15 research outputs found
Size versus truthfulness in the House Allocation problem
We study the House Allocation problem (also known as the Assignment problem),
i.e., the problem of allocating a set of objects among a set of agents, where
each agent has ordinal preferences (possibly involving ties) over a subset of
the objects. We focus on truthful mechanisms without monetary transfers for
finding large Pareto optimal matchings. It is straightforward to show that no
deterministic truthful mechanism can approximate a maximum cardinality Pareto
optimal matching with ratio better than 2. We thus consider randomised
mechanisms. We give a natural and explicit extension of the classical Random
Serial Dictatorship Mechanism (RSDM) specifically for the House Allocation
problem where preference lists can include ties. We thus obtain a universally
truthful randomised mechanism for finding a Pareto optimal matching and show
that it achieves an approximation ratio of . The same bound
holds even when agents have priorities (weights) and our goal is to find a
maximum weight (as opposed to maximum cardinality) Pareto optimal matching. On
the other hand we give a lower bound of on the approximation
ratio of any universally truthful Pareto optimal mechanism in settings with
strict preferences. In the case that the mechanism must additionally be
non-bossy with an additional technical assumption, we show by utilising a
result of Bade that an improved lower bound of holds. This
lower bound is tight since RSDM for strict preference lists is non-bossy. We
moreover interpret our problem in terms of the classical secretary problem and
prove that our mechanism provides the best randomised strategy of the
administrator who interviews the applicants.Comment: To appear in Algorithmica (preliminary version appeared in the
Proceedings of EC 2014
Size versus truthfulness in the house allocation problem
We study the House Allocation problem (also known as the Assignment problem), i.e., the problem of allocating a set of objects among a set of agents, where each agent has ordinal preferences (possibly involving ties) over a subset of the objects. We focus on truthful mechanisms without monetary transfers for finding large Pareto optimal matchings. It is straightforward to show that no deterministic truthful mechanism can approximate a maximum cardinality Pareto optimal matching with ratio better than 2. We thus consider randomized mechanisms. We give a natural and explicit extension of the classical Random Serial Dictatorship Mechanism (RSDM) specifically for the House Allocation problem where preference lists can include ties. We thus obtain a universally truthful randomized mechanism for finding a Pareto optimal matching and show that it achieves an approximation ratio of eovere-1. The same bound holds even when agents have priorities (weights) and our goal is to find a maximum weight (as opposed to maximum cardinality) Pareto optimal matching. On the other hand we give a lower bound of 18 over 13 on the approximation ratio of any universally truthful Pareto optimal mechanism in settings with strict preferences. In the case that the mechanism must additionally be non-bossy, an improved lower bound of eovere-1 holds. This lower bound is tight given that RSDM for strict preference lists is non-bossy. We moreover interpret our problem in terms of the classical secretary problem and prove that our mechanism provides the best randomized strategy of the administrator who interviews the applicants
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
Social Welfare in One-Sided Matching Mechanisms
We study the Price of Anarchy of mechanisms for the well-known problem of
one-sided matching, or house allocation, with respect to the social welfare
objective. We consider both ordinal mechanisms, where agents submit preference
lists over the items, and cardinal mechanisms, where agents may submit
numerical values for the items being allocated. We present a general lower
bound of on the Price of Anarchy, which applies to all
mechanisms. We show that two well-known mechanisms, Probabilistic Serial, and
Random Priority, achieve a matching upper bound. We extend our lower bound to
the Price of Stability of a large class of mechanisms that satisfy a common
proportionality property, and show stronger bounds on the Price of Anarchy of
all deterministic mechanisms
Manipulation Strategies for the Rank Maximal Matching Problem
We consider manipulation strategies for the rank-maximal matching problem. In
the rank-maximal matching problem we are given a bipartite graph such that denotes a set of applicants and a set of posts. Each
applicant has a preference list over the set of his neighbours in
, possibly involving ties. Preference lists are represented by ranks on the
edges - an edge has rank , denoted as , if post
belongs to one of 's -th choices. A rank-maximal matching is one in which
the maximum number of applicants is matched to their rank one posts and subject
to this condition, the maximum number of applicants is matched to their rank
two posts, and so on. A rank-maximal matching can be computed in time, where denotes the number of applicants, the
number of edges and the maximum rank of an edge in an optimal solution.
A central authority matches applicants to posts. It does so using one of the
rank-maximal matchings. Since there may be more than one rank- maximal matching
of , we assume that the central authority chooses any one of them randomly.
Let be a manipulative applicant, who knows the preference lists of all
the other applicants and wants to falsify his preference list so that he has a
chance of getting better posts than if he were truthful. In the first problem
addressed in this paper the manipulative applicant wants to ensure that
he is never matched to any post worse than the most preferred among those of
rank greater than one and obtainable when he is truthful. In the second problem
the manipulator wants to construct such a preference list that the worst post
he can become matched to by the central authority is best possible or in other
words, wants to minimize the maximal rank of a post he can become matched
to
Pareto optimal matchings in many-to-many markets with ties
We consider Pareto optimal matchings (POMs) in a many-to-many market of applicants
and courses where applicants have preferences, which may include ties, over
individual courses and lexicographic preferences over sets of courses. Since this is the
most general setting examined so far in the literature, our work unifies and generalizes
several known results. Specifically, we characterize POMs and introduce the Generalized
Serial Dictatorship Mechanism with Ties (GSDT) that effectively handles ties
via properties of network flows. We show that GSDT can generate all POMs using
different priority orderings over the applicants, but it satisfies truthfulness only for
certain such orderings. This shortcoming is not specific to our mechanism; we show
that any mechanism generating all POMs in our setting is prone to strategic manipulation.
This is in contrast to the one-to-one case (with or without ties), for which
truthful mechanisms generating all POMs do exist
Pareto Optimal Matchings in Many-to-Many Markets with Ties
We consider Pareto-optimal matchings (POMs) in a many-to-many market of
applicants and courses where applicants have preferences, which may include
ties, over individual courses and lexicographic preferences over sets of
courses. Since this is the most general setting examined so far in the
literature, our work unifies and generalizes several known results.
Specifically, we characterize POMs and introduce the \emph{Generalized Serial
Dictatorship Mechanism with Ties (GSDT)} that effectively handles ties via
properties of network flows. We show that GSDT can generate all POMs using
different priority orderings over the applicants, but it satisfies truthfulness
only for certain such orderings. This shortcoming is not specific to our
mechanism; we show that any mechanism generating all POMs in our setting is
prone to strategic manipulation. This is in contrast to the one-to-one case
(with or without ties), for which truthful mechanisms generating all POMs do
exist
Truthful Facility Assignment with Resource Augmentation: An Exact Analysis of Serial Dictatorship
We study the truthful facility assignment problem, where a set of agents with
private most-preferred points on a metric space are assigned to facilities that
lie on the metric space, under capacity constraints on the facilities. The goal
is to produce such an assignment that minimizes the social cost, i.e., the
total distance between the most-preferred points of the agents and their
corresponding facilities in the assignment, under the constraint of
truthfulness, which ensures that agents do not misreport their most-preferred
points.
We propose a resource augmentation framework, where a truthful mechanism is
evaluated by its worst-case performance on an instance with enhanced facility
capacities against the optimal mechanism on the same instance with the original
capacities. We study a very well-known mechanism, Serial Dictatorship, and
provide an exact analysis of its performance. Although Serial Dictatorship is a
purely combinatorial mechanism, our analysis uses linear programming; a linear
program expresses its greedy nature as well as the structure of the input, and
finds the input instance that enforces the mechanism have its worst-case
performance. Bounding the objective of the linear program using duality
arguments allows us to compute tight bounds on the approximation ratio. Among
other results, we prove that Serial Dictatorship has approximation ratio
when the capacities are multiplied by any integer . Our
results suggest that even a limited augmentation of the resources can have
wondrous effects on the performance of the mechanism and in particular, the
approximation ratio goes to 1 as the augmentation factor becomes large. We
complement our results with bounds on the approximation ratio of Random Serial
Dictatorship, the randomized version of Serial Dictatorship, when there is no
resource augmentation
Pareto Optimal Matchings of Students to Courses in the Presence of Prerequisites
We consider the problem of allocating applicants to courses, where each applicant
has a subset of acceptable courses that she ranks in strict order of preference. Each
applicant and course has a capacity, indicating the maximum number of courses and
applicants they can be assigned to, respectively. We thus essentially have a many-tomany
bipartite matching problem with one-sided preferences, which has applications
to the assignment of students to optional courses at a university.
We consider additive preferences and lexicographic preferences as two means of extending
preferences over individual courses to preferences over bundles of courses.
We additionally focus on the case that courses have prerequisite constraints: we will
mainly treat these constraints as compulsory, but we also allow alternative prerequisites.
We further study the case where courses may be corequisites.
For these extensions to the basic problem, we present the following algorithmic results,
which are mainly concerned with the computation of Pareto optimal matchings
(POMs). Firstly, we consider compulsory prerequisites. For additive preferences, we
show that the problem of finding a POM is NP-hard. On the other hand, in the
case of lexicographic preferences we give a polynomial-time algorithm for finding a
POM, based on the well-known sequential mechanism. However we show that the
problem of deciding whether a given matching is Pareto optimal is co-NP-complete.
We further prove that finding a maximum cardinality (Pareto optimal) matching is
NP-hard. Under alternative prerequisites, we show that finding a POM is NP-hard
for either additive or lexicographic preferences. Finally we consider corequisites. We
prove that, as in the case of compulsory prerequisites, finding a POM is NP-hard
for additive preferences, though solvable in polynomial time for lexicographic preferences.
In the latter case, the problem of finding a maximum cardinality POM is
NP-hard and very difficult to approximate