Random paths to P-stability in the roommate problem

Iñarra, Elena; Larrea, Concepción; Molis, Elena

doi:10.1007/s00182-007-0089-y

Cited by 29 publications

(20 citation statements)

References 11 publications

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“…We use the latter to give an alternative proof for the theorems of Diamantoudi et al [8] and Inarra et al [13]. That is, we show that starting from an arbitrary matching of a solvable sr instance one can always find a stable matching by successively satisfying blocking pairs; and that starting from an arbitrary matching of an instance of sr (solvable or unsolvable) one can always find an h-stable matching by successively satisfying blocking pairs.…”

Section: Convergence To Stability An Alternative Proofmentioning

confidence: 81%

“…Modifying the Tan-Hsueh algorithm slightly (with h-stable matchings rather than with stable half-matchings), we can obtain an alternative proof for the following theorem of Inarra et al [13], with an upper bound on the number of steps needed to reach an h-stable matching.…”

Section: Convergence To Stability An Alternative Proofmentioning

confidence: 99%

“…In Section 2 we give an alternative proof for the theorems of Diamantoudi et al [8] and Inarra et al [13]. This new proof, which is based on the Tan-Hsueh algorithm, is not only shorter and simpler than the originals, but also provides upper bounds on the number of steps needed to reach a stable (or h-stable) matching.…”

Section: Introductionmentioning

confidence: 99%

“…Inarra et al [13] define an h-stable matching M relative to a stable halfmatching h as follows. Let M contain every edge that has weight 1 in h, every second edge from each even half-weighted cycle of h (if there were any), and k (disjoint) edges from each odd half-weighted cycle of length 2k + 1 in h. 6 They show that, starting from an arbitrary matching, one can get an h-stable matching by successively satisfying blocking pairs for a given instance of sr.…”

Section: Introductionmentioning

confidence: 99%

“…In Section 2, we give an alternative short proof for the theorem of Inarra et al [13] by using the Tan-Hsueh algorithm, by providing also upper bounds for the steps needed to reach a desired matching. In another paper, Inarra et al [14] define the absorbing sets for an instance of sr as follows.…”

Section: Introductionmentioning

confidence: 99%

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Analysis of stochastic matching markets

Bíró

Norman

2012

Int J Game Theory

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Abstract. Suppose that the agents of a matching market contact each other randomly and form new pairs if is in their interest. Does such a process always converge to a stable matching if one exists? If so, how quickly? Are some stable matchings more likely to be obtained by this process than others? In this paper we are going to provide answers to these and similar questions, posed by economists and computer scientists. In the first part of the paper we give an alternative proof for the theorems by Diamantoudi et al. and Inarra et al. which imply that the corresponding stochastic processes are absorbing Markov chains. Our proof is not only shorter, but also provides upper bounds for the number of steps needed to stabilise the system. The second part of the paper proposes new techniques to analyse the behaviour of matching markets. We introduce the Stable Marriage and Stable Roommates Automaton and show how the probabilistic model checking tool PRISM may be used to predict the outcomes of stochastic interactions between myopic agents.In particular, we demonstrate how one can calculate the probabilities of reaching different matchings in a decentralised market and determine the expected convergence time of the stochastic process concerned. We illustrate the usage of this technique by studying some well-known marriage and roommates instances and randomly generated instances.

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Section: Convergence To Stability An Alternative Proofmentioning

confidence: 81%

Section: Convergence To Stability An Alternative Proofmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Analysis of stochastic matching markets

Bíró

Norman

2012

Int J Game Theory

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Balancing stability and efficiency in team formation as a generalized roommate problem

Yekta

Bergman

Day

2022

Naval Research Logistics

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The assignment of personnel to teams is a fundamental managerial function typically involving several objectives and a variety of idiosyncratic practical constraints. Despite the prevalence of this task in practice, the process is seldom approached as an optimization problem over the reported preferences of all agents. This is due in part to the underlying computational complexity that occurs when intra‐team interpersonal interactions are taken into consideration, and also due to game‐theoretic considerations, when those taking part in the process are self‐interested agents. Variants of this fundamental decision problem arise in a number of settings, including, for example, human resources and project management, military platooning, ride sharing, data clustering, and in assigning students to group projects. In this article, we study an analytical approach to “team formation” focused on the interplay between two of the most common objectives considered in the related literature: economic efficiency (i.e., the maximization of social welfare) and game‐theoretic stability (e.g., finding a core solution when one exists). With a weighted objective across these two goals, the problem is modeled as a bi‐level binary optimization problem, and transformed into a single‐level, exponentially sized binary integer program. We then devise a branch‐cut‐and‐price algorithm and demonstrate its efficacy through an extensive set of simulations, with favorable comparisons to other algorithms from the literature.

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Matching Dynamics with Constraints

Hoefer

Wagner

2014

Web and Internet Economics

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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.

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Random paths to P-stability in the roommate problem

Abstract: Roommate problem, Random paths to stability, C78,

Cited by 29 publications

References 11 publications

Analysis of stochastic matching markets

Analysis of stochastic matching markets

Balancing stability and efficiency in team formation as a generalized roommate problem

Matching Dynamics with Constraints

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