Locally Stable Marriage with Strict Preferences

We study two-sided matching markets with locality of information and control. Each male (female) agent has an arbitrary strict preference list over all female (male) agents. In addition, each agent is a node in a fixed network. Agents learn about possible partners dynamically based on their current network neighborhood. We consider convergence of dynamics to locally stable matchings that are stable with respect to their imposed information structure in the network. While existence of such states is guaranteed, we show that reachability becomes NP-hard to decide. This holds even when the network exists only among one side. In contrast, if only one side has no network and agents remember a previous match every round, reachability 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 memory with the most recent partner, but not for memory with the best partner. Also, it is crucial which partition of the agents has memory. Finally, we conclude with results on approximating maximum locally stable matchings.

[1]  Akihisa Tamura,et al.  Transformation from Arbitrary Matchings to Stable Matchings , 1993, J. Comb. Theory, Ser. A.

[2]  Ramamohan Paturi,et al.  Jealousy Graphs: Structure and Complexity of Decentralized Stable Matching , 2013, WINE.

[3]  Kemal Akkaya,et al.  Autonomous actor positioning in wireless sensor and actor networks using stable-matching , 2010, Int. J. Parallel Emergent Distributed Syst..

[4]  Elena Molis,et al.  Random paths to P-stability in the roommate problem , 2008, Int. J. Game Theory.

[5]  Zoltán Király.  Better and Simpler Approximation Algorithms for the Stable Marriage Problem , 2008, ESA.

[6]  David Manlove,et al.  Algorithmics of Matching Under Preferences , 2013, Bull. EATCS.

[7]  U. Rothblum,et al.  Vacancy Chains and Equilibration in Senior-Level Labor Markets , 1997 .

[8]  Roger Wattenhofer,et al.  On Finding Better Friends in Social Networks , 2012, SSS.

[9]  Róbert F. Veszteg,et al.  Decentralized matching markets : a laboratory experiment , 2012 .

[10]  Uriel G. Rothblum,et al.  "Timing Is Everything" and Marital Bliss , 2002, J. Econ. Theory.

[11]  Leeat Yariv,et al.  An Experimental Study of Decentralized Matching , 2012 .

[12]  David Manlove,et al.  The Stable Roommates Problem with Ties , 2002, J. Algorithms.

[13]  Subhash Khot,et al.  Inapproximability of Vertex Cover and Independent Set in Bounded Degree Graphs , 2009, 2009 24th Annual IEEE Conference on Computational Complexity.

[14]  Alvin E. Roth,et al.  Two-Sided Matching: A Study in Game-Theoretic Modeling and Analysis , 1990 .

[15]  Vahab S. Mirrokni,et al.  Uncoordinated two-sided matching markets , 2009, SECO.

[16]  Robert W. Irving,et al.  The Stable marriage problem - structure and algorithms , 1989, Foundations of computing series.

[17]  Marina Thottan,et al.  Market sharing games applied to content distribution in ad hoc networks , 2004, IEEE Journal on Selected Areas in Communications.

[18]  Sergei Vassilvitskii,et al.  Social Networks and Stable Matchings in the Job Market , 2009, WINE.

[19]  David Manlove,et al.  Hard variants of stable marriage , 2002, Theor. Comput. Sci..

[20]  Martin Hoefer,et al.  Friendship and Stable Matching , 2013, ESA.

[21]  Eric McDermid,et al.  Maximum Locally Stable Matchings , 2013, Algorithms.

[22]  Eric McDermid.  A 3/2-Approximation Algorithm for General Stable Marriage , 2009, ICALP.

[23]  Martin Hoefer,et al.  Matching Dynamics with Constraints , 2014, WINE.

[24]  Alvin E. Roth,et al.  Pairwise Kidney Exchange , 2004, J. Econ. Theory.

[25]  David Manlove,et al.  Socially Stable Matchings in the Hospitals/Residents Problem , 2013, WADS.

[26]  Harry R. Lewis,et al.  Review of "Mariages stables et leur relations avec d'autre problèmes combinatoires: introduction à l'analyze mathématique des algorithmes" by Donald E. Knuth. Les Presses de l'Université de Montréal. , 1978, SIGA.

[27]  Elena Molis,et al.  The stability of the roommate problem revisited , 2010 .

[28]  A. Roth,et al.  Random paths to stability in two-sided matching , 1990 .

[29]  Tamás Fleiner,et al.  The dynamics of stable matchings and half-matchings for the stable marriage and roommates problems , 2008, Int. J. Game Theory.

[30]  Péter Biró,et al.  Analysis of stochastic matching markets , 2013, Int. J. Game Theory.

[31]  David Manlove,et al.  Approximability results for stable marriage problems with ties , 2003, Theor. Comput. Sci..

[32]  Bettina Klaus,et al.  Stochastic stability for roommate markets , 2010, J. Econ. Theory.

[33]  F. Mathieu.  Acyclic Preference-Based Systems , 2010 .

[34]  Martin Hoefer,et al.  Hedonic Coalition Formation in Networks , 2015, AAAI.

[35]  Xuemin Shen,et al.  Handbook of Peer-to-Peer Networking , 2009 .

[36]  Martin Hoefer,et al.  Local matching dynamics in social networks , 2011, Inf. Comput..

[37]  Robert W. Irving.  An Efficient Algorithm for the "Stable Roommates" Problem , 1985, J. Algorithms.

[38]  Chung-Piaw Teo,et al.  The Geometry of Fractional Stable Matchings and Its Applications , 1998, Math. Oper. Res..

[39]  Fabien Mathieu,et al.  Self-stabilization in preference-based systems , 2008, Peer-to-Peer Netw. Appl..

[40]  Eiichi Miyagawa,et al.  Random paths to stability in the roommate problem , 2004, Games Econ. Behav..

[41]  Hiroki Yanagisawa.  Approximation algorithms for stable marriage problems , 2007 .