Indivisible Items Research Articles

An improved approximation algorithm for maximin shares

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 [1,3–6] 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 (34−ϵ)-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.

Approximating nash social welfare under rado valuations

The Nash social welfare problem asks for an allocation of indivisible items to agents in order to maximize the geometric mean of agents' valuations. We give an overview of the constant-factor approximation algorithm for the problem when agents have Rado valuations [Garg et al. 2021]. Rado valuations are a common generalization of the assignment (OXS) valuations and weighted matroid rank functions. Our approach also gives the first constant-factor approximation algorithm for the asymmetric Nash social welfare problem under the same valuations, provided that the maximum ratio between the weights is bounded by a constant.

Fair and Truthful Mechanisms for Dichotomous Valuations

We consider the problem of allocating a set on indivisible items to players with private preferences in an efficient and fair way. We focus on valuations that have dichotomous marginals, in which the added value of any item to a set is either 0 or 1, and aim to design truthful allocation mechanisms (without money) that maximize welfare and are fair. For the case that players have submodular valuations with dichotomous marginals, we design such a deterministic truthful allocation mechanism. The allocation output by our mechanism is Lorenz dominating, and consequently satisfies many desired fairness properties, such as being envy-free up to any item (EFX), and maximizing the Nash Social Welfare (NSW). We then show that our mechanism with random priorities is envy-free ex-ante, while having all the above properties ex-post. Furthermore, we present several impossibility results precluding similar results for the larger class of XOS valuations.

Almost Envy-freeness, Envy-rank, and Nash Social Welfare Matchings

Envy-freeness up to one good (EF1) and envy-freeness up to any good (EFX) are two well-known extensions of envy-freeness for the case of indivisible items. It is shown that EF1 can always be guaranteed for agents with subadditive valuations. In sharp contrast, it is unknown whether or not an EFX allocation always exists, even for four agents and additive valuations. In addition, the best approximation guarantee for EFX is (φ − 1) ≃ 0.61 by Amanitidis et al.. In order to find a middle ground to bridge this gap, in this paper we suggest another fairness criterion, namely envy-freeness up to a random good or EFR, which is weaker than EFX, yet stronger than EF1. For this notion, we provide a polynomial-time 0.73-approximation allocation algorithm. For our algorithm we introduce Nash Social Welfare Matching which makes a connection between Nash Social Welfare and envy freeness.

On the PTAS for Maximin Shares in an Indivisible Mixed Manna

We study fair allocation of indivisible items, both goods and chores, under the popular fairness notion of maximin share (MMS). The problem is well-studied when there are only goods (or chores), where a PTAS to compute the MMS values of agents is well-known. In contrast, for the mixed manna, a recent result showed that finding even an approximate MMS value of an agent up to any approximation factor in (0,1] is NP-hard for general instances. In this paper, we complement the hardness result by obtaining a PTAS to compute the MMS value when its absolute value is at least 1/p times either the total value of all the goods or total cost of all the chores, for some constant p valued at least 1.

Probabilistic serial mechanism for multi-type resource allocation

In multi-type resource allocation (MTRA) problems, there are $$d\ge 2$$ types of items, and n agents who each demand one unit of items of each type and have strict linear preferences over bundles consisting of one item of each type. For MTRAs with indivisible items, our first result is an impossibility theorem that is in direct contrast to the single type ( $$d=1$$ ) setting: no mechanism, the output of which is always decomposable into a probability distribution over discrete assignments (where no item is split between agents), can satisfy both sd-efficiency and sd-envy-freeness. We show that this impossibility result is circumvented under the natural assumption of lexicographic preferences. We provide lexicographic probabilistic serial (LexiPS) as an extension of the probabilistic serial (PS) mechanism for MTRAs with lexicographic preferences, and prove that LexiPS satisfies sd-efficiency and sd-envy-freeness, retaining the desirable properties of PS. Moreover, LexiPS satisfies sd-weak-strategyproofness when agents are not allowed to misreport their importance orders. For MTRAs with divisible items, we show that the existing multi-type probabilistic serial (MPS) mechanism satisfies the stronger efficiency notion of lexi-efficiency, and is sd-envy-free under strict linear preferences and sd-weak-strategyproof under lexicographic preferences. We also prove that MPS can be characterized both by leximin-optimality and by item-wise ordinal fairness, and the family of eating algorithms which MPS belongs to can be characterized by lexi-efficiency.

Barter markets, indivisibilities, and Markovian core

AbstractWe study a general barter market where every agent is endowed with several heterogeneous indivisible items and wishes to exchange. There is no medium of exchange like money. Agents have general preferences over their interested bundles of items and may acquire several items. We propose a practical and sensible solution called a Markovian core, generalizing the classical notion of the core. A Markovian core allocation is individually rational, Pareto‐efficient, and stable against any coalition deviation by comparison with their current assignments instead of their initial endowments and is shown to be a natural outcome of a decentralized market process.

Allocating contiguous blocks of indivisible chores fairly

In the cake cutting problem, a cake needs to be divided among players with possibly different valuations for different pieces of the cake. We consider a variation where the cake represents chores that need to be allocated. We assume the players to be work-averse, i.e., intending to get as little of the cake as possible. We further assume that the chores are indivisible and that they are ordered along a line. We seek a fair allocation of the chores.We study the loss in social welfare due to fairness using the price of fairness concept. Previous work has considered fair division of cakes and chores with non-connected and connected pieces as well as fair division of indivisible contiguous blocks of indivisible items (with positive utility) on a line. We complete the picture by providing price of fairness results for the allocation of contiguous blocks of indivisible chores.

The Fair Division of Hereditary Set Systems

We consider the fair division of indivisible items using the maximin shares measure. Recent work on the topic has focused on extending results beyond the class of additive valuation functions. In this spirit, we study the case where the items form a hereditary set system. We present a simple algorithm that allocates each agent a bundle of items whose value is at least 0.3666 times the maximin share of the agent. This improves upon the current best known guarantee of 0.2 due to Ghodsi et al. The analysis of the algorithm is almost tight; we present an instance where the algorithm provides a guarantee of at most 0.3738. We also show that the algorithm can be implemented in polynomial time given a valuation oracle for each agent.

Two-Person Fair Division of Indivisible Items when Envy-Freeness is Impossible

Assume two players, A and B, must divide a set of indivisible items that each strictly ranks from best to worst. If the number of items is even, assume that the players desire that the allocations be balanced (each player gets half the items), item-wise envy-free (EF), and Pareto-optimal (PO). Meeting this ideal is frequently impossible. If so, we find a balanced maximal partial allocation of items to the players that is EF, though it may not be PO. Then we show how to augment it in a way that makes it a complete allocation that is EF for one player (say, A) and almost-EF for the other player (B) in the sense that it is EF for B except for one item – it would be EF for B if a specific item assigned to A were removed. Moreover, we show how low-ranked that exceptional item can be for B, thereby finding an almost-EF allocation that is as close as possible to EF – as well as complete, balanced, and PO. We provide algorithms to find such almost-EF allocations, adapted from algorithms that apply when complete balanced EF-PO allocations are possible.

Two-Person Fair Division of Indivisible Items: Compatibilities and Incompatibilities

Suppose two players wish to divide a finite set of indivisible items, over which each distributes a specified number of points. Assuming the utility of a player’s bundle is the sum of the points it assigns to the items it contains, we analyze what divisions are fair. We show that if there is an envy-free (EF) allocation of the items, two other desirable properties — Pareto-optimality (PO) and maximinality (MM) — can also be satisfied, rendering these three properties compatible, but other properties — balance (BL), maximum Nash welfare (MNW), maximum total welfare (MTW), and lexicographic optimality (LO) — may fail. If there is no EF division, as is likely, it is always possible to satisfy EFx, a weaker form of EF, but an EFx allocation may not be PO, BL, MNW, MTW, or LO. Moreover, if one player considers an item worthless (i.e., assigns zero points to it), an EFx division may be Pareto dominated by a nonEFx allocation that is MNW. Although these incompatibilities suggest that there is no “perfect” 2-person fair division of indivisible items, EFx and MNW divisions — if they give different allocations when there is no EF-PO-MM division — seem the most compelling alternatives, with EFx, we conjecture, satisfying the Rawlsian objective of helping the worse-off player and MNW, a modification of MTW, suggesting a more Benthamite view.

Contiguous Cake Cutting: Hardness Results and Approximation Algorithms

We study the fair allocation of a cake, which serves as a metaphor for a divisible resource, under the requirement that each agent should receive a contiguous piece of the cake. While it is known that no finite envy-free algorithm exists in this setting, we exhibit efficient algorithms that produce allocations with low envy among the agents. We then establish NP-hardness results for various decision problems on the existence of envy-free allocations, such as when we fix the ordering of the agents or constrain the positions of certain cuts. In addition, we consider a discretized setting where indivisible items lie on a line and show a number of hardness results extending and strengthening those from prior work. Finally, we investigate connections between approximate and exact envy-freeness, as well as between continuous and discrete cake cutting.

Manipulations of Equivalent Preference in Parallel Allocation Mechanism

The parallel distribution is a non-centralized mechanism for distributing indivisible items to agents, which can take into account computing efficiency, economic benefits and social equality. However, as most decentralized allocation mechanism, the parallel protocol is not strategy-proof. In this paper, supposed the manipulator has additive preferences with possible indifferences between single objects, we study the most basic manipulation problem under the parallel allocation mechanism. For any given set of items, we proved that the agent1 can determine whether all objects in the set can be guaranteed in polynomial time. In addition, we given an algorithm for finding pessimism and proved the correctness, completeness and time complexity of polynomial.

Computing the Subgame Perfect Nash Equilibriums in Parallel Allocation Indivisible Items

Multi-Agent resource allocation has also become a research hotspot in the field of artificial intelligence in recent years. A lot of related work aims to design a procedural resource allocation system with execution efficiency. This paper Studied a parallel allocation method for allocating indivisible items between agents in which every agent asks for an object among those that re-main in every round, and every reported item will be allocated randomly to an agent reporting it. In order to depicts the entire distribution process, the authors propose a method of constructing a game tree of parallel allocation. Study the corresponding relationship between the total number of branches of the game tree and items. In the last part, prof the parallel allocation mechanism is dynamic game and show that the subgame perfect Nash equilibrium (SPNE) can be found by backward introduction the game trees. Provides a method to compute the unique subgame perfect Nash equilibrium in linear time.

A polynomial-time algorithm for computing a Pareto optimal and almost proportional allocation

We consider fair allocation of indivisible items under additive utilities. We show that there exists a strongly polynomial-time algorithm that always computes an allocation satisfying Pareto optimality and proportionality up to one item even if the utilities are mixed and the agents have asymmetric weights. The result does not hold if either of Pareto optimality or PROP1 is replaced with slightly stronger concepts.

On the number of almost envy-free allocations

Envy-freeness is a standard benchmark of fairness in resource allocation. Since it cannot always be satisfied when the resource consists of indivisible items even when there are two agents, the relaxations envy-freeness up to one item (EF1) and envy-freeness up to any item (EFX) are often considered. We establish tight lower bounds on the number of allocations satisfying each of these benchmarks in the case of two agents. In particular, while there can be as few as two EFX allocations for any number of items, the number of EF1 allocations is always exponential in the number of items. Our results apply a version of the vertex isoperimetric inequality on the hypercube and help explain the large gap in terms of robustness between the two notions.

Contiguous Cake Cutting: Hardness Results and Approximation Algorithms

We study the fair allocation of a cake, which serves as a metaphor for a divisible resource, under the requirement that each agent should receive a contiguous piece of the cake. While it is known that no finite envy-free algorithm exists in this setting, we exhibit efficient algorithms that produce allocations with low envy among the agents. We then establish NP-hardness results for various decision problems on the existence of envy-free allocations, such as when we fix the ordering of the agents or constrain the positions of certain cuts. In addition, we consider a discretized setting where indivisible items lie on a line and show a number of hardness results strengthening those from prior work.

Algorithms for Manipulating Sequential Allocation

Sequential allocation is a simple and widely studied mechanism to allocate indivisible items in turns to agents according to a pre-specified picking sequence of agents. At each turn, the current agent in the picking sequence picks its most preferred item among all items having not been allocated yet. This problem is well-known to be not strategyproof, i.e., an agent may get more utility by reporting an untruthful preference ranking of items. It arises the problem: how to find the best response of an agent? It is known that this problem is polynomially solvable for only two agents and NP-complete for an arbitrary number of agents. The computational complexity of this problem with three agents was left as an open problem. In this paper, we give a novel algorithm that solves the problem in polynomial time for each fixed number of agents. We also show that an agent can always get at least half of its optimal utility by simply using its truthful preference as the response.

Approximate Random Allocation Mechanisms

Abstract We generalize the scope of random allocation mechanisms, in which the mechanism first identifies a feasible “expected allocation” and then implements it by randomizing over nearby feasible integer allocations. The previous literature has shown that the cases in which this is possible are sharply limited. We show that if some of the feasibility constraints can be treated as goals rather than hard constraints, then, subject to weak conditions that we identify, any expected allocation that satisfies all the constraints and goals can be implemented by randomizing among nearby integer allocations that satisfy all the hard constraints exactly and the goals approximately. By defining ex post utilities as goals, we are able to improve the ex post properties of several classic assignment mechanisms, such as the random serial dictatorship. We use the same approach to prove the existence of $\epsilon$-competitive equilibrium in large markets with indivisible items and feasibility constraints.

When Do Envy-Free Allocations Exist?

We consider a fair division setting in which $m$ indivisible items are to be allocated among $n$ agents, where the agents have additive utilities and the agents' utilities for individual items are independently sampled from a distribution. Previous work has shown that an envy-free allocation is likely to exist when $m=\Omega(n\log n)$ but not when $m=n+o(n)$, and left open the question of determining where the phase transition from non-existence to existence occurs. We show that, surprisingly, there is in fact no universal point of transition---instead, the transition is governed by the divisibility relation between $m$ and $n$. On the one hand, if $m$ is divisible by $n$, an envy-free allocation exists with high probability as long as $m\geq 2n$. On the other hand, if $m$ is not "almost" divisible by $n$, an envy-free allocation is unlikely to exist even when $m=\Theta(n\log n/\log\log n)$.