Efficiency, Sequenceability and Deal-Optimality in Fair Division of Indivisible Goods

National audienceIn fair division of indivisible goods, using sequences of sincere choices (or picking sequences) is a natural way to allocate the objects. The idea is as follows: at each stage, a designated agent picks one object among those that remain. Another intuitive way to obtain an allocation is to give objects to agents in the rst place, and to let agents exchange them as long as such "deals" are bene cial. This paper investigates these notions, when agents have additive preferences over objects, and unveils surprising connections between them, and with other e ciency and fairness notions. In particular, we show that an allocation is sequenceable if and only if it is optimal for a certain type of deals, namely cycle deals involving a single object. Furthermore, any Pareto-optimal allocation is sequenceable, but not the converse. Regarding fairness, we show that an allocation can be envy-free and non-sequenceable, but that every competitive equilibrium with equal incomes is sequenceable. To complete the picture, we show how some domain restrictions may a ect the relations between these notions. Finally, we experimentally explore the links between the scales of e ciency and fairness

Approximating Maximin Share Allocations

We study the problem of fair allocation of M indivisible items among N agents using the popular notion of maximin share as our measure of fairness. The maximin share of an agent is the largest value she can guarantee herself if she is allowed to choose a partition of the items into N bundles (one for each agent), on the condition that she receives her least preferred bundle. A maximin share allocation provides each agent a bundle worth at least their maximin share. While it is known that such an allocation need not exist [Procaccia and Wang, 2014; Kurokawa et al., 2016], a series of work [Procaccia and Wang, 2014; David Kurokawa et al., 2018; Amanatidis et al., 2017; Barman and Krishna Murthy, 2017] provided 2/3 approximation algorithms in which each agent receives a bundle worth at least 2/3 times their maximin share. Recently, [Ghodsi et al., 2018] improved the approximation guarantee to 3/4. Prior works utilize intricate algorithms, with an exception of [Barman and Krishna Murthy, 2017] which is a simple greedy solution but relies on sophisticated analysis techniques. In this paper, we propose an alternative 2/3 maximin share approximation which offers both a simple algorithm and straightforward analysis. In contrast to other algorithms, our approach allows for a simple and intuitive understanding of why it works

Efficiency and Sequenceability in Fair Division of Indivisible Goods with Additive Preferences

International audienceIn fair division of indivisible goods, using sequences of sincere choices (or picking sequences) is a natural way to allocate the objects. The idea is the following: at each stage, a designated agent picks one object among those that remain. This paper, restricted to the case where the agents have numerical additive preferences over objects, revisits to some extent the seminal paper by Brams and King [9] which was specific to ordinal and linear order preferences over items. We point out similarities and differences with this latter context. In particular, we show that any Pareto-optimal allocation (under additive preferences) is sequenceable, but that the converse is not true anymore. This asymmetry leads naturally to the definition of a " scale of efficiency " having three steps: Pareto-optimality, sequenceability without Pareto-optimality, and non-sequenceability. Finally, we investigate the links between these efficiency properties and the " scale of fairness " we have described in an earlier work [7]: we first show that an allocation can be envy-free and non-sequenceable, but that every competitive equilibrium with equal incomes is sequenceable. Then we experimentally explore the links between the scales of efficiency and fairness