7 research outputs found
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
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