11 research outputs found
On the Complexity of Chore Division
We study the proportional chore division problem where a protocol wants to
divide an undesirable object, called chore, among different players. The
goal is to find an allocation such that the cost of the chore assigned to each
player be at most of the total cost. This problem is the dual variant of
the cake cutting problem in which we want to allocate a desirable object.
Edmonds and Pruhs showed that any protocol for the proportional cake cutting
must use at least queries in the worst case, however,
finding a lower bound for the proportional chore division remained an
interesting open problem. We show that chore division and cake cutting problems
are closely related to each other and provide an lower bound
for chore division
Developments in Multi-Agent Fair Allocation
Fairness is becoming an increasingly important concern when designing
markets, allocation procedures, and computer systems. I survey some recent
developments in the field of multi-agent fair allocation
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
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
3 Persons, 2 Cuts: A Maximin Envy-Free and a Maximally Equitable Cake-Cutting Algorithm
We describe a 3-person, 2-cut envy-free cake-cutting algorithm, inspired by a continuous moving-knife procedure, that does not require that the players continuously move knifes across the cake. By having the players submit their value functions over the cake to a referee—rather than move knives according to these functions—the referee can ensure that the division is not only envy-free but also maximin. In addition, the referee can use the value functions to find a maximally equitable division, whereby the players receive equally valued shares that are maximal, but this allocation may not be envy-free
Algorithms for Competitive Division of Chores
We study the problem of allocating divisible bads (chores) among multiple
agents with additive utilities, when money transfers are not allowed. The
competitive rule is known to be the best mechanism for goods with additive
utilities and was recently extended to chores by Bogomolnaia et al (2017). For
both goods and chores, the rule produces Pareto optimal and envy-free
allocations. In the case of goods, the outcome of the competitive rule can be
easily computed. Competitive allocations solve the Eisenberg-Gale convex
program; hence the outcome is unique and can be approximately found by standard
gradient methods. An exact algorithm that runs in polynomial time in the number
of agents and goods was given by Orlin.
In the case of chores, the competitive rule does not solve any convex
optimization problem; instead, competitive allocations correspond to local
minima, local maxima, and saddle points of the Nash Social Welfare on the
Pareto frontier of the set of feasible utilities. The rule becomes multivalued
and none of the standard methods can be applied to compute its outcome.
In this paper, we show that all the outcomes of the competitive rule for
chores can be computed in strongly polynomial time if either the number of
agents or the number of chores is fixed. The approach is based on a combination
of three ideas: all consumption graphs of Pareto optimal allocations can be
listed in polynomial time; for a given consumption graph, a candidate for a
competitive allocation can be constructed via explicit formula; and a given
allocation can be checked for being competitive using a maximum flow
computation as in Devanur et al (2002).
Our algorithm immediately gives an approximately-fair allocation of
indivisible chores by the rounding technique of Barman and Krishnamurthy
(2018).Comment: 38 pages, 4 figure