We study fair allocation of indivisible items, where the items are furnished with a set of conflicts, and agents are not permitted to receive conflicting items. This kind of constraint captures, for example, participating in events that overlap in time, or taking on roles in the presence of conflicting interests. We demonstrate, both theoretically and experimentally, that fairness characterizations such as EF1, MMS and MNW still are applicable and useful under item conflicts. Among other existence, non-existence and computability results, we show that a $1/Δ$-approximate MMS allocation for $n$ agents may be found in polynomial time when $n>Δ>2$, for any conflict graph with maximum degree $Δ$, and that, if $n > Δ$, a 1/3-approximate MMS allocation always exists.
We study a fair allocation problem of indivisible items under additive externalities in which each agent also receives values from items that are assigned to other agents. We propose several new fairness concepts. We extend the well-studied envy-freeness up to one item (EF1) and envy-freeness up to any item (EFX) to this setting, and we propose a new fairness concept called general fair share (GFS). We undertake a detailed study and present algorithms for finding fair allocations.
In this paper, we study the classic problem of fairly allocating indivisible items with the extra feature that the items lie on a line. Our goal is to find a fair allocation that is contiguous, meaning that the bundle of each agent forms a contiguous block on the line. While allocations satisfying the classical fairness notions of proportionality, envy-freeness, and equitability are not guaranteed to exist even without the contiguity requirement, we show the existence of contiguous allocations satisfying approximate versions of these notions that do not degrade as the number of agents or items…
Incorporating fairness criteria in optimization problems comes at a certain cost, which is measured by the so-called price of fairness. Here we consider the allocation of indivisible goods. For envy-freeness as fairness criterion it is known from literature that the price of fairness can increase linearly in terms of the number of agents. For the constructive lower bound a quadratic number of items was used. In practice this might be inadequately large. So we introduce the price of fairness in terms of both the number of agents and items, i.e., key parameters which generally may be considered …
The theory of algorithmic fair allocation is within the center of multi-agent systems and economics in the last decade due to its industrial and social importance. At a high level, the problem is to assign a set of items that are either goods or chores to a set of agents so that every agent is happy with what she obtains. Particularly, in this survey, we focus on indivisible items, for which absolute fairness such as envy-freeness and proportionality cannot be guaranteed. One main theme in the recent research agenda is about designing algorithms that approximately achieve the fairness criteria…
We study the problem of dynamically allocating $T$ indivisible items to $n$ agents with the restriction that the allocation is fair all the time. Due to the negative results to achieve fairness when allocations are irrevocable, we allow adjustments to make fairness attainable with the objective to minimize the number of adjustments. For restricted additive or general identical valuations, we show that envy-freeness up to one item (EF1) can be achieved with no adjustments. For additive valuations, we give an EF1 algorithm that requires $O(mT)$ adjustments, improving the previous result of $O(nm…
We study the fair division of items to agents supposing that agents can form groups. We thus give natural generalizations of popular concepts such as envy-freeness and Pareto efficiency to groups of fixed sizes. Group envy-freeness requires that no group envies another group. Group Pareto efficiency requires that no group can be made better off without another group be made worse off. We study these new group properties from an axiomatic viewpoint. We thus propose new fairness taxonomies that generalize existing taxonomies. We further study near versions of these group properties as allocation…
We consider the problem of fairly and efficiently allocating indivisible items (goods or bads) under capacity constraints. In this setting, we are given a set of categorized items. Each category has a capacity constraint (the same for all agents), that is an upper bound on the number of items an agent can receive from each category. Our main result is a polynomial-time algorithm that solves the problem for two agents with additive utilities over the items. When each category contains items that are all goods (positively evaluated) or all chores (negatively evaluated) for each of the agents, ou…
In this work, we revisit the problem of fairly allocating a number of indivisible items that are located on a line to multiple agents. A feasible allocation requires that the allocated items to each agent are connected on the line. The items can be goods on which agents have non-negative utilities, or chores on which the utilities are non-positive. Our objective is to understand the extent to which welfare is inevitably sacrificed by enforcing the allocations to be fair, i.e., price of fairness (PoF). We study both egalitarian and utilitarian welfare. Previous works by Suksompong [Discret. App…
We investigate the tradeoffs between fairness and efficiency when allocating indivisible items over time. Suppose T items arrive over time and must be allocated upon arrival, immediately and irrevocably, to one of n agents. Agent i assigns a value v_{it} in [0,1] to the t-th item to arrive and has an additive valuation function. If the values are chosen by an adaptive adversary it is known that the algorithm that minimizes maximum pairwise envy simply allocates each item uniformly at random; the maximum pairwise envy is then sublinear in T. If the values are independently and identically drawn…
We consider the problem of approximate maximin share (MMS) allocation of indivisible items among three agents with additive valuation functions. For goods, we show that an $\frac{11}{12}$ - MMS allocation always exists, improving over the previously known bound of $\frac{8}{9}$ . Moreover, in our allocation, we can prespecify an agent that is to receive her full proportional share (PS); we also present examples showing that for such allocations the ratio of $\frac{11}{12}$ is best possible. For chores, we show that a $\frac{19}{18}$-MMS allocation always exists. Also in this case, we can presp…
We consider a novel setting where a set of items are matched to the same set of agents repeatedly over multiple rounds. Each agent gets exactly one item per round, which brings interesting challenges to finding efficient and/or fair {\em repeated matchings}. A particular feature of our model is that the value of an agent for an item in some round depends on the number of rounds in which the item has been used by the agent in the past. We present a set of positive and negative results about the efficiency and fairness of repeated matchings. For example, when items are goods, a variation of the …
We consider the problem of fairly dividing a set of items. Much of the fair division literature assumes that the items are `goods' i.e., they yield positive utility for the agents. There is also some work where the items are `chores' that yield negative utility for the agents. In this paper, we consider a more general scenario where an agent may have negative or positive utility for each item. This framework captures, e.g., fair task assignment, where agents can have both positive and negative utilities for each task. We show that whereas some of the positive axiomatic and computational result…
In fair division applications, agents may have unequal entitlements reflecting their different contributions. Moreover, the contributions of agents may depend on the allocation itself. Previous fairness notions designed for agents with equal or pre-determined entitlement fail to characterize fairness in these collaborative allocation scenarios. We propose a novel fairness notion of average envy-freeness (AEF), where the envy of agents is defined on the average value of items in the bundles. Average envy-freeness provides a reasonable comparison between agents based on the items they receive an…
In the assignment problem, the goal is to assign indivisible items to agents who have ordinal preferences, efficiently and fairly, in a strategyproof manner. In practice, first-choice maximality, i.e., assigning a maximal number of agents their top items, is often identified as an important efficiency criterion and measure of agents' satisfaction. In this paper, we propose a natural and intuitive efficiency property, favoring-eagerness-for-remaining-items (FERI), which requires that each item is allocated to an agent who ranks it highest among remaining items, thereby implying first-choice max…
We consider the fair division problem of indivisible items. It is well-known that an envy-free allocation may not exist, and a relaxed version of envy-freeness, envy-freeness up to one item (EF1), has been widely considered. In an EF1 allocation, an agent may envy others' allocated shares, but only up to one item. In many applications, we may wish to specify a subset of prioritized agents where strict envy-freeness needs to be guaranteed from these agents to the remaining agents, while ensuring the whole allocation is still EF1. Prioritized agents may be those agents who are envious in a previ…
We study the problem of fairly allocating a multiset $M$ of $m$ indivisible items among $n$ agents with additive valuations. Specifically, we introduce a parameter $t$ for the number of distinct types of items and study fair allocations of multisets that contain only items of these $t$ types, under two standard notions of fairness: 1. Envy-freeness (EF): For arbitrary $n$, $t$, we show that a complete EF allocation exists when at least one agent has a unique valuation and the number of items of each type exceeds a particular finite threshold. We give explicit upper and lower bounds on this thr…
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 fair…
We consider fair allocation of indivisible items under an additional constraint: there is an undirected graph describing the relationship between the items, and each agent's share must form a connected subgraph of this graph. This framework captures, e.g., fair allocation of land plots, where the graph describes the accessibility relation among the plots. We focus on agents that have additive utilities for the items, and consider several common fair division solution concepts, such as proportionality, envy-freeness and maximin share guarantee. While finding good allocations according to these …
We study a new but simple model for online fair division in which indivisible items arrive one-by-one and agents have monotone utilities over bundles of the items. We consider axiomatic properties of mechanisms for this model such as strategy-proofness, envy-freeness, and Pareto efficiency. We prove a number of impossibility results that justify why we consider relaxations of the properties, as well as why we consider restricted preference domains on which good axiomatic properties can be achieved. We propose two mechanisms that have good axiomatic fairness properties on restricted but common …
Tight Asymptotic Bounds for Fair Division With Externalities
McGill University（麦吉尔大学）；Indian Institute of Technology Bombay（印度理工学院孟买分校）；AGH University of Krakow（克拉科夫AGH科技大学）；Czech Technical University in Prague（捷克布拉格工业大学）
We consider item allocation to individual agents who have additive valuations, in settings in which there are protected groups, and the allocation needs to give each protected group its "fair" share of the total welfare. Informally, within each protected group we consider the total welfare that the allocation gives the members of the group, and compare it to the maximum possible welfare that an allocation can give to the group members. An allocation is fair towards the group if the ratio between these two values is no worse then the relative size of the group. For divisible items, our formal d…
Several fairness concepts have been proposed recently in attempts to approximate envy-freeness in settings with indivisible goods. Among them, the concept of envy-freeness up to any item (EFX) is arguably the closest to envy-freeness. Unfortunately, EFX allocations are not known to exist except in a few special cases. We make significant progress in this direction. We show that for every instance with additive valuations, there is an EFX allocation of a subset of items with a Nash welfare that is at least half of the maximum possible Nash welfare for the original set of items. That is, after d…
One of the important yet insufficiently studied subjects in fair allocation is the externality effect among agents. For a resource allocation problem, externalities imply that a bundle allocated to an agent may affect the utilities of other agents. In this paper, we conduct a study of fair allocation of indivisible goods when the externalities are not negligible. We present a simple and natural model, namely \emph{network externalities}, to capture the externalities. To evaluate fairness in the network externalities model, we generalize the idea behind the notion of maximin-share ($\MMS$) to a…
We consider a fair division setting where indivisible items are allocated to agents. Each agent in the setting has strictly negative, zero or strictly positive utility for each item. We, thus, make a distinction between items that are good for some agents and bad for other agents (i.e. mixed), good for everyone (i.e. goods) or bad for everyone (i.e. bads). For this model, we study axiomatic concepts of allocations such as jealousy-freeness up to one item, envy-freeness up to one item and Pareto-optimality. We obtain many new possibility and impossibility results in regard to combinations of th…
We consider the problem of fair allocation of indivisible items among $n$ agents with additive valuations, when agents have equal entitlements to the goods, and there are no transfers. Best-of-Both-Worlds (BoBW) fairness mechanisms aim to give all agents both an ex-ante guarantee (such as getting the proportional share in expectation) and an ex-post guarantee. Prior BoBW results have focused on ex-post guarantees that are based on the "up to one item" paradigm, such as envy-free up to one item (EF1). In this work we attempt to give every agent a high value ex-post, and specifically, a constant…
In the classical version of online bipartite matching, there is a given set of offline vertices (aka agents) and another set of vertices (aka items) that arrive online. When each item arrives, its incident edges -- the agents who like the item -- are revealed and the algorithm must irrevocably match the item to such agents. We initiate the study of class fairness in this setting, where agents are partitioned into a set of classes and the matching is required to be fair with respect to the classes. We adopt popular fairness notions from the fair division literature such as envy-freeness (up to …
We study the following multiagent variant of the knapsack problem. We are given a set of items, a set of voters, and a value of the budget; each item is endowed with a cost and each voter assigns to each item a certain value. The goal is to select a subset of items with the total cost not exceeding the budget, in a way that is consistent with the voters' preferences. Since the preferences of the voters over the items can vary significantly, we need a way of aggregating these preferences, in order to select the socially best valid knapsack. We study three approaches to aggregating voters' prefe…
University of Oxford（牛津大学）；Alan Turing Institute（阿兰·图灵研究所）；City University of Hong Kong（香港城市大学）；University of Edinburgh（爱丁堡大学）；Agency for Science, Technology and Research（科技研究局）
