101 research outputs found
Locally Stable Marriage with Strict Preferences
We study stable matching problems with locality of information and control.
In our model, each agent is a node in a fixed network and strives to be matched
to another agent. An agent has a complete preference list over all other agents
it can be matched with. Agents can match arbitrarily, and they learn about
possible partners dynamically based on their current neighborhood. We consider
convergence of dynamics to locally stable matchings -- states that are stable
with respect to their imposed information structure in the network. In the
two-sided case of stable marriage in which existence is guaranteed, we show
that the existence of a path to stability becomes NP-hard to decide. This holds
even when the network exists only among one partition of agents. In contrast,
if one partition has no network and agents remember a previous match every
round, a path to stability is guaranteed and random dynamics converge with
probability 1. We characterize this positive result in various ways. For
instance, it holds for random memory and for cache memory with the most recent
partner, but not for cache memory with the best partner. Also, it is crucial
which partition of the agents has memory. Finally, we present results for
centralized computation of locally stable matchings, i.e., computing maximum
locally stable matchings in the two-sided case and deciding existence in the
roommates case.Comment: Conference version in ICALP 2013; to appear in SIAM J. Disc Mat
Socially stable matchings in the hospitals / residents problem
In the Hospitals/Residents (HR) problem, agents are partitioned into hospitals and residents. Each agent wishes to be matched to an agent in the other set and has a strict preference over these potential matches. A matching is stable if there are no blocking pairs, i.e., no pair of agents that prefer each other to their assigned matches. Such a situation is undesirable as it could lead to a deviation in which the blocking pair form a private arrangement outside the matching. This however assumes that the blocking pair have social ties or communication channels to facilitate the deviation. Relaxing the stability definition to take account of the potential lack of social ties between agents can yield larger stable matchings.
In this paper, we define the Hospitals/Residents problem under Social Stability (HRSS) which takes into account social ties between agents by introducing a social network graph to the HR problem. Edges in the social network graph correspond to resident-hospital pairs in the HR instance that know one another. Pairs that do not have corresponding edges in the social network graph can belong to a matching M but they can never block M. Relative to a relaxed stability definition for HRSS, called social stability, we show that socially stable matchings can have different sizes and the problem of finding a maximum socially stable matching is NP-hard, though approximable within 3/2. Furthermore we give polynomial time algorithms for three special cases of the problem
Matching Dynamics with Constraints
We study uncoordinated matching markets with additional local constraints
that capture, e.g., restricted information, visibility, or externalities in
markets. Each agent is a node in a fixed matching network and strives to be
matched to another agent. Each agent has a complete preference list over all
other agents it can be matched with. However, depending on the constraints and
the current state of the game, not all possible partners are available for
matching at all times. For correlated preferences, we propose and study a
general class of hedonic coalition formation games that we call coalition
formation games with constraints. This class includes and extends many recently
studied variants of stable matching, such as locally stable matching, socially
stable matching, or friendship matching. Perhaps surprisingly, we show that all
these variants are encompassed in a class of "consistent" instances that always
allow a polynomial improvement sequence to a stable state. In addition, we show
that for consistent instances there always exists a polynomial sequence to
every reachable state. Our characterization is tight in the sense that we
provide exponential lower bounds when each of the requirements for consistency
is violated. We also analyze matching with uncorrelated preferences, where we
obtain a larger variety of results. While socially stable matching always
allows a polynomial sequence to a stable state, for other classes different
additional assumptions are sufficient to guarantee the same results. For the
problem of reaching a given stable state, we show NP-hardness in almost all
considered classes of matching games.Comment: Conference Version in WINE 201
The Stability of the Roommate Problem Revisited
The lack of stability in some matching problems suggests that alternative solution concepts to the core might be applied to find predictable matchings. We propose the absorbing sets as a solution for the class of roommate problems with strict preferences. This solution, which always exists, either gives the matchings in the core or predicts some other matchings when the core is empty. Furthermore, it satisfies an interesting property of outer stability. We also characterize the absorbing sets, determine their number and, in case of multiplicity, we find that they all share a similar structure.roommate problem, core, absorbing sets
A new solution concept for the roommate problem
Abstract The aim of this paper is to propose a new solution concept for the roommate problem with strict preferences. We introduce maximum irreversible matchings and consider almost stable matchings (Abraham et al., 2006) and maximum stable matchings (Tan 1990, 1991b). These solution concepts are all core consistent. We find that almost stable matchings are incompatible with the other two concepts. Hence, to solve the roommate problem we propose matchings that lie at the intersection of the maximum irreversible matchings and maximum stable matchings, which we call Q -stable matchings. We construct an efficient algorithm for computing one element of this set for any roommate problem. We also show that the outcome of our algorithm always belongs to an absorbing set (Inarra et al., 2013)
A new solution for the roommate problem
The aim of this paper is to propose a new solution
for the roommate problem with strict
references. We introduce the solution of maximum ir
reversibility and consider almost stable
matchings (Abraham et al. [2]) and maximum stable m
atchings (Tan [30] [32]). We find that
almost stable matchings are incompatible with the o
ther two solutions. Hence, to solve the
roommate problem we propose matchings that lie at t
he intersection of the maximum
irreversible matchings and maximum stable matchings
, which are called Q-stable matchings.
These matchings are core consistent and we offer an
efficient algorithm for computing one of
them. The outcome of the algorithm belongs to an ab
sorbing set
The stability of the roommate problem revisited
The lack of stability in some matching problems suggests that alternative solution concepts to the core might be a step towards furthering our understanding of matching market performance. We propose absorbing sets as a solution for the class of roommate problems with strict preferences. This solution, which always exists, either gives the matchings in the core or predicts other matchings when the core is empty. Furthermore, it satisfies the interesting property of outer stability. We also determine the matchings in absorbing sets and find that in the case of multiple absorbing sets a similar structure is shared by all.roommate problem, core, absorbing sets
A new solution for the roommate problem: The Q-stable matchings
The aim of this paper is to propose a new solution for the roommate problem with strict preferences. We introduce the solution of maximum irreversibility and consider almost stable matchings (Abraham et al. [2])and maximum stable matchings (Ta [30] [32]). We find that almost stable matchings are incompatible with the other two solutions. Hence, to solve the roommate problem we propose matchings that lie at the
intersection of the maximum irreversible matchings and maximum stable matchings, which are called Q-stable matchings. These matchings are core consistent and we offer an effi cient algorithm for computing one of
them. The outcome of the algorithm belongs to an absorbing set.This research is supported by the Spanish Ministry of Science and Innovation (ECO2010-
17049 and ECO2012-31346), co-funded by ERDF, by Basque Government IT-568-13 and by the Government of Andalusia Project for Excellence in Research (P07.SEJ.02547). P eter Bir o also acknowledges the support from the Hungarian Academy of Sciences under its Momentum Programme (LD-004/2010), and the Hungarian Scientific Research Fund,OTKA, grant no.K108673
Pairing games and markets
Pairing Games or Markets studied here are the non-two-sided NTU generalization of assignment games. We show that the Equilibrium Set is nonempty, that it is the set of stable allocations or the set of semistable allocations, and that it has has several notable structural properties. We also introduce the solution concept of pseudostable allocations and show that they are in the Demand Bargaining Set. We give a dynamic Market Procedure that reaches the Equilibrium Set in a bounded number of steps. We use elementary tools of graph theory and a representation theorem obtained here
Two-Sided Random Matching Markets: Ex-Ante Equivalence of the Deferred Acceptance Procedures
Stable matching in a community consisting of men and women is a
classical combinatorial problem that has been the subject of intense
theoretical and empirical study since its introduction in 1962 in a seminal
paper by Gale and Shapley.
When the input preference profile is generated from a distribution, we study
the output distribution of two stable matching procedures:
women-proposing-deferred-acceptance and men-proposing-deferred-acceptance. We
show that the two procedures are ex-ante equivalent: that is, under certain
conditions on the input distribution, their output distributions are identical.
In terms of technical contributions, we generalize (to the non-uniform case)
an integral formula, due to Knuth and Pittel, which gives the probability that
a fixed matching is stable. Using an inclusion-exclusion principle on the set
of rotations, we give a new formula which gives the probability that a fixed
matching is the women/men-optimal stable matching. We show that those two
probabilities are equal with an integration by substitution.Comment: Accepted for publication in the 21st ACM Conference on Economics and
Computation (EC'20
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