For the allocation of public goods, it can be shown that the fractional MNW outcome can be irrational despite rational inputs [1], preventing an exact algorithm. For our approximation results, a fractional solution that approximately preserves the utility to each agent would suffice, and we prove the following theorem in Appendix B.1.

Theorem 4.

For any , we can compute a fractional -core outcome in time polynomial in the input size and .

3 Matroid Constraints

We now consider public goods allocation with matroid constraints. In particular, we show that when the feasibility constraints encode independent sets of a matroid , maximizing the smooth Nash welfare objective in Equation (2) with yields a -core outcome. However, optimizing this objective is known to be NP-hard [12]. We also show that given , a local search procedure for this objective function (given below) yields a -core outcome in polynomial time, which proves Theorem 1.

3.1 Algorithm

Fix . Let , where is the number of elements. Recall that there are agents.

  1. Start with an arbitrary basis of .

  2. Compute .

  3. Let a swap be a pair such that , , and is also a basis of .

  4. Find a swap such that .

    • If such a swap exists, then perform the swap, i.e., update , and go to Step (2).

    • If no such swap exists, then output as the final outcome.

3.2 Analysis

First, we show that the local search algorithm runs in time polynomial in , , and . Note that because in our normalization, each agent can have utility at most . Thus, the number of iterations is . Finally, each iteration can be implemented in time by iterating over all pairs and computing the change in the smooth Nash welfare objective.

Next, let denote the outcome maximizing the smooth Nash welfare objective with , and denote the outcome returned by the local search algorithm. We show that is a -core outcome, while is a -core outcome.

For outcome , define . Fix an arbitrary outcome . For an agent with , we have that for every element :

This holds because for and . Summing this over all gives

For an agent with , we trivially have . Summing over all agents, we have that for every outcome :

(4)

We now use the following result:

Lemma 1 ([22]).

For every pair of bases and of a matroid , there is a bijection such that for every , is also a basis.

Using the above lemma, combined with the fact that for and , we have that for all :

(5)

We now provide almost similar proofs for the approximations achieved by the global optimum and the local optimum .

Global optimum. Suppose for contradiction that is not a -core outcome. Then, there exist a subset of agents and an outcome such that for all ,

and at least one inequality is strict. Rearranging the terms and summing over all , we obtain:

Combining this with Equation (5), and subtracting Equation (4) yields:

This implies existence of a pair such that , which contradicts the optimality of because is also a basis of .

Local optimum. Similarly, suppose for contradiction that is not a -core outcome. Then, there exist a subset of agents and an outcome such that for all ,

Here, the final transition holds because . Again, rearranging and summing over all , we obtain:

Once again, combining this with Equation (5), and subtracting Equation (4) yields:

This implies existence of a pair such that , which violates local optimality of because is also a basis of .

Lower Bound.

While a -core always outcome exists, we show in the following example that a -core outcome is not guaranteed to exist for any .

Lemma 2.

For and matroid constraints, -core outcomes are not guaranteed to exist.

Proof.

Consider the following instance of public decision making where we have several issues and must choose a single alternative for each issue, a special case of matroid constraints. There are agents, where is even. There are issues. The first issues correspond to unit-value private goods, i.e., each such issue has alternatives, and each alternative gives utility to a unique agent and utility to others. The remaining issues are “pair issues”; each such issue has alternatives, one corresponding to every pair of agents that gives both agents in the pair utility and all other agents utility .

It is easy to see that every integer allocation gives utility at most to at least two agents. Consider the deviating coalition consisting of these two agents. They can choose the alternative that gives them each utility on every pair issue, and split the private goods equally. Thus, they each get utility . For the outcome to be a -core outcome, we need . As , this requires . Hence, for any , a -core outcome is not guaranteed to exist. ∎

Note that Theorem 1 shows existence of a -core outcome, which is therefore tight up to a unit additive relaxation. Whether a -core outcome always exists under matroid constraints remains an important open question. Interestingly, we show that such an outcome always exists for the special case of private goods allocation, and, in fact, can be achieved by maximizing the smooth Nash welfare objective.

Lemma 3.

For private goods allocation, maximizing the smooth Nash welfare objective with returns a -core outcome.

Proof.

There is a set of agents and a set of private goods . Each agent has a utility function . Utilities are additive, so for all . For simplicity, we denote . Without loss of generality, we normalize the utility of each agent such that for each . An allocation is a partition of the set of goods among the agents; let denote the bundle of goods received by agent . We want to show that an allocation maximizing the objective is a -core outcome.

Let denote an allocation maximizing the smooth Nash welfare objective with . We assume without loss of generality that every good is positively valued by at least one agent. Hence, must imply .

For agents with (hence ), and good , moving to should not increase the objective function. Hence, for each , we have

Using additivity of utilities, this simplifies to

(6)

For every agent with and good , define . Abusing the notation a little, for a set define . Then, from Equation (6), we have that for all players and goods ,

(7)

Suppose for contradiction that is not a -core outcome. Then, there exists a set of agents and an allocation of the set of all goods to agents in such that for every agent , and at least one inequality is strict. Rearranging the terms and summing over , we have

(8)

We now derive a contradiction. For agent , summing Equation (7) over , we get

Summing this over , we get

However, this contradicts Equation (8). ∎

4 Matching Constraints

We now present the algorithm proving Theorem 2. We show that if the elements are edges of an undirected graph , and the feasibility constraints encode a matching, then for constant , a -core always exists and is efficiently computable. The idea is to again run a local search on the smooth Nash welfare objective in Equation (2), but this time with .

Algorithm.

Recall that there are agents. Let and . Let . For simplicity, assume . Our algorithm is inspired by the PRAM algorithm for approximate maximum weight matchings due to [19], and we follow their terminology. Given a matching , an augmentation with respect to is a matching . The size of the augmentation is . Let denote the subset of edges of that have a vertex which is matched under . Then, the matching is called the augmentation of using .

  1. Start with an arbitrary matching of .

  2. Compute .

  3. Let be the set of all augmentations with respect to of size at most .

    • If there exists such that , perform this augmentation (i.e., let ) and go to Step (2).

    • Otherwise, output as the final outcome.

Analysis.

The outline of the analysis is similar to the analysis for matroid constraints. First, we show that the algorithm runs in polynomial time. Again, recall that each agent has utility at most . Thus, . Because each improvement increases the objective value by at least , the number of iterations is . Each iteration can be implemented by naïvely going over all subsets of edges of size at most , checking if they are valid augmentations with respect to , and whether they improve the objective function by more than . The local search therefore runs in polynomial time for constant .

Let denote the outcome returned by the algorithm. We next show that is indeed a -core outcome. Suppose for contradiction that this is not true. Then, there exist a subset of agents and a matching such that for all ,

and at least one inequality is strict (the last inequality is because ). Rearranging and summing over all , we obtain

(9)

For , define and . Let , and . It is easy to check that

(10)

where the latter follows from Equation (9). Further note that for all .

For an augmentation with respect to , define . The next lemma is a simple generalization of the analysis in [19]; we give the adaptation here for completeness.

Lemma 4.

Assuming weights for all edges , for any integer and matchings and , there exists a multiset of augmentations with respect to such that:

  • For each , and ;

  • ; and

  • .

Proof.

We follow [19] in the construction the multiset of augmentations with respect to out of edges in . Let be the symmetric difference of matchings and consisting of alternating paths and cycles. For every cycle or path , let be be set of edges . For all with , just add to OPT times (note that is a multiset, not a set). For with , we break up into multiple smaller augmentations. To do so, index the edges in from to and add different augmentations to by considering starting at every index in and including the next edges in with wrap-around from to .

Now we must argue that as we have constructed it satisfies the conditions of the lemma. The first point, that and , follows trivially from the construction. The second point also follows easily from observing that we add augmentations to for every , and graph has vertices.

To see the third point, note that every edge in is contained in at least augmentations in . On the other hand, for every edge , there are no more than augmentations such that (recall are the edges of with a vertex matched under ). This can happen, for example, if happens to be a path of length . Finally, for the edges , the weight . Putting these facts together, the third point of the lemma follows. ∎

Consider the set of augmentations from Lemma 4. For augmentation , we have:

Here, the second transition holds because for all and , and the third transition holds due to and . Therefore, we have:

where the second transition follows from Lemma 4, and the third transition follows from Equation (10). Since , there exists an augmentation with , which violates local optimality of . This completes the proof of Theorem 2.

Lower Bound.

We give a stronger lower bound for matchings than the lower bound for matroids in Lemma 2.

Lemma 5.

A -core outcome is not guaranteed to exist for matching constraints, for any and .

Proof.

This example shows that Consider the graph (the complete bipartite graph with two vertices on each side). This graph has four edges, and two disjoint perfect matchings.

Let there be two agents. Agent has unit utility for the edges of one matching, while agent has unit utility for the edges of the other matching. Any integer outcome gives zero utility to one of these agents. This agent can deviate and obtain utility . Hence, for an outcome to be a -core outcome, we need , which is impossible for any and . ∎

5 General Packing Constraints

In this section, we study approximation to the core under general packing constraints of the form . Recall that there are elements, is the maximum possible utility that agent can receive from a feasible outcome, and . We prove a statement slightly more general than Theorem 3. We first need the following concept.

5.1 Maximal Proportionally Fair Outcome

Given an instance of public goods allocation subject to packing constraints, we define the notions of an -proportionally fair (-PF) outcome, a maximally proportionally fair (MPF) outcome, and the MPF value of the instance.

Definition 5 (MPF Outcome).

For , we say that a fractional outcome is -proportionally fair (-PF) if it satisfies:

The maximally proportionally fair (MPF) value of an instance is the least value such that there exists an -PF outcome. For simplicity, we say that an -PF outcome is a maximally proportionally fair (MPF) outcome.

This concept is crucial to stating and deriving our approximation results. In words, an -PF outcome gives each agent an fraction of its maximum possible utility (which can be thought of as the fair share guarantee of the agent), if the agent is given unit of utility for free. Thus, a smaller value of indicates a better solution. The MPF value denotes the best possible guarantee. The additive in Def. 5 can be replaced by any positive constant; we choose for simplicity.

We now show an upper bound for that holds for all instances. Recall from Equation (1) that is the width of the instance.

Lemma 6.

, and an MPF outcome is computable in polynomial time.

Proof.

To show that is well-defined, note that for , an -PF outcome simply requires , which is trivially achieved by every outcome. Therefore, is well-defined, and . Next, follows from the fact that there exist fractional outcomes satisfying proportionality (e.g., the outcome obtained by taking the uniform convex combination of the outcomes that give optimal to each individual agent). Finally, to show , consider the outcome in which for each element . Clearly, for all . Further, is satisfied trivially due to the fact that is the width of the packing constraints.

To compute the value of as well as an MPF outcome, we first note that the value of for each agent can be computed by solving a separate LP. Then, we consider the following LP:

(11)

Here, are the packing constraints of the instance, and is a variable representing . Thus, maximizing minimizes , which yields an MPF outcome. This can be accomplished by solving linear programs, which can be done in polynomial time. ∎

Our main result in this section uses any -PF outcome, and provides a guarantee in terms of . Thus, we do not need to necessarily compute an exact MPF outcome. We note that an MPF outcome can be very different from a core outcome. Yet, an MPF outcome gives each agent a large fraction of its maximum possible utility, subject to a small additive relaxation. As we show below, this helps us find integral outcomes that provide good approximations of the core.

5.2 Result and Proof Idea

Our main result for this section (Theorem 3) can be stated in a refined way as follows. Recall that