Abstract
We study a multicast game in communication networks in which a source sends the same message or service to a set of destinations and the cost of the used links is divided among the receivers according to given cost sharing methods. Assuming a selfish and rational behavior, each receiving user is willing to select a strategy yielding the minimum shared cost. A Nash equilibrium is a solution in which no user can decrease its payment by adopting a different strategy, and the price of anarchy is defined as the worst case ratio between the overall communication cost yielded by an equilibrium and the minimum possible one. Nash equilibria requiring an excessive number of steps to be reached or being hard to compute or not existing at all, we are interested in the determination of the price of anarchy reached in a limited number of rounds, each of which containing at least one move per receiving user. We consider different reasonable cost sharing methods, including the well-known Shapley and egalitarian ones, and investigate their performances versus two possible global criteria: the overall cost of the used links and the maximum shared cost of users. We show that, even in case of two receivers making the best possible move at each step, the number of steps needed to reach a Nash equilibrium can be arbitrarily large. Moreover, we determine the cost sharing methods for which a single round is already sufficient to get a price of anarchy comparable to the one at equilibria, and the ones not satisfying such a property. Finally, we show that finding the sequence of moves leading to the best possible global performance after one-round is already an intractable problem, i.e., NP-hard.
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Archer, A., Feigenbaum, J., Krishnamurthy, A., Sami, R., Shenker, S.: Approximation and collusion in multicast cost sharing. Games and Economic Behavior 47(1), 36–71 (2004)
Bilò, V., Flammini, M., Melideo, G., Moscardelli, L.: On nash equilibria for multicast transmissions in ad-hoc wireless networks. In: Fleischer, R., Trippen, G. (eds.) ISAAC 2004. LNCS, vol. 3341, pp. 172–183. Springer, Heidelberg (2004)
Bilò, V., Flammini, M., Moscardelli, L.: On nash equilibria in non-cooperative all-optical networks. In: Diekert, V., Durand, B. (eds.) STACS 2005. LNCS, vol. 3404, pp. 448–459. Springer, Heidelberg (2005)
Bilò, V., Di Francescomarino, C., Flammini, M., Melideo, G.: Sharing the cost of multicast transmissions in wireless networks. In: SPAA, pp. 180–187. ACM Press, New York (2004)
Bilò, V., Moscardelli, L.: The price of anarchy in all-optical networks. In: Kralovic, R., Sýkora, O. (eds.) SIROCCO 2004. LNCS, vol. 3104, pp. 13–22. Springer, Heidelberg (2004)
Christodoulou, G., Mirrokni, V.S., Sidiropoulos, A.: Convergence and approximation in potential games. In: Durand, B., Thomas, W. (eds.) STACS 2006. LNCS, vol. 3884, pp. 349–360. Springer, Heidelberg (2006)
Conitzer, V., Sandholm, T.: Complexity results about nash equilibria. In: IJCAI, pp. 765–771. Morgan Kaufmann, San Francisco (2003)
Deering, S., CHeriton, D.: Multicast routing in datagram internetworks and extended lans. ACM Transactions on Computer Systems 8, 85–110 (1990)
Dutta, B., Ray, D.: A concept of egalitarianism under participation constraints. Econometrica 57, 615–635 (1989)
Even-Dar, E., Kesselman, A., Mansour, Y.: Convergence time to nash equilibria. In: Baeten, J.C.M., Lenstra, J.K., Parrow, J., Woeginger, G.J. (eds.) ICALP 2003. LNCS, vol. 2719, pp. 502–513. Springer, Heidelberg (2003)
Fabrikant, A., Luthra, A., Maneva, E., Papadimitriou, C.H., Shenker, S.: On a network creation game. In: PODC, pp. 347–351 (2003)
Fabrikant, A., Papadimitriou, C.H., Talwar, K.: The complexity of pure nash equilibria. In: STOC, pp. 604–612. ACM Press, New York (2004)
Feigenbaum, J., Krishnamurthy, A., Sami, R., Shenker, S.: Hardness results for multicast cost sharing. Journal of Public Economics 304(1-3), 215–236 (2003)
Feigenbaum, J., Papadimitriou, C., Shenker, S.: Sharing the cost of multicast transmissions. In: Proceedings of 32nd ACM Symposium on Theory of Computing (STOC), pp. 218–227. ACM Press, New York (2000)
Fotakis, D., Kontogiannis, S.C., Koutsoupias, E., Mavronicolas, M., Spirakis, P.G.: The structure and complexity of nash equilibria for a selfish routing game. In: Widmayer, P., Triguero, F., Morales, R., Hennessy, M., Eidenbenz, S., Conejo, R. (eds.) ICALP 2002. LNCS, vol. 2380, pp. 123–134. Springer, Heidelberg (2002)
Goemans, M.X., Li, E.L., Mirrokni, V.S., Thottan, M.: Market sharing games applied to content distribution in ad-hoc networks. In: MobiHoc, pp. 55–66. ACM Press, New York (2004)
Goemans, M.X., Mirrokni, V.S., Vetta, A.: Sink equilibria and convergence. In: FOCS, pp. 142–154. IEEE Computer Society Press, Los Alamitos (2005)
Jain, K., Vazirani, V.V.: Applications of approximation algorithms to cooperative games. In: STOC, pp. 364–372 (2001)
Kesselman, A., Kowalski, D., Segal, M.: Energy efficient communication in ad hoc networks from user’s and designer’s perspective. SIGMOBILE Mob. Comput. Commun. Rev. 9(1), 15–26 (2005)
Koutsoupias, E., Papadimitriou, C.H.: Worst-case equilibria. In: Meinel, C., Tison, S. (eds.) STACS 1999. LNCS, vol. 1563, pp. 404–413. Springer, Heidelberg (1999)
Mavronicolas, M., Spirakis, P.G.: The price of selfish routing. In: STOC, pp. 510–519 (2001)
Milchtaich, I.: Congestion games with player-specific payoff functions. Games and Economic Behavior 13, 111–124 (1996)
Mirrokni, V.S., Vetta, A.: Convergence issues in competitive games. In: Jansen, K., Khanna, S., Rolim, J.D.P., Ron, D. (eds.) RANDOM 2004 and APPROX 2004. LNCS, vol. 3122, pp. 183–194. Springer, Heidelberg (2004)
Nash, J.F.: Equilibrium points in n-person games. In: Proceedings of the National Academy of Sciences, vol. 36, pp. 48–49 (1950)
Penna, P., Ventre, C.: More powerful and simpler cost-sharing methods. In: Persiano, G., Solis-Oba, R. (eds.) WAOA 2004. LNCS, vol. 3351, pp. 97–110. Springer, Heidelberg (2005)
Penna, P., Ventre, C.: Sharing the cost of multicast transmissions in wireless networks. In: Kralovic, R., Sýkora, O. (eds.) SIROCCO 2004. LNCS, vol. 3104, pp. 255–266. Springer, Heidelberg (2004)
Penna, P., Ventre, C.: Free-riders in steiner tree cost-sharing games. In: Pelc, A., Raynal, M. (eds.) SIROCCO 2005. LNCS, vol. 3499, pp. 231–245. Springer, Heidelberg (2005)
Rosenthal, R.W.: A class of games possessing pure-strategy nash equilibria. International Journal of Game Theory 2, 65–67 (1973)
Roughgarden, T., Tardos, E.: How bad is selfish routing? Journal of ACM 49(2), 236–259 (2002)
Shapley, L.S.: The value of n-person games. In: Contributions to the theory of games, pp. 31–40. Princeton University Press, Princeton (1953)
Vetta, A.: Nash equilibria in competitive societies, with applications to facility location, traffic routing and auctions. In: FOCS, pp. 416–425. IEEE Computer Society Press, Los Alamitos (2002)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2006 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Fanelli, A., Flammini, M., Melideo, G., Moscardelli, L. (2006). Multicast Transmissions in Non-cooperative Networks with a Limited Number of Selfish Moves. In: Královič, R., Urzyczyn, P. (eds) Mathematical Foundations of Computer Science 2006. MFCS 2006. Lecture Notes in Computer Science, vol 4162. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11821069_32
Download citation
DOI: https://doi.org/10.1007/11821069_32
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-37791-7
Online ISBN: 978-3-540-37793-1
eBook Packages: Computer ScienceComputer Science (R0)