Fairly Allocating Contiguous Blocks of Indivisible Items

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 increases. We also study the efficiency loss of contiguous allocations due to fairness constraints.Comment: Appears in the 10th International Symposium on Algorithmic Game Theory (SAGT), 201

On black-box transformations in downward-closed environments

Black-box transformations have been extensively studied in algorithmic mechanism design as a generic tool for converting algorithms into truthful mechanisms without degrading the approximation guarantees. While such transformations have been designed for a variety of settings, Chawla et al. showed that no fully general black-box transformation exists for single-parameter environments. In this paper, we investigate the potentials and limits of black-box transformations in the prior-free (i.e., non-Bayesian) setting in downward-closed single-parameter environments, a large and important class of environments in mechanism design. On the positive side, we show that such a transformation can preserve a constant fraction of the welfare at every input if the private valuations of the agents take on a constant number of values that are far apart, while on the negative side, we show that this task is not possible for general private valuations

Asymptotic existence of proportionally fair allocations

Fair division has long been an important problem in the economics literature. In this note, we consider the existence of proportionally fair allocations of indivisible goods, i.e., allocations of indivisible goods in which every agent gets at least her proportionally fair share according to her own utility function. We show that when utilities are additive and utilities for individual goods are drawn independently at random from a distribution, proportionally fair allocations exist with high probability if the number of goods is a multiple of the number of agents or if the number of goods grows asymptotically faster than the number of agents

Asymptotic existence of proportionally fair allocations

Fair division has long been an important problem in the economics literature. In this note, we consider the existence of proportionally fair allocations of indivisible goods, i.e., allocations of indivisible goods in which every agent gets at least her proportionally fair share according to her own utility function. We show that when utilities are additive and utilities for individual goods are drawn independently at random from a distribution, proportionally fair allocations exist with high probability if the number of goods is a multiple of the number of agents or if the number of goods grows asymptotically faster than the number of agents

On black-box transformations in downward-closed environments

Black-box transformations have been extensively studied in algorithmic mechanism design as a generic tool for converting algorithms into truthful mechanisms without degrading the approximation guarantees. While such transformations have been designed for a variety of settings, Chawla et al. showed that no fully general black-box transformation exists for single-parameter environments. In this paper, we investigate the potentials and limits of black-box transformations in the prior-free (i.e., non-Bayesian) setting in downward-closed single-parameter environments, a large and important class of environments in mechanism design. On the positive side, we show that such a transformation can preserve a constant fraction of the welfare at every input if the private valuations of the agents take on a constant number of values that are far apart, while on the negative side, we show that this task is not possible for general private valuations

Asymptotic existence of fair divisions for groups

The problem of dividing resources fairly occurs in many practical situations and is therefore an important topic of study in economics. In this paper, we investigate envy-free divisions in the setting where there are multiple players in each interested party. While all players in a party share the same set of resources, each player has her own preferences. Under additive valuations drawn randomly from probability distributions, we show that when all groups contain an equal number of players, a welfare-maximizing allocation is likely to be envy-free if the number of items exceeds the total number of players by a logarithmic factor. On the other hand, an envy-free allocation is unlikely to exist if the number of items is less than the total number of players. In addition, we show that a simple truthful mechanism, namely the random assignment mechanism, yields an allocation that satisfies the weaker notion of approximate envy-freeness with high probability

Computing a small agreeable set of indivisible items

We study the problem of assigning a small subset of indivisible items to a group of agents so that the subset is agreeable to all agents, meaning that all agents value the subset as least as much as its complement. For an arbitrary number of agents and items, we derive a tight worst-case bound on the number of items that may need to be included in such a set. We then present polynomial-time algorithms that find an agreeable set whose size matches the worst-case bound when there are two or three agents. We also show that finding small agreeable sets is possible even when we only have access to the agents' preferences on single items. Furthermore, we investigate the problem of efficiently computing an agreeable set whose size approximates the size of the smallest agreeable set for any given instance. We consider two well-known models for representing the preferences of the agents—the value oracle model and additive utilities—and establish tight bounds on the approximation ratio that can be obtained by algorithms running in polynomial time in each of these models

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

Asymptotic existence of fair divisions for groups

The problem of dividing resources fairly occurs in many practical situations and is therefore an important topic of study in economics. In this paper, we investigate envy-free divisions in the setting where there are multiple players in each interested party. While all players in a party share the same set of resources, each player has her own preferences. Under additive valuations drawn randomly from probability distributions, we show that when all groups contain an equal number of players, a welfare-maximizing allocation is likely to be envy-free if the number of items exceeds the total number of players by a logarithmic factor. On the other hand, an envy-free allocation is unlikely to exist if the number of items is less than the total number of players. In addition, we show that a simple truthful mechanism, namely the random assignment mechanism, yields an allocation that satisfies the weaker notion of approximate envy-freeness with high probability

Corrigendum to “The impossibility of extending random dictatorship to weak preferences” [Econom. Lett. 141 (2016) 44–47]

The present note corrects an error in the proof of Theorem 1 of Brandl et al. (2016). The proof for the case of [display formula] and [display formula] and the reduction of the case [display formula] to the case [display formula] are correct. However, the reduction of the case [display formula] to the case [display formula] is incorrect, since [display formula] as constructed in this step is not necessarily an extension of [display formula]. To fix this, let [display formula] be defined whenever for each agent, the set of his most-preferred alternatives either is a singleton or contains all alternatives that are uniquely most-preferred by some other agent. Additionally we require that at least one agent has a unique most-preferred alternative. The lottery returned by [display formula] is obtained by choosing an agent with a unique most-preferred alternative uniformly at random and returning that agent's most-preferred alternative. With this definition of [display formula] the proof of Theorem 1 is correct. All other statements are unaffected. The authors thank Manuel Eberl for bringing this mistake to their attention