9 research outputs found
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
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
Network Creation Games: Think Global - Act Local
We investigate a non-cooperative game-theoretic model for the formation of
communication networks by selfish agents. Each agent aims for a central
position at minimum cost for creating edges. In particular, the general model
(Fabrikant et al., PODC'03) became popular for studying the structure of the
Internet or social networks. Despite its significance, locality in this game
was first studied only recently (Bil\`o et al., SPAA'14), where a worst case
locality model was presented, which came with a high efficiency loss in terms
of quality of equilibria. Our main contribution is a new and more optimistic
view on locality: agents are limited in their knowledge and actions to their
local view ranges, but can probe different strategies and finally choose the
best. We study the influence of our locality notion on the hardness of
computing best responses, convergence to equilibria, and quality of equilibria.
Moreover, we compare the strength of local versus non-local strategy-changes.
Our results address the gap between the original model and the worst case
locality variant. On the bright side, our efficiency results are in line with
observations from the original model, yet we have a non-constant lower bound on
the price of anarchy.Comment: An extended abstract of this paper has been accepted for publication
in the proceedings of the 40th International Conference on Mathematical
Foundations on Computer Scienc
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
Complexité des dynamiques de jeux
La th eorie de la complexit e permet de classi er les problemes en fonction de leur di cult e. Le cadre classique dans lequel elle s applique est celui d un algorithme centralis e qui dispose de toutes les informations. Avec l essor des r eseaux et des architectures d ecentralis ees, l algo- rithmique distribu ee a et e etudi ee. Dans un grand nombre de problemes, en optimisation et en economie, les d ecisions et les calculs sont e ectu es par des agents ind ependants qui suivent des objectifs di erents dont la r ealisation d epend des d ecisions des autres agents. La th eorie des jeux est un cadre naturel pour analyser les solutions de tels problemes. Elle propose des concepts de stabilit e, le plus classique etant l equilibre de Nash.Une maniere naturelle de calculer de telles solutions est de faire r eagir les agents ; si un agent voit quelles sont les d ecisions des autres joueurs ou plus g en eralement un etat du jeu , il peut d ecider de changer sa d ecision pour atteindre son objectif faisant ainsi evoluer l etat du jeu. On dit que ces algorithmes sont des dynamiques .On sait que certaines dynamiques convergent vers un concept de solution. On s int eresse a la vitesse de convergence des dynamiques. Certains concepts de solutions sont m eme complets pour certaines classes de complexit e ce qui rend peu vraisemblable l existence de dynamiques simples qui convergent rapidement vers ces solutions. On a utilis e alors trois approches pour obtenir une convergence rapide : am eliorer la dynamique (en utilisant par exemple des bits al eatoires), restreindre la structure du probleme, et rechercher une solution approch ee.Sur les jeux de congestion, on a etendu les r esultats de convergence rapide vers un equilibre de Nash approch e aux jeux n egatifs. Cependant, on a montr e que sur les jeux sans contrainte de signe, calculer un equilibre de Nash approch e est PLS-complet. Sur les jeux d appariement, on a etudi e la vitesse de dynamiques concurrentes lorsque les joueurs ont une information partielle param etr ee par un r eseau social. En particulier, on a am elior e des dynamiques naturelles a n qu elles atteignent un equilibre enO(log(n)) tours (avec n le nombre de joueurs).Complexity theory allows to classify problems by their algorithmic hardness. The classical framework in which it applies is the one of a centralized algorithm that knows every informa- tion. With the development of networks and decentralized architectures, distributed dynamics was studied. In many problems, in optimization or economy, actions and computations are made by independant agents that don t share the same objective whose realization depends on the actions of other agents. Game theory is a natural framework to study solutions of this kind of problem. It provides solution concepts such as the Nash equilibrium.A natural way to compute these solutions is to make the agents react ; if an agent sees the actions of the other player, or more generally the state of the game, he can decide to change his decision to reach his objective and updates the state of the game. We call dynamics this kind of algorithms.We know some dynamics converges to a stable solution. We are interested by the speed of convergence of these dynamics. Some solution concepts are even complete for some complexity classes which make unrealistic the existence of fast converging dynamics. We used three ways to obtain a fast convergence : improving dynamics (using random bits), nding simple subcases, and nding an approximate solution.We extent fast convergence results to an approximate Nash equilibria in negative congestion games. However, we proved that nding an approximate Nash equilibrium in a congestion games without sign restriction is PLS-complete. On matching game, we studied the speed of concurrent dynamics when players have partial information that depends on a social network. Especially, we improved natural dynamics for them to reach an equilibrium inO(log(n)) rounds (with n is the number of players).PARIS11-SCD-Bib. électronique (914719901) / SudocSudocFranceF
Efficient algorithms for optimal matching problems under preferences
In this thesis we consider efficient algorithms for matching problems involving preferences,
i.e., problems where agents may be required to list other agents that they find
acceptable in order of preference. In particular we mainly study the Stable Marriage
problem (SM), the Hospitals / Residents problem (HR) and the Student / Project Allocation
problem (SPA), and some of their variants. In some of these problems the aim
is to find a stable matching which is one that admits no blocking pair. A blocking pair
with respect to a matching is a pair of agents that prefer to be matched to each other
than their assigned partners in the matching if any.
We present an Integer Programming (IP) model for the Hospitals / Residents problem
with Ties (HRT) and use it to find a maximum cardinality stable matching. We also
present results from an empirical evaluation of our model which show it to be scalable
with respect to real-world HRT instance sizes.
Motivated by the observation that not all blocking pairs that exist in theory will lead
to a matching being undermined in practice, we investigate a relaxed stability criterion
called social stability where only pairs of agents with a social relationship have the
ability to undermine a matching. This stability concept is studied in instances of
the Stable Marriage problem with Incomplete lists (smi) and in instances of hr. We
show that, in the smi and hr contexts, socially stable matchings can be of varying
sizes and the problem of finding a maximum socially stable matching (max smiss and
max hrss respectively) is NP-hard though approximable within 3/2. Furthermore we
give polynomial time algorithms for three special cases of the problem arising from
restrictions on the social network graph and the lengths of agents’ preference lists.
We also consider other optimality criteria with respect to social stability and establish
inapproximability bounds for the problems of finding an egalitarian, minimum regret
and sex equal socially stable matching in the sm context.
We extend our study of social stability by considering other variants and restrictions
of max smiss and max hrss. We present NP-hardness results for max smiss even
under certain restrictions on the degree and structure of the social network graph as
well as the presence of master lists. Other NP-hardness results presented relate to the
problem of determining whether a given man-woman pair belongs to a socially stable
matching and the problem of determining whether a given man (or woman) is part of
at least one socially stable matching. We also consider the Stable Roommates problem
with Incomplete lists under Social Stability (a non-bipartite generalisation of smi under
social stability). We observe that the problem of finding a maximum socially stable
matching in this context is also NP-hard. We present efficient algorithms for three
special cases of the problem arising from restrictions on the social network graph and
the lengths of agents’ preference lists. These are the cases where (i) there exists a
constant number of acquainted pairs (ii) or a constant number of unacquainted pairs
or (iii) each preference list is of length at most 2.
We also present algorithmic results for finding matchings in the spa context that are
optimal with respect to profile, which is the vector whose ith component is the number
of students assigned to their ith-choice project. We present an efficient algorithm for
finding a greedy maximum matching in the spa context — this is a maximum matching
whose profile is lexicographically maximum. We then show how to adapt this algorithm
to find a generous maximum matching — this is a matching whose reverse profile is
lexicographically minimum. We demonstrate how this approach can allow additional
constraints, such as lecturer lower quotas, to be handled flexibly. We also present
results of empirical evaluations carried out on both real world and randomly generated
datasets. These results demonstrate the scalability of our algorithms as well as some
interesting properties of these profile-based optimality criteria.
Practical applications of spa motivate the investigation of certain special cases of the
problem. For instance, it is often desired that the workload on lecturers is evenly distributed
(i.e. load balanced). We enforce this by either adding lower quota constraints
on the lecturers (which leads to the potential for infeasible problem instances) or adding
a load balancing optimisation criterion. We present efficient algorithms in both cases.
Another consideration is the fact that certain projects may require a minimum number
of students to become viable. This can be handled by enforcing lower quota constraints
on the projects (which also leads to the possibility of infeasible problem instances). A
technique of handling this infeasibility is the idea of closing projects that do not meet
their lower quotas (i.e. leaving such project completely unassigned). We show that the
problem of finding a maximum matching subject to project lower quotas where projects
can be closed is NP-hard even under severe restrictions on preference lists lengths and
project upper and lower quotas. To offset this hardness, we present polynomial time
heuristics that find large feasible matchings in practice. We also present ip models
for the spa variants discussed and show results obtained from an empirical evaluation
carried out on both real and randomly generated datasets. These results show that
our algorithms and heuristics are scalable and provide good matchings with respect to
profile-based optimalit