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 an hereditary set system. We present a simple algorithm that allocates each agent a bundle of items whose value is at least $0.3667$ 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.Comment: 22 pages, 1 figure, full version of WINE 2018 submissio

Approximate Maximin Shares for Groups of Agents

We investigate the problem of fairly allocating indivisible goods among interested agents using the concept of maximin share. Procaccia and Wang showed that while an allocation that gives every agent at least her maximin share does not necessarily exist, one that gives every agent at least $2/3$ of her share always does. In this paper, we consider the more general setting where we allocate the goods to groups of agents. The agents in each group share the same set of goods even though they may have conflicting preferences. For two groups, we characterize the cardinality of the groups for which a constant factor approximation of the maximin share is possible regardless of the number of goods. We also show settings where an approximation is possible or impossible when there are several groups.Comment: To appear in the 10th International Symposium on Algorithmic Game Theory (SAGT), 201

Simplification and Improvement of MMS Approximation

We consider the problem of fairly allocating a set of indivisible goods among $n$ agents with additive valuations, using the popular fairness notion of maximin share (MMS). Since MMS allocations do not always exist, a series of works provided existence and algorithms for approximate MMS allocations. The current best approximation factor, for which the existence is known, is $(\frac{3}{4} + \frac{1}{12n})$ [Garg and Taki, 2021]. Most of these results are based on complicated analyses, especially those providing better than $2/3$ factor. Moreover, since no tight example is known of the Garg-Taki algorithm, it is unclear if this is the best factor of this approach. In this paper, we significantly simplify the analysis of this algorithm and also improve the existence guarantee to a factor of $(\frac{3}{4} + \min(\frac{1}{36}, \frac{3}{16n-4}))$. For small $n$, this provides a noticeable improvement. Furthermore, we present a tight example of this algorithm, showing that this may be the best factor one can hope for with the current techniques