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On Nash equilibria for multicast transmissions in ad-hoc wireless networks

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Abstract

We study a multicast game in ad-hoc wireless networks in which a source sends the same message or service to a set of receiving stations via multi-hop communications and the overall transmission cost is divided among the receivers according to given cost sharing methods. We assume that each receiver gets a certain utility from the transmission and enjoys a benefit equal to the difference between his utility and the shared cost he is asked to pay. Assuming a selfish and rational behavior, each user is willing to receive the transmission if and only if his shared cost does not exceed his utility. Moreover, given the strategies of the other users, he wants to select a strategy of minimum shared cost. A Nash equilibrium is a solution in which no user can increase his benefit by choosing to adopt a different strategy. We consider the following reasonable cost sharing methods: egalitarian, semi-egalitarian next-hop-proportional, path-proportional, egalitarian-path-proportional and Shapley value. We prove that, while the first five cost sharing methods in general do not admit a Nash equilibrium, the Shapley value yields games always converging to a Nash equilibrium. We then turn our attention to the special case in which the receivers’ set R is part of the input (that is only the stations belonging to R have a positive utility which is set equal to infinity) and show that in such a case also the egalitarian and the egalitarian-path-proportional methods yield convergent games. In such a framework, we show that the price of anarchy is unbounded for the game yielded by the egalitarian method and provide matching upper and lower bounds for the price of anarchy of the other two convergent games with respect to two different global cost functions, that is the overall cost of the power assignment, that coincides with the sum of all the shared costs, and the maximum shared cost paid by the receivers. Finally, in all cases we show that finding the best Nash equilibrium is computationally intractable, that is NP-hard.

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References

  1. A. Archer, J. Feigenbaum, A. Krishnamurthy, R. Sami and S. Shenker, Approximation and collusion in multicast cost sharing. Games and Economic Behavior 47(1) (2004) 36–71.

    Article  MATH  MathSciNet  Google Scholar 

  2. V. Bilò, M. Flammini and L. Moscardelli, On nash equilibria in non-cooperative all-optical networks, in: Proceedings of the 22nd Annual Symposium on Theoretical Aspects of Computer Science (STACS), Vol. 3404 of LNCS, Springer (2005) pp. 448–459.

  3. V. Bilò, C. Di Francescomarino, M. Flammini and G. Melideo, Sharing the cost of multicast transmissions in wireless networks, in: Proceedings of the 16th ACM Symposium on Parallelism in Algorithms and Architectures (SPAA) ACM Press (2004) pp. 180–187.

  4. V. Bilò and L. Moscardelli, The price of anarchy in all-optical networks, in: Proceedings of the 11th Colloquium on Structural Information and Communication Complexity (SIROCCO), Vol. 3104 of LNCS, Springer (2004) pp. 13–22.

  5. A.E.F. Clementi, H. Huiban, P. Penna, G. Rossi and P. Vocca, Some recent theoretical advances and open questions on energy consumption in static ad-hoc wireless networks, in: Proceedings of the 3rd International Workshop on Approximation and Randomization (ARACNE) (2002) pp. 23–38.

  6. V. Conitzer and T. Sandholm, Complexity, results about nash equilibria, in: Proceedings of the 18th International Joint Conference on Artificial Intelligence (IJCAI) (2003) pp. 765–771.

  7. S. Eidenbenz, V.S.A. Kumar and S. Zust, Equilibria in topology control games for ad hoc networks, in: Proceedings of the 2003 Joint Workshop on Foundations of Mobile Computing, DIALM-POMC’03, ACM Press (2003) pp. 2–11.

  8. E. Even-Dar, A. Kesselman and Y. Mansour, Convergence time to nash equilibria, in: Proceedings of the 30th Annual International Colloquium on Automata, Languages and Programming (ICALP), Vol. 2719 of LNCS Springer (2003) pp. 502–513.

  9. A. Fabrikant, A. Luthra, E. Maneva, C.H. Papadimitriou and S. Shenker, On a network creation game, in: Proceedings of the 22nd ACM Symposium on Principles of Distributed Computing (PODC) (2003) pp. 347–351.

  10. A. Fabrikant, C.H. Papadimitriou and K. Talwar, The complexity of pure equilibria, in: Proceedings of the 36th ACM Symposium on Theory of Computing (STOC) ACM Press (2004) pp. 604–612.

  11. J. Feigenbaum, A. Krishnamurthy, R. Sami and S. Shenker, Hardness results for multicast cost sharing, Journal of Public Economics 304(1–3) (2003) 215–236.

    MATH  MathSciNet  Google Scholar 

  12. J. Feigenbaum, C. Papadimitriou and S. Shenker, Sharing the cost of multicast transmissions, in: Proceedings of 32nd ACM Symposium on Theory of Computing (STOC) ACM (2000) pp. 218–227.

  13. D. Fotakis, S. Kontogiannis, E. Koutsoupias, M. Mavronicolas and P. Spirakis, The structure and complexity of nash equilibria for a selfish routing game, in: Proceedings of the 29th International Colloquium on Automata. Languages and Programming (ICALP) LNCS, Springer (2002) pp. 123–134.

  14. M.R. Garey and D.S. Johnson, Computers and Intractability: A Guide to the Theory of NP-Completeness. W.H. Freeman and Co. (1979).

  15. K. Jain and V.V. Vazirani, Applications of approximation algorithms to cooperative games, in: Proceedings of 33rd ACM Symposium on Theory of Computing (STOC) ACM (2001) pp. 364–372.

  16. A. Kesselman, D. Kowaalski and M. Segal, Energy efficient communication in ad hoc networks from user’s and designer’s perspective, SIGMOBILE Mob. Comput. Commun. Rev. 9(1) (2005) 15–26.

    Google Scholar 

  17. E. Koutsoupias and C.H. Papadimitriou, Worst-case equilibria, in: Proceedings of the 16th Annual Symposium on Theoretical Aspects of Computer Science (STACS), Vol. 1563 of LNCS (1999) pp. 387–396.

  18. M. Mavronicolas and P. Spirakis, The price of selfish routing, in: Proceedings of the 33rd Annual ACM Symposium on the Theory of Computing (STOC) (2001) pp. 510–519.

  19. I. Milchtaich, Congestion games with player-specific payoff functions, Games and Economic Behavior 13 (1996) 111–124.

    Google Scholar 

  20. J.F. Nash, Equilibrium points in n-person games, in: Proceedings of the National Academy of Sciences, Vol. 36 (1950) pp. 48–49.

  21. P. Penna and C. Ventre, More powerful and simpler cost-sharing methods, in: Proceedings of the 2nd International Workshop on Approximation and Online Algorithms (WAOA), Vol. 3351 of LNCS, Springer (2004) pp. 97–110.

  22. P. Penna and C. Ventre, Sharing the cost of multicast transmissions in wireless networks, in: Proceedings of the 11th Colloquium on Structural Information and Communication Complexity (SIROCCO), Vol. 3104 of LNCS, Springer (2004) pp. 255–266.

  23. P. Penna and C. Ventre, Free-riders in steiner tree cost-sharing games, in: Proceedings of the 12th Colloquium on Structural Information and Communication Complexity (SIROCCO), Vol. 3499 of LNCS, Springer (2005) pp. 231–245.

  24. T.S. Rappaport, Wireless Communications: Principles and Practice (Prentice-Hall, Englewood Cliffs, NY, 1996).

  25. R.W. Rosenthal, A class of games possessing pure-strategy nash equilibria, International Journal of Game Theory 2 (1973) 65–67.

    Google Scholar 

  26. T. Roughgarden and E. Tardos, How bad is selfish routing?, Journal of ACM 49(2) (2002) 236–259.

    Google Scholar 

  27. L.S. Shapley, The value of n-person games, Contributions to the Theory of Games (Princeton University Press, 1953) pp. 31–40.

  28. A. Vetta, Nash equilibria in competitive societies, with applications to facility location, traffic routing and auctions, in: Proceedings of the 43rd Annual IEEE Symposium on Foundations of Computer Science (FOCS) (2002) pp. 416–425.

  29. J.E. Wieselthier, G.D. Nguyen and A. Ephremides, On the construction of energy-efficient broadcast and multicast trees in wireless networks, in: Proceedings of the 19th Annual Joint Conference of the IEEE Computer and Communications Societies (INFOCOM), IEEE Computer Society (2000) pp. 585–594.

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Authors and Affiliations

  1. Dipartimento di Matematica “Ennio De Giorgi”, Università di Lecce, Provinciale Lecce-Arnesano, P.O. Box 193, 73100, Lecce, Italy

    Vittorio Bilò

  2. Dipartimento di Informatica, Università di L’Aquila, Via Vetoio, Coppito, 67100, L’Aquila, Italy

    Vittorio Bilò, Michele Flammini, Giovanna Melideo & Luca Moscardelli

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  1. Vittorio Bilò
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  2. Michele Flammini
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  3. Giovanna Melideo
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  4. Luca Moscardelli
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Correspondence to Vittorio Bilò.

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Vittorio Bilò received the degree in Computer Science and the Ph.D. degree in Computer Science at the University of L’Aquila in 2001 and 2005 respectively. He is currently an assistant professor at the Department of Mathematics “Ennio De Giorgi” of the University of Lecce. His research interests include algorithms and computational complexity, communication problems in interconnection networks and game theoretical issues in non-cooperative networks.

Michele Flammini received the degree in Computer Science at the University of L’Aquila in 1990 and the Ph.D. degree in Computer Science at the University of Rome “La Sapienza” in 1995. He is full professor at the Computer Science Department of the University of L’Aquila since March 2005. His research interests include algorithms and computational complexity, game theory, communication problems in interconnection networks and routing. He has authored and co-authored more than 70 papers in his fields of interest published in the most reputed international conferences and journals.

Giovanna Melideo received the Laurea degree in computer science in 1997 from the University of L’Aquila (Italy) and the Ph.D. degree in computer engineering in 2001 from the University of Rome “La Sapienza”. From November 1999 to February 2000 she was visitor at IRISA/INRIA, University of Rennes 1, Rennes (France) in the ADP group. She was research fellow from June to October 2001 at the University of L’Aquila, where she is currently an assistant professor in the Computer Science Department. Her research interests include algorithms and complexity, algorithmic game theory, wireless networks, models for information integration and cooperative information systems, certification and security in e-service, distributed protocols and dependability. She has been a member of the scientific and organizing committee of the IFIP international workshop on Certification and Security in E-Services (CSES 2002), Montreal, Canada.

Luca Moscardelli received the degree in Computer Science at the University of L’Aquila in 2004. He is currently a Ph.D. student in Computer Science at the University of L’Aquila. His current research interests include optimization problems in social and communication networks, and the analysis of the interaction between selfish agents in non-cooperative networks.

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Bilò, V., Flammini, M., Melideo, G. et al. On Nash equilibria for multicast transmissions in ad-hoc wireless networks. Wireless Netw 14, 147–157 (2008). https://doi.org/10.1007/s11276-006-8817-y

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