3,521 research outputs found
Pareto Optimal Allocation under Uncertain Preferences
The assignment problem is one of the most well-studied settings in social
choice, matching, and discrete allocation. We consider the problem with the
additional feature that agents' preferences involve uncertainty. The setting
with uncertainty leads to a number of interesting questions including the
following ones. How to compute an assignment with the highest probability of
being Pareto optimal? What is the complexity of computing the probability that
a given assignment is Pareto optimal? Does there exist an assignment that is
Pareto optimal with probability one? We consider these problems under two
natural uncertainty models: (1) the lottery model in which each agent has an
independent probability distribution over linear orders and (2) the joint
probability model that involves a joint probability distribution over
preference profiles. For both of the models, we present a number of algorithmic
and complexity results.Comment: Preliminary Draft; new results & new author
Counting Houses of Pareto Optimal Matchings in the House Allocation Problem
Let with and be two sets. We assume that every
element has a reference list over all elements from . We call an
injective mapping from to a matching. A blocking coalition of
is a subset of such that there exists a matching that
differs from only on elements of , and every element of
improves in , compared to according to its preference list. If
there exists no blocking coalition, we call the matching an exchange
stable matching (ESM). An element is reachable if there exists an
exchange stable matching using . The set of all reachable elements is
denoted by . We show This is
asymptotically tight. A set is reachable (respectively exactly
reachable) if there exists an exchange stable matching whose image
contains as a subset (respectively equals ). We give bounds for the
number of exactly reachable sets. We find that our results hold in the more
general setting of multi-matchings, when each element of is matched
with elements of instead of just one. Further, we give complexity
results and algorithms for corresponding algorithmic questions. Finally, we
characterize unavoidable elements, i.e., elements of that are used by all
ESM's. This yields efficient algorithms to determine all unavoidable elements.Comment: 24 pages 2 Figures revise
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
A Generalization of the AL method for Fair Allocation of Indivisible Objects
We consider the assignment problem in which agents express ordinal
preferences over objects and the objects are allocated to the agents based
on the preferences. In a recent paper, Brams, Kilgour, and Klamler (2014)
presented the AL method to compute an envy-free assignment for two agents. The
AL method crucially depends on the assumption that agents have strict
preferences over objects. We generalize the AL method to the case where agents
may express indifferences and prove the axiomatic properties satisfied by the
algorithm. As a result of the generalization, we also get a speedup on
previous algorithms to check whether a complete envy-free assignment exists or
not. Finally, we show that unless P=NP, there can be no polynomial-time
extension of GAL to the case of arbitrary number of agents
Size versus fairness in the assignment problem
When not all objects are acceptable to all agents, maximizing the number of objects
actually assigned is an important design concern. We compute the guaranteed size ratio
of the Probabilistic Serial mechanism, i.e., the worst ratio of the actual expected size to
the maximal feasible size. It converges decreasingly to 1 − 1 e 63.2% as the maximal size
increases. It is the best ratio of any Envy-Free assignment mechanism
"Almost-stable" matchings in the Hospitals / Residents problem with Couples
The Hospitals / Residents problem with Couples (hrc) models the allocation of intending junior doctors to hospitals where couples are allowed to submit joint preference lists over pairs of (typically geographically close) hospitals. It is known that a stable matching need not exist, so we consider min bp hrc, the problem of finding a matching that admits the minimum number of blocking pairs (i.e., is “as stable as possible”). We show that this problem is NP-hard and difficult to approximate even in the highly restricted case that each couple finds only one hospital pair acceptable. However if we further assume that the preference list of each single resident and hospital is of length at most 2, we give a polynomial-time algorithm for this case. We then present the first Integer Programming (IP) and Constraint Programming (CP) models for min bp hrc. Finally, we discuss an empirical evaluation of these models applied to randomly-generated instances of min bp hrc. We find that on average, the CP model is about 1.15 times faster than the IP model, and when presolving is applied to the CP model, it is on average 8.14 times faster. We further observe that the number of blocking pairs admitted by a solution is very small, i.e., usually at most 1, and never more than 2, for the (28,000) instances considered
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