Bargaining over multiple issues in finite horizon alternating-offers protocol



In this paper we study multi issue alternating-offers bargaining in a perfect information finite horizon setting, we determine the pertinent subgame perfect equilibrium, and we provide an algorithm to compute it. The equilibrium is determined by making a novel use of backward induction together with convex programming techniques in multi issue settings. We show that the agents reach an agreement immediately and that such an agreement is Pareto efficient. Furthermore, we prove that, when the multi issue utility functions are linear, the problem of computing the equilibrium is tractable and the related complexity is polynomial with the number of issues and linear with the deadline of bargaining.


game theory multiagent systems automated negotiation convex programming 

Mathematics Subject Classifications (2000)

90C25 91A05 91A18 


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  1. 1.
    Cramton, P.C., Ausubel, L.M., Deneckere, R.J.: Handbook of Game Theory, vol. 3, Chap. Bargaining with Incomplete Information, pp. 1897–1945, Elsevier Science, Dordrecht, The Netherlands (2002)Google Scholar
  2. 2.
    Faratin, P., Sierra, C., Jennings, N.R.: Negotiation decision functions for autonomous agents. Robot. Auton. Syst. 24(3-4), 159–182 (1998)CrossRefGoogle Scholar
  3. 3.
    Fatima, S.S., Wooldridge, M., Jennings, N.R.: An agenda-based framework for multi-issue negotiation. Artif. Intel. 152, 1–45 (2004)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Fudenberg, D., Tirole, J.: Game Theory. MIT Press, Cambridge, MA (1991)Google Scholar
  5. 5.
    Harsanyi, J.C., Selten, R.: A generalized Nash solution for two-person bargaining games with incomplete information. Manag. Sci. 18, 80–106 (1972)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Hiriart-Urruty, J.-B., Lemaréchal, C.: Convex Analysis and Minimization Algorithms I. Springer, Berlin Heidelberg New York (1996)Google Scholar
  7. 7.
    In, Y., Serrano, R.: Agenda restrictions in multi-issue bargaining. J. Econ. Behav. Organ. 53, 385–399 (2004)CrossRefGoogle Scholar
  8. 8.
    Karmarkar, N.: A new polynomial-time algorithm for linear programming. Combinatorica 4(4), 373–395 (1984)MATHMathSciNetGoogle Scholar
  9. 9.
    Kraus, S.: Strategic Negotiation in Multiagent Environments. MIT Press, Cambridge, MA (2001)MATHGoogle Scholar
  10. 10.
    Lai, G., Li, C., Sycara, K., Giampapa, J.: Literature review on Multi-attribute Negotiations. Technical Report CMU–RI–TR–04–66, Carnegie Mellon University, (2004)Google Scholar
  11. 11.
    Napel, S.: Bilateral Bargaining: Theory and Applications. Springer, Berlin Heidelberg New York (2002)MATHGoogle Scholar
  12. 12.
    Osborne, M.J., Rubinstein, A.: Bargaining and Markets. Academic, San Diego, CA (1990)MATHGoogle Scholar
  13. 13.
    Peters, H.: Simultaneity of issues and additivity in bargaining. Econometrica 54(1), 153–170 (1986)MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Raiffa, H.: The Art and Science of Negotiation. Harvard University Press, Cambridge, MA (1982)Google Scholar
  15. 15.
    Rosenschein, J.S., Zlotkin, G.: Rules of Encounter. In: Designing Conventions for Automated Negotiations among Computers. MIT Press, Cambridge, MA (1994)Google Scholar
  16. 16.
    Rubinstein, A.: Perfect equilibrium in a bargaining model. Econometrica 50(1), 97–109 (1982)MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Sandholm, T.: Agents in electronic commerce: component technologies for automated negotiation and coalition formation. Autonomous Agents and Multi-agent Systems 3(1), 73–96 (2000)CrossRefGoogle Scholar
  18. 18.
    Stahl, I.: Bargaining Theory. Stockholm School of Economics, Stockholm, Sweden (1972)Google Scholar

Copyright information

© Springer Science+Business Media, Inc. 2007

Authors and Affiliations

  1. 1.Dipartimento di Elettronica e InformazionePolitecnico di MilanoMilanoItaly

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