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

Dividing bads under additive utilities

We compare the Egalitarian rule (aka Egalitarian Equivalent) and the Competitive rule (aka Comeptitive Equilibrium with Equal Incomes) to divide bads (chores). They are both welfarist: the competitive disutility profile(s) are the critical points of their Nash product on the set of efficient feasible profiles. The C rule is Envy Free, Maskin Monotonic, and has better incentives properties than the E rule. But, unlike the E rule, it can be wildly multivalued, admits no selection continuous in the utility and endowment parameters, and is harder to compute. Thus in the division of bads, unlike that of goods, no rule normatively dominates the other

Multi-unit assignment under dichotomous preferences

I study the problem of allocating objects among agents without using money. Agents can receive several objects and have dichotomous preferences, meaning that they either consider objects to be acceptable or not. In this set-up, the egalitarian solution is more appealing than the competitive equilibrium with equal incomes because it is Lorenz dominant, unique in utilities, and group strategy- proof. Both solutions are disjoint

Efficient Fair Division with Minimal Sharing

A collection of objects, some of which are good and some are bad, is to be divided fairly among agents with different tastes, modeled by additive utility-functions. If the objects cannot be shared, so that each of them must be entirely allocated to a single agent, then a fair division may not exist. What is the smallest number of objects that must be shared between two or more agents in order to attain a fair and efficient division? We focus on Pareto-optimal, envy-free and/or proportional allocations. We show that, for a generic instance of the problem -- all instances except of a zero-measure set of degenerate problems -- a fair Pareto-optimal division with the smallest possible number of shared objects can be found in polynomial time, assuming that the number of agents is fixed. The problem becomes computationally hard for degenerate instances, where agents' valuations are aligned for many objects.Comment: Add experiments with Spliddit.org dat