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The Max-Distance Network Creation Game on General Host Graphs

  • Davide Bilò
  • Luciano Gualà
  • Stefano Leucci
  • Guido Proietti
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7695)

Abstract

In this paper we study a generalization of the classic network creation game to the scenario in which the n players sit on a given arbitrary host graph, which constrains the set of edges a player can activate at a cost of α ≥ 0 each. This finds its motivations in the physical limitations one can have in constructing links in practice, and it has been studied in the past only when the routing cost component of a player is given by the sum of distances to all the other nodes. Here, we focus on another popular routing cost, namely that which takes into account for each player its maximum distance to any other player. For this version of the game, we first analyze some of its computational and dynamic aspects, and then we address the problem of understanding the structure of associated pure Nash equilibria. In this respect, we show that the corresponding price of anarchy (PoA) is fairly bad, even for several basic classes of host graphs. More precisely, we first exhibit a lower bound of \(\Omega (\sqrt{ n / (1+\alpha)})\) for any α = o(n). Notice that this implies a counter-intuitive lower bound of \(\Omega(\sqrt{n})\) for the case α = 0 (i.e., edges can be activated for free). Then, we show that when the host graph is restricted to be either k-regular (for any constant k ≥ 3), or a 2-dimensional grid, the PoA is still \(\Omega(1+\min\{\alpha, \frac{n}{\alpha}\})\), which is proven to be tight for \(\alpha=\Omega(\sqrt{n})\). On the positive side, if α ≥ n, we show the PoA is O(1). Finally, in the case in which the host graph is very sparse (i.e., |E(H)| = n − 1 + k, with k = O(1)), we prove that the PoA is O(1), for any α.

Keywords

Cost Function Nash Equilibrium White Vertex Pure Nash Equilibrium Pure Strategy Nash Equilibrium 
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.

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References

  1. 1.
    Alon, N., Demaine, E.D., Hajiaghayi, M., Leighton, T.: Basic network creation games. In: Proceedings of the 22nd ACM Symposium on Parallelism in Algorithms and Architectures (SPAA 2010), pp. 106–113. ACM Press (2010)Google Scholar
  2. 2.
    Biló, D., Gualá, L., Proietti, G.: Bounded-Distance Network Creation Games. In: Goldberg, P.W., Guo, M. (eds.) WINE 2012. LNCS, vol. 7695, pp. 72–85. Springer, Heidelberg (2012)Google Scholar
  3. 3.
    Demaine, E.D., Hajiaghayi, M., Mahini, H., Zadimoghaddam, M.: The price of anarchy in network creation games. In: Proceedings of the 36th Annual ACM Symposium on Principles of Distributed Computing (PODC 2007), pp. 292–298. ACM Press (2007)Google Scholar
  4. 4.
    Demaine, E.D., Hajiaghayi, M., Mahini, H., Zadimoghaddam, M.: The price of anarchy in cooperative network creation games. In: Proceedings of the 26th International Symposium on Theoretical Aspects of Computer Science (STACS 2009). Leibniz International Proceedings in Informatics, vol. 3, pp. 301–312 (2009)Google Scholar
  5. 5.
    Ehsani, S., Fazli, M., Mehrabian, A., Sadeghabad, S.S., Saghafian, M., Shokatfadaee, S., Safari, M.: On a bounded budget network creation game. In: Proceedings of the 23rd ACM Symposium on Parallelism in Algorithms and Architectures (SPAA 2011). ACM Press (2011) (in press)Google Scholar
  6. 6.
    Fabrikant, A., Luthra, A., Maneva, E., Papadimitriou, C.H., Shenker, S.: On a network creation game. In: Proceedings of the 22nd Symposium on Principles of Distributed Computing (PODC 2003), pp. 347–351. ACM Press (2003)Google Scholar
  7. 7.
    Jackson, M.O., Wolinsky, A.: A strategic model of social and economic networks. Journal of Economic Theory 71(1), 44–74 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Laoutaris, N., Poplawski, L.J., Rajaraman, R., Sundaram, R., Teng, S.-H.: Bounded budget connection (BBC) games or how to make friends and influence people, on a budget. In: Proceedings of the 27th ACM Symposium on Principles of Distributed Computing (PODC 2008), pp. 165–174. ACM Press (2008)Google Scholar
  9. 9.
    Mihalák, M., Schlegel, J.C.: The Price of Anarchy in Network Creation Games Is (Mostly) Constant. In: Kontogiannis, S., Koutsoupias, E., Spirakis, P.G. (eds.) SAGT 2010. LNCS, vol. 6386, pp. 276–287. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  10. 10.
    Schoone, A.A., Bodlaender, H.L., van Leeuwen, J.: Improved Diameter Bounds for Altered Graphs. In: Tinhofer, G., Schmidt, G. (eds.) WG 1986. LNCS, vol. 246, pp. 227–236. Springer, Heidelberg (1987)CrossRefGoogle Scholar
  11. 11.
    Lenzner, P.: On Dynamics in Basic Network Creation Games. In: Persiano, G. (ed.) SAGT 2011. LNCS, vol. 6982, pp. 254–265. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  12. 12.
    Mihalák, M., Schlegel, J.C.: Asymmetric Swap-Equilibrium: A Unifying Equilibrium Concept for Network Creation Games. In: Rovan, B., Sassone, V., Widmayer, P. (eds.) MFCS 2012. LNCS, vol. 7464, pp. 693–704. Springer, Heidelberg (2012)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Davide Bilò
    • 1
  • Luciano Gualà
    • 2
  • Stefano Leucci
    • 2
  • Guido Proietti
    • 3
    • 4
  1. 1.Dipartimento di Scienze Umanistiche e SocialiUniversità di SassariItaly
  2. 2.Dipartimento di Ingegneria dell’ImpresaUniversità di Roma “Tor Vergata”Italy
  3. 3.Dipartimento di Ingegneria e Scienze dell’Informazione e MatematicaUniversità degli Studi dell’AquilaItaly
  4. 4.Istituto di Analisi dei Sistemi ed InformaticaCNRRomeItaly

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