Elsevier

Artificial Intelligence

Volume 300, November 2021, 103547
Artificial Intelligence

An improved approximation algorithm for maximin shares,☆☆

https://doi.org/10.1016/j.artint.2021.103547Get rights and content

Abstract

Fair division is a fundamental problem in various multi-agent settings, where the goal is to divide a set of resources among agents in a fair manner. We study the case where m indivisible items need to be divided among n agents with additive valuations using the popular fairness notion of maximin share (MMS). An MMS allocation provides each agent a bundle worth at least her maximin share. While it is known that such an allocation need not exist [1], [2], a series of remarkable work [1], [3], [4], [5], [6] provided approximation algorithms for a 23-MMS allocation in which each agent receives a bundle worth at least 23 times her maximin share. More recently, Ghodsi et al. [7] showed the existence of a 34-MMS allocation and a PTAS to find a (34ϵ)-MMS allocation for an ϵ>0. Most of the previous works utilize intricate algorithms and require agents' approximate MMS values, which are computationally expensive to obtain.

In this paper, we develop a new approach that gives a simple algorithm for showing the existence of a 34-MMS allocation. Furthermore, our approach is powerful enough to be easily extended in two directions: First, we get a strongly polynomial time algorithm to find a 34-MMS allocation, where we do not need to approximate the MMS values at all. Second, we show that there always exists a (34+112n)-MMS allocation, improving the best previous factor. This improves the approximation guarantee, most notably for small n. We note that 34 was the best factor known for n>4.

Introduction

Fair division is a fundamental problem in various multi-agent settings, where the goal is to divide a set of resources among agents in a fair manner. It has been a subject of intense study since the seminal work of Steinhaus [9] where he introduced the cake-cutting problem for n>2 agents: Given a heterogeneous (divisible) cake and a set of agents with different valuation functions, the problem is to find a fair allocation. The two most well-studied notions of fairness are: 1) Envy-freeness, where each agent prefers her own share of cake over any other agents' share, and 2) Proportionality, where each agent receives a share that is worth at least 1/n of her value for the entire cake.

We study the discrete fair division problem where m indivisible items need to be divided among n agents with additive valuations. For this setting, no algorithm can provide either envy-freeness or proportionality, in general, e.g., consider allocating a single item among n>1 agents. This necessitated an alternate concept of fairness. Budish [10] introduced an intriguing option called maximin share, which has attracted a lot of attention [1], [2], [7], [6], [3], [5], [11], [4]. The idea is a straightforward generalization of the popular cut and choose protocol in the cake-cutting problem and a natural relaxation of proportionality. Suppose we ask an agent i to partition the items into n bundles (one for each agent), with the condition that the other n1 agents get to choose a bundle before her. In the worst case, i receives the least preferred bundle. Clearly, in such a situation, the agent will choose a partition that maximizes the value of her least preferred bundle. This maximum possible value is called i's maximin share (MMS) value. In fact, when all agents have the same valuations, i cannot guarantee more than the MMS value.

Each agent's MMS value is a specific objective that gives her an intuitive measure of the fairness of an allocation. For example, Gates et al. [12] showed that in real-life experiments maximin metric is preferred by participating agents over others. This raises a natural question: Is there an allocation where each agent receives a bundle worth at least her MMS value? An allocation satisfying this property is said to be maximin share allocation (MMS allocation), and if it exists, it provides strong fairness guarantees to each individual agent. However, Procaccia and Wang [1], through a clever counter-example, showed that MMS allocation might not exist but a 23-MMS allocation always exists, i.e., an allocation where each agent receives a bundle worth at least 23 of their MMS value. Later, Ghodsi et al. [7] improved the factor by showing the existence of a 34-MMS allocation using a sophisticated technique with a very challenging analysis.

We note that these are primarily existential results that do not provide any efficient algorithm to find such an allocation. The main issue in these techniques is the need for agents' MMS values. The problem of finding the MMS value of an agent is NP-hard,1 but a polynomial-time approximation scheme (PTAS) exists [13]. Theoretically, one can use PTAS to find a (34ϵ)-MMS allocation for an ϵ>0 in polynomial time. However, for practical purposes, such algorithms are not very useful for small ϵ. Hence, finding an efficient algorithm to compute a 34-MMS allocation remained open.

In this paper, we develop a new approach that gives a simple algorithm for showing the existence of a 34-MMS allocation. Furthermore, our approach is powerful enough to be easily extended in two directions: First, we get a strongly polynomial time algorithm to find a 34-MMS allocation, where we do not need to use the PTAS in [13] to approximate the MMS values at all. Second, we show that there always exists a (34+112n)-MMS allocation, improving the best previous factor by Ghodsi et al. [7]. This improves the approximation guarantee, most notably for small n. We note that there are works, e.g., [4], [7], exploring better approximation factors for a small number of agents, and 34 was the best factor known for n>4.

Our algorithms are extremely simple. We first describe the basic algorithm, given in Section 3, that shows the existence of a 34-MMS allocation. We assume that MMS values are known for all agents. Since the MMS problem is scale-invariant (shown in Lemma 2.4), we scale valuations to make each agent's MMS value 1. Then, we assign high-value items (e.g., a single item that some agent values at least 34) to agents, who value them at least 34, with a simple greedy approach based on the pigeonhole principle. We remove the assigned items and the agents receiving these items from further consideration. This reduces the number of high-value items to be at most 2n, where n is the number of remaining agents. These greedy assignments massively simplify allocation of high-value items, which was the most challenging part of previous algorithms. Next, we prepare n bags, one for each remaining agent, and put at most two high-value items in each bag. Then, we add low-value items on top of each of these bags one by one using a bag filling procedure until the value of bag for some agent is at least 34. The main technical challenge here is to show that there are enough low-value items to give every agent a bag they value at least 34.

In Section 4, we extend the basic algorithm to compute a 34-MMS allocation in strongly polynomial time without any need to compute the actual MMS values (using the PTAS in [13]). Here, we define a notion of tentative assignments and a novel way for updating the MMS upper bound. For each agent, we use the average value, that is the value of all items divided by the number of agents, as an upper bound of her MMS value. The only change from the basic algorithm is that some of the greedy assignments are tentative, i.e., they are valid only if the current upper bound of the MMS values is tight enough. We show that this can be checked by using the total valuation of low-value items. If the upper bounds are not tight enough for some agents, then we update the MMS value of such an agent and repeat. We show that we do not need to update the MMS upper bounds more than O(n3) times before we have a good upper bound on all MMS values. Then, we show that the same bag filling procedure, as in the basic algorithm, satisfy every remaining agent. The running time of the entire algorithm is O(nm(n4+logm)).

In Section 5, we show that our basic algorithm also yields a better bound of the existence of a (34+112n)-MMS allocation. The entire algorithm remains exactly the same but with an involved analysis. The analysis is tricky in this case, so we add a set of dummy items to make proofs easier. We use these items to make up for the extra loss for the remaining agents due to the additional factor, and, of course, these items are not assigned to any agent in the algorithm.

Maximin share is a popular fairness notion of allocating indivisible items among agents. Bouveret and Lemaître [14] showed that an MMS allocation always exists in some restricted cases, e.g., when there are only two agents or if agents' valuations for items are either 0 or 1, but left the general case as an open problem. As mentioned earlier, Procaccia and Wang [1] showed that MMS allocation might not exist, but a 23-MMS allocation always exists. They also provided a polynomial time algorithm to find a 23-MMS allocation when the number of agents n is constant. For the special case of four agents, their algorithm finds a 34-MMS allocation. Amanatidis et al. [4] improved this result by addressing the requirement for a constant number of agents, obtaining a PTAS that finds a (23ϵ)-MMS allocation for an arbitrary number of agents; see [3] for an alternate proof. In [4], they also showed that a 78 MMS allocation always exists when there are three agents. This factor was later improved to 89 in [15].

Taking a different approach, Barman and Krishnamurthy [5] obtained a greedy algorithm to find a 23-MMS allocation. While their algorithm is fairly simple, the analysis is not. More recently, Garg et al. [6] obtained a simple algorithm to find a 23-MMS allocation that also has a simple analysis.

Ghodsi et al. [7] improved these results by showing the existence of a 34-MMS allocation and a PTAS to find a (34ϵ) MMS allocation.

Maximin share fairness has also been studied in many different setting, e.g., for asymmetric agents (i.e., agents with different entitlements) [11], for group fairness [16], [17], beyond additive valuations [5], [7], [18], in matroids [15], with additional constraints [15], [19], for agents with externalities [20], [21], with graph constraints [22], [23], for allocating chores [24], [5], [25], and with strategic agents [26], [27], [28], [29].

Section snippets

The MMS problem and its properties

We consider the fair allocation of a set M of m indivisible items among a set N of n agents with additive valuations, using the popular notion of maximin share (MMS) as our measure of fairness. Let vij denote agent i's value for item j, and i's valuation of any bundle SM of items is given by vi(S)=jSvij. Let V=(v1,,vn) denote the set of all valuation functions.

An agent's MMS value is defined as the maximum value she can guarantee herself if she is allowed to choose a partition of items into

Existence of 34-MMS allocation

In this section, we present a simple proof of the existence of a 34-MMS allocation for a given instance I=N,M,V. We assume that the MMS value μi of each agent i is given. Finding the exact μi is an NP-Hard problem, however a PTAS exists [13]. This implies a PTAS to compute a (34ϵ)-MMS allocation for any ϵ>0. Using the properties stated in Section 2.1, we normalize valuations so that μin(M)=1,i (Lemma 2.4) and assume that I is an ordered instance, i.e., vi1vim,i (Lemma 2.5). Our proof is

Algorithm for 34-MMS allocation

The existence proof of 34-MMS allocation in Section 3 requires the knowledge of the exact MMS value μi's of all agents, e.g., the proof of valid reduction for bundle S4 in Lemma 3.1 needs this assumption. Finding an exact μi of an agent i is an NP-Hard problem, however a PTAS exists [13]. This implies a PTAS to compute a (34ϵ)-MMS allocation for an ϵ>0. However, for small ϵ, this PTAS is computationally very expensive and may not be practical. In this section, we show that our algorithmic

Existence of (34+112n)-MMS allocation

In this section, we show that our approach in Section 3 can be extended to obtain the existence of a (34+γ)-MMS allocation for any given instance I=N,M,V where γ=112n. We note that γ is a constant for the given instance, where n:=|N|. We assume that the MMS value μi of each agent i is given. Finding an exact μi is an NP-Hard problem, however a PTAS exists [13]. This implies a PTAS to compute a (34+γϵ)-MMS allocation for any ϵ>0. Using the properties shown in Section 2.1, we normalize

Conclusions

We developed a new approach that gives a simple algorithm for showing the existence of a 34-MMS allocation. Furthermore, we showed that our approach is powerful enough to be easily extended to obtain (i) a strongly polynomial time algorithm to find a 34-MMS allocation, and (ii) the existence of a (34+112n)-MMS allocation, improving the best previous factor. Consequently, this gives a PTAS for finding a (34+112nϵ)-MMS allocation for any ϵ>0. An interesting question is to find the maximum γ for

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

We would like to thank anonymous referees for their comments and suggestions that have helped to improve the presentation of the paper. Work on this paper supported by NSF Grant CCF-1942321 (CAREER).

References (30)

  • J. Garg et al.

    An improved approximation algorithm for maximin shares

  • H. Steinhaus

    The problem of fair division

    Econometrica

    (1948)
  • E. Budish

    The combinatorial assignment problem: approximate competitive equilibrium from equal incomes

    J. Polit. Econ.

    (2011)
  • A. Farhadi et al.

    Fair allocation of indivisible goods to asymmetric agents

    J. Artif. Intell. Res.

    (2019)
  • V. Gates et al.

    How to be helpful to multiple people at once

    Cogn. Sci.

    (2020)
  • Cited by (42)

    View all citing articles on Scopus

    This paper is a participant in the 2020 ACM Conference on Economics and Computation (EC) Forward-to-Journal Program.

    ☆☆

    A two-page abstract of this work appeared in the proceedings of ACM EC'20 [8].

    View full text