Chore division on a graph

The paper considers fair allocation of indivisible nondisposable items that generate disutility (chores). We assume that these items are placed in the vertices of a graph and each agent's share has to form a connected subgraph of this graph. Although a similar model has been investigated before for goods, we show that the goods and chores settings are inherently different. In particular, it is impossible to derive the solution of the chores instance from the solution of its naturally associated fair division instance. We consider three common fair division solution concepts, namely proportionality, envy-freeness and equitability, and two individual disutility aggregation functions: additive and maximum based. We show that deciding the existence of a fair allocation is hard even if the underlying graph is a path or a star. We also present some efficiently solvable special cases for these graph topologies

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

Chore division on a graph

Le PDF est une version non publiée datant de 2018.International audienceThe paper considers fair allocation of indivisible nondisposable items that generate disutility (chores). We assume that these items are placed in the vertices of a graph and each agent’s share has to form a connected subgraph of this graph. Although a similar model has been investigated before for goods, we show that the goods and chores settings are inherently different. In particular, it is impossible to derive the solution of the chores instance from the solution of its naturally associated fair division instance. We consider three common fair division solution concepts, namely proportionality, envy-freeness and equitability, and two individual disutility aggregation functions: additive and maximum based. We show that deciding the existence of a fair allocation is hard even if the underlying graph is a path or a star. We also present some efficiently solvable special cases for these graph topologies

Competitive division of a mixed manna

A mixed manna contains goods (that everyone likes) and bads (that everyone dislikes), as well as items that are goods to some agents, but bads or satiated to others. If all items are goods and utility functions are homogeneous of degree 1 and concave (and monotone), the competitive division maximizes the Nash product of utilities (Gale–Eisenberg): hence it is welfarist (determined by the set of feasible utility profiles), unique, continuous, and easy to compute. We show that the competitive division of a mixed manna is still welfarist. If the zero utility profile is Pareto dominated, the competitive profile is strictly positive and still uniquely maximizes the product of utilities. If the zero profile is unfeasible (for instance, if all items are bads), the competitive profiles are strictly negative and are the critical points of the product of disutilities on the efficiency frontier. The latter allows for multiple competitive utility profiles, from which no single-valued selection can be continuous or resource monotonic. Thus the implementation of competitive fairness under linear preferences in interactive platforms like SPLIDDIT will be more difficult when the manna contains bads that overwhelm the goods

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

Chore Cutting: Envy and Truth

We study the fair division of divisible bad resources with strategic agents who can manipulate their private information to get a better allocation. Within certain constraints, we are particularly interested in whether truthful envy-free mechanisms exist over piecewise-constant valuations. We demonstrate that no deterministic truthful envy-free mechanism can exist in the connected-piece scenario, and the same impossibility result occurs for hungry agents. We also show that no deterministic, truthful dictatorship mechanism can satisfy the envy-free criterion, and the same result remains true for non-wasteful constraints rather than dictatorship. We further address several related problems and directions.Comment: arXiv admin note: text overlap with arXiv:2104.07387 by other author