Fair assignment of indivisible objects under ordinal preferences

We consider the discrete assignment problem in which agents express ordinal preferences over objects and these objects are allocated to the agents in a fair manner. We use the stochastic dominance relation between fractional or randomized allocations to systematically define varying notions of proportionality and envy-freeness for discrete assignments. The computational complexity of checking whether a fair assignment exists is studied for these fairness notions. We also characterize the conditions under which a fair assignment is guaranteed to exist. For a number of fairness concepts, polynomial-time algorithms are presented to check whether a fair assignment exists. Our algorithmic results also extend to the case of unequal entitlements of agents. Our NP-hardness result, which holds for several variants of envy-freeness, answers an open question posed by Bouveret, Endriss, and Lang (ECAI 2010). We also propose fairness concepts that always suggest a non-empty set of assignments with meaningful fairness properties. Among these concepts, optimal proportionality and optimal weak proportionality appear to be desirable fairness concepts.Comment: extended version of a paper presented at AAMAS 201

An EF2X Allocation Protocol for Restricted Additive Valuations

We study the problem of fairly allocating a set of $m$ indivisible goods to aset of $n$ agents. Envy-freeness up to any good (EFX) criteria -- whichrequires that no agent prefers the bundle of another agent after removal of anysingle good -- is known to be a remarkable analogous of envy-freeness when theresource is a set of indivisible goods. In this paper, we investigate EFXnotion for the restricted additive valuations, that is, every good has somenon-negative value, and every agent is interested in only some of the goods. We introduce a natural relaxation of EFX called EFkX which requires that noagent envies another agent after removal of any $k$ goods. Our maincontribution is an algorithm that finds a complete (i.e., no good is discarded)EF2X allocation for the restricted additive valuations. In our algorithm wedevise new concepts, namely "configuration" and "envy-elimination" that mightbe of independent interest. We also use our new tools to find an EFX allocation for restricted additivevaluations that discards at most $\lfloor n/2 \rfloor -1$ goods. This improvesthe state of the art for the restricted additive valuations by a factor of $2$.<br

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