Nash Equilibria with Minimum Potential in Undirected Broadcast Games

  • Yasushi Kawase
  • Kazuhisa Makino
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7157)


In this paper, we consider undirected network design games with fair cost allocation. We introduce two concepts Potential-Optimal Price of Anarchy (POPoA) and Potential-Optimal Price of Stability (POPoS), where POPoA is the ratio between the worst cost of Nash equilibria with optimal potential and the minimum social cost, and POPoS is the ratio between the best cost of Nash equilibria with optimal potential and the minimum social cost, and show that

  • The POPoA and POPoS for undirected broadcast games with n players are \(\mathrm{O}(\sqrt{\log n})\).

  • The POPoA and POPoS for undirected broadcast games with |V| vertices are O(log|V|).

  • There exists an undirected broadcast game with n players such that POPoA, \(\mathrm{POPoS} = \Omega(\sqrt{\log\log n})\).

  • There exists an undirected broadcast game with |V| vertices such that POPoA,POPoS = Ω(log|V|).


Nash Equilibrium Span Tree Minimum Potential Congestion Game Potential Game 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Albers, S.: On the value of coordination in network design. SIAM Journal on Computing 38, 2273–2302 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Anshelevich, E., Dasgupta, A., Kleinberg, J.M., Tardos, É., Wexler, T., Roughgarden, T.: The price of stability for network design with fair cost allocation. SIAM Journal on Computing 38(4), 1602–1623 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Anshelevich, E., Dasgupta, A., Tardos, É., Wexler, T.: Near-optimal network design with selfish agents. Theory of Computing 4(1), 77–109 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Asadpour, A., Saberi, A.: On the Inefficiency Ratio of Stable Equilibria in Congestion Games. In: Leonardi, S. (ed.) WINE 2009. LNCS, vol. 5929, pp. 545–552. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  5. 5.
    Bilò, V., Caragiannis, I., Fanelli, A., Monaco, G.: Improved Lower Bounds on the Price of Stability of Undirected Network Design Games. In: Kontogiannis, S., Koutsoupias, E., Spirakis, P.G. (eds.) SAGT 2010. LNCS, vol. 6386, pp. 90–101. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  6. 6.
    Blume, L.E.: The statistical mechanics of strategic interaction. Games and Economic Behavior 5, 387–424 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Chekuri, C., Chuzhoy, J., Lewin-Eytan, L., Naor, J., Orda, A.: Non-cooperative multicast and facility location games. IEEE Journal on Selected Areas in Communications 25(6), 1193–1206 (2007)CrossRefGoogle Scholar
  8. 8.
    Chen, H.L., Roughgarden, T.: Network design with weighted players. In: Proceedings of the 18th Annual ACM Symposium on Parallelism in Algorithms and Architectures, pp. 29–38 (2006)Google Scholar
  9. 9.
    Chen, H.-L., Roughgarden, T., Valiant, G.: Designing network protocols for good equilibria. SIAM Journal on Computing 39(5), 1799–1832 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Fabrikant, A., Papadimitriou, C.H., Talwar, K.: The complexity of pure Nash equilibria. In: Proceedings of the 36th Annual ACM Symposium on Theory of Computing, pp. 604–612 (2004)Google Scholar
  11. 11.
    Fiat, A., Kaplan, H., Levy, M., Olonetsky, S., Shabo, R.: On the price of stability for designing undirected networks with fair cost allocations. In: Proceedings of the 33rd Annual International Colloquium on Automata, Languages, and Programming, pp. 608–618 (2006)Google Scholar
  12. 12.
    Koutsoupias, E., Papadimitriou, C.H.: Worst-case equilibria. In: Proceedings of the 16th Annual Symposium on Theoretical Aspects of Computer Science, pp. 404–413 (1999)Google Scholar
  13. 13.
    Li, J.: An \(\mathrm{O}\left(\frac{\log n}{\log \log n}\right)\) upper bound on the price of stability for undirected Shapley network design games. Information Processing Letters 109(15), 876–878 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Monderer, D., Shapley, L.S.: Potential games. Games and Economic Behavior 14, 124–143 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Rosenthal, R.W.: A class of games possessing pure-strategy Nash equilibria. International Journal of Game Theory 2, 65–67 (1973)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Roughgarden, T.: Potential functions and the inefficiency of equilibria. In: Proceedings of International Congress of Mathematicians (2006)Google Scholar
  17. 17.
    Ui, T.: Robust equilibria of potential games. Econometrica 69(5), 1373–1380 (2001)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Yasushi Kawase
    • 1
  • Kazuhisa Makino
    • 1
  1. 1.University of TokykoJapan

Personalised recommendations