Abstract
We study a network extension to the Nash bargaining game, as introduced by Kleinberg and Tardos [6], where the set of players corresponds to vertices in a graph G = (V,E) and each edge ij ∈ E represents a possible deal between players i and j. We reformulate the problem as a cooperative game and study the following question: Given a game with an empty core (i.e. an unstable game) is it possible, through minimal changes in the underlying network, to stabilize the game? We show that by removing edges in the network that belong to a blocking set we can find a stable solution in polynomial time. This motivates the problem of finding small blocking sets. While it has been previously shown that finding the smallest blocking set is NP-hard [2], we show that it is possible to efficiently find approximate blocking sets in sparse graphs.
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References
Bateni, M., Hajiaghayi, M., Immorlica, N., Mahini, H.: The Cooperative Game Theory Foundations of Network Bargaining Games. In: Abramsky, S., Gavoille, C., Kirchner, C., Meyer auf der Heide, F., Spirakis, P.G. (eds.) ICALP 2010, Part I. LNCS, vol. 6198, pp. 67–78. Springer, Heidelberg (2010)
Biró, P., Kern, W., Paulusma, D.: On Solution Concepts for Matching Games. In: Kratochvíl, J., Li, A., Fiala, J., Kolman, P. (eds.) TAMC 2010. LNCS, vol. 6108, pp. 117–127. Springer, Heidelberg (2010)
Chalkiadakis, G., Elkind, E., Wooldridge, M.: Computational Aspects of Cooperative Game Theory. Synthesis Lectures on Artificial Intelligence and Machine Learning. Morgan & Claypool Publishers (2011)
Deng, X., Ibaraki, T., Nagamochi, H.: Algorithmic aspects of the core of combinatorial optimization games. Math. Oper. Res. 24(3), 751–766 (1999)
Faigle, U., Kern, W., Kuipers, J.: An efficient algorithm for nucleolus and prekernel computation in some classes of tu-games. Tech. Rep. 1464, U. of Twente (1998)
Kleinberg, J.M., Tardos, É.: Balanced outcomes in social exchange networks. In: Proceedings of ACM Symposium on Theory of Computing, pp. 295–304 (2008)
Könemann, J., Larson, K., Steiner, D.: Network bargaining: Using approximate blocking sets to stabilize unstable instances. Tech. Rep. submit/0522859, arXiV (full version, 2012)
Lau, L.C., Ravi, R., Singh, M.: Iterative Methods in Combinatorial Optimization. Cambridge University Press (2011)
Nash, J.: The bargaining problem. Econometrica 18, 155–162 (1950)
Peleg, B., Sudhölter, P.: Introduction to the Theory of Cooperative Games. Springer (2003)
Schrijver, A.: Theory of Linear and Integer Programming. Wiley, Chichester (1986)
Shapley, L.S., Shubik, M.: The assignment game: the core. International Journal of Game Theory 1(1), 111–130 (1971), http://dx.doi.org/10.1007/BF01753437
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Könemann, J., Larson, K., Steiner, D. (2012). Network Bargaining: Using Approximate Blocking Sets to Stabilize Unstable Instances. In: Serna, M. (eds) Algorithmic Game Theory. SAGT 2012. Lecture Notes in Computer Science, vol 7615. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-33996-7_19
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DOI: https://doi.org/10.1007/978-3-642-33996-7_19
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