An FPTAS for Bargaining Networks with Unequal Bargaining Powers

  • Yashodhan Kanoria
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6484)

Abstract

Bargaining networks model social or economic situations in which agents seek to form the most lucrative partnership with another agent from among several alternatives. There has been a flurry of recent research studying Nash bargaining solutions (also called ‘balanced outcomes’) in bargaining networks, so that we now know when such solutions exist, and that they can be computed efficiently, even by market agents behaving in a natural manner.

In this work we study a generalization of Nash bargaining, that models the possibility of unequal ‘bargaining powers’. This generalization was introduced in [12], where it was shown that the corresponding ‘unequal division’ (UD) solutions exist if and only if Nash bargaining solutions exist, and also that a certain local dynamics converges to UD solutions when they exist. However, the convergence time for that dynamics was exponential in network size for the unequal division case. Other approaches, such as the one of Kleinberg and Tardos, do not generalize to the unsymmetrical case. Thus, the question of computational tractability of UD solutions has remained open.

In this paper, we provide an FPTAS for the computation of UD solutions, when such solutions exist. On a graph G = (V,E) with weights (i.e. pairwise profit opportunities) uniformly bounded above by 1, our FPTAS finds an ε-UD solution in time polynomial in the input and 1/ε. We also provide a fast local algorithm for finding ε-UD solution.

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References

  1. 1.
    Azar, Y., Birnbaum, B., Celis, L.E., Devanur, N.R., Peres, Y.: Convergence of Local Dynamics to Balanced Outcomes in Exchange Networks. In: 50th IEEE Symp. Foundations of Computer Science, Atlanta (2009)Google Scholar
  2. 2.
    Baillon, J., Bruck, R.E.: The rate of asymptotic regularity is \(O(1/{\sqrt{n}})\). In: Kartsatos, A.G. (ed.) Theory and applications of nonlinear operators of accretive and monotone type. Lecture Notes in Pure and Appl. Math, vol. 178, pp. 51–81. Marcel Dekker, Inc., New York (1996)Google Scholar
  3. 3.
    Bateni, M., Hajiaghayi, M., Immorlica, N., Mahini, H.: The cooperative game theory foundations of network bargaining games. In: Intl. Colloquium on Automata, Languages and Programming (2010)Google Scholar
  4. 4.
    Bayati, M., Shah, D., Sharma, M.: Max-Product for Maximum Weight Matching: Convergence, Correctness, and LP Duality. IEEE Trans. Inform. Theory 54, 1241–1251 (2008)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Bayati, M., Borgs, C., Chayes, J., Zecchina, R.: On the exactness of the cavity method for Weighted b-Matchings on Arbitrary Graphs and its Relation to Linear Programs. arXiv:0807.3159 (2007)Google Scholar
  6. 6.
    Chakraborty, T., Judd, S., Kearns, M., Tan, J.: A Behavioral Study of Bargaining in Social Networks. In: Proc. 11th ACM Conf. Electronic Commerce (2010)Google Scholar
  7. 7.
    Cook, K.S., Yamagishi, T.: Power exchange in networks: A power-dependence formulation. Social Networks 14, 245–265 (1992)CrossRefGoogle Scholar
  8. 8.
    Faigle, U., Kern, W., Kuipers, J.: On the computation of the nucleolus of a cooperative game. Intl. Journal of Game Theory 30, 79–98 (2001)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Gabow, H.N., Tarjan, R.E.: Faster scaling algorithms for general graph-matching problems. J. ACM 38(4), 815–853 (1991)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Ishikawa, S.: Fixed points and iteration of a nonexpansive mapping in a Banach space. Proc. American Mathematical Society 59(1) (1976)Google Scholar
  11. 11.
    Kanoria, Y.: An FPTAS for Bargaining Networks with Unequal Bargaining Powers. Full version of this paper: arXiv:1008.0212 (2010)Google Scholar
  12. 12.
    Kanoria, Y., Bayati, M., Borgs, C., Chayes, J., Montanari, A.: Fast Convergence of Natural Bargaining Dynamics in Exchange Networks. arXiv:1004.2079 (April 2010); To appear in Proc. ACM-SIAM Symp. Discrete Algorithms (2011)Google Scholar
  13. 13.
    Kleinberg, J., Tardos, E.: Balanced outcomes in social exchange networks. In: Proc. 40th ACM Symposium on Theory of Computing (2008)Google Scholar
  14. 14.
    Lucas, J.W., Younts, C.W., Lovaglia, M.J., Markovsky, B.: Lines of power in exchange networks. Social Forces 80, 185–214 (2001)CrossRefGoogle Scholar
  15. 15.
    Nash, J.: The bargaining problem. Econometrica 18, 155–162 (1950)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Rochford, S.C.: Symmetric pairwise-bargained allocations in an assignment market. J. Economic Theory 34, 262–281 (1984)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Rubinstein, A.: Perfect equilibrium in a bargaining model. Econometrica 50, 97–109 (1982)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Sanghavi, S., Malioutov, D., Willsky, A.: Linear Programming Analysis of Loopy Belief Propagation for Weighted Matching. In: Neural Information Processing Systems, NIPS (2007)Google Scholar
  19. 19.
    Skvoretz, J., Willer, D.: Exclusion and power: A test of four theories of power in exchange networks. American Sociological Review 58, 801–818 (1993)CrossRefGoogle Scholar
  20. 20.
    Sotomayor, M.: On The Core Of The One-Sided Assignment Game (2005), http://www.usp.br/feaecon/media/fck/File/one_sided_assignment_game.pdf
  21. 21.
    Willer, D. (ed.): Network Exchange Theory. Praeger (1999)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Yashodhan Kanoria
    • 1
  1. 1.Department of Electrical EngineeringStanford UniversityUSA

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