Advertisement

Solutions of Partition Function-Based TU Games for Cooperative Communication Networking

  • Giovanni RossiEmail author
Conference paper
Part of the Lecture Notes in Networks and Systems book series (LNNS, volume 70)

Abstract

In networked communications nodes choose among available actions and benefit from exchanging information through edges, while continuous technological progress fosters system functionings that increasingly often rely on cooperation. Growing attention is being placed on coalition formation, where each node chooses what coalition to join, while the surplus generated by cooperation is an amount of TU (transferable utility) quantified by a real-valued function defined on partitions -or even embedded coalitions- of nodes. A TU-sharing rule is thus essential, as how players are rewarded determines their behavior. This work offers a new option for distributing partition function-based surpluses, dealing with cooperative game theory in terms of both global games and games in partition function form, namely lattice functions, while the sharing rule is a point-valued solution or value. The novelty is grounded on the combinatorial definition of such solutions as lattice functions whose Möbius inversion lives only on atoms, i.e. on the first level of the lattice. While rephrasing the traditional solution concept for standard coalitional games, this leads to distribute the surplus generated by partitions across the edges of the network, as the atoms among partitions are unordered pairs of players. These shares of edges are further divided between nodes, but the corresponding Shapley value is very different from the traditional one and leads to two alternative forms, obtained by focusing either on marginal contributions along maximal chains, or else on the uniform division of Harsanyi dividends. The core is also addressed, and supermodularity is no longer sufficient for its non-emptiness.

Keywords

Networked communications Cooperative game theory Partition function Shapley value Lattice Möbius inversion 

References

  1. 1.
    Han, Z., Niyato, D., Saad, W., Başar, T., Hjørungnes, A.: Game Theory for Wireless and Communication Networks. Cambridge University Press, Cambridge (2012)zbMATHGoogle Scholar
  2. 2.
    Saad, W., Han, Z., Debbah, M., Hjørungnes, A., Başar, T.: Coalitional game theory for communication networks: a tutorial, pp. 1–26. arXiv:0905.4057v1 (2009)CrossRefGoogle Scholar
  3. 3.
    Slikker, M.: Coalition formation and potential games. Games Econ. Behav. 37, 436–448 (2001)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Aigner, M.: Combinatorial Theory. Springer, Berlin (1997)CrossRefGoogle Scholar
  5. 5.
    Roth, A. (ed.): The Shapley Value-Essays in Honor of Lloyd S. Shapley. Cambridge University Press, Cambridge (1988)zbMATHGoogle Scholar
  6. 6.
    Akyildiz, I., Lo, B., Balakrishnan, R.: Cooperative spectrum sensing in cognitive radio networks: a survey. Phys. Commun. 4, 40–62 (2011)CrossRefGoogle Scholar
  7. 7.
    Laneman, J., Wornell, G.: Distributed space-time-coded protocols for exploiting cooperative diversity in wireless network. IEEE Trans. Inf. Theory 49, 2415–2425 (2003)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Letaief, K., Zhang, W.: Cooperative spectrum sensing. In: Hossain, E., Bhargava, V. (eds.) Cognitive Wireless Communication Networks, pp. 115–138. Springer, Berlin (2007)CrossRefGoogle Scholar
  9. 9.
    Zhou, Z., Zhou, S., Cui, S., Cui, J.H.: Energy-effiecient cooperative communication in clustered wireless sensor networks. IEEE Trans. Veh. Technol. 57, 3618–3628 (2008)CrossRefGoogle Scholar
  10. 10.
    Al-Karaki, J.L., Kamal, A.E.: Routing techniques in wireless sensor networks: a survey. IEEE Wirel. Commun. 11, 6–28 (2004)CrossRefGoogle Scholar
  11. 11.
    Cai, J., Pooch, U.: Allocate fair payoff for cooperation in wireless ad hoc networks using Shapley value. In: Proceedings of IPDPS 04, pp. 219–226 (2004)Google Scholar
  12. 12.
    Cavalcanti, D., Agrawal, D., Cordero, C., Xie, B., Kumar, A.: Issues in integrating cellular networks, WLANs and MANETs: a futuristic heterogeneous wireless network. IEEE Wirel. Commun. 12, 30–41 (2005)CrossRefGoogle Scholar
  13. 13.
    Cho, J.H., Swami, A., Chen, I.R.: A survey on trust management for mobile ad hoc networks. IEEE Commun. Surv. Tutorials 13(4), 562–583 (2011)CrossRefGoogle Scholar
  14. 14.
    Chow, C.Y., Leong, H.V., Chan, A.T.S.: GroCoca: group-based peer-to-peer cooperative caching in mobile environment. IEEE J. SAC 25(1), 179–191 (2007)Google Scholar
  15. 15.
    Huang, X., Zhai, H., Fang, Y.: Robust cooperative routing protocol in mobile wireless sensor networks. IEEE Trans. Wirel. Commun. 7(12), 5278–5285 (2008)CrossRefGoogle Scholar
  16. 16.
    Ye, M., Li, C., Chen, G., Wu, J.: An energy efficient clustering scheme in wireless sensor networks. Ad Hoc Sens. Wirel. Netw. 3, 99–119 (2007)Google Scholar
  17. 17.
    Nandan, A., Das, S., Pau, G., Gerla, M., Sanadidi, M.: Co-operative downloading in vehicular ad-hoc wireless networks. In Proceedings of WONS 05, pp. 32–41 (2005)Google Scholar
  18. 18.
    Vardhe, K., Reynolds, D., Woerner, B.: Joint power allocation and relay selection for multiuser cooperative communication. IEEE Trans. Wirel. Commun. 9(4), 1255–1260 (2010)CrossRefGoogle Scholar
  19. 19.
    Wang, B., Han, Z., Liu, K.: Distributed relay selection and power control for multiuser cooperative communication networks using buyer/seller game. In: IEEE INFOCOM 2007 Proceedings, pp. 544–552 (2007)Google Scholar
  20. 20.
    Wang, T., Giannakis, G.B.: Complex field network coding for multiuser cooperative communications. IEEE J. SAC 26(3), 561–571 (2008)Google Scholar
  21. 21.
    Younis, O., Fahmy, S.: HEED a hybrid, energy-efficient, distributed clustering approach for ad hoc sensor networks. IEEE Trans. Mob. Comput. 3(4), 366–379 (2004)CrossRefGoogle Scholar
  22. 22.
    Younis, O., Krunz, M., Ramasubramanian, S.: Node clustering in wireless sensor networks: recent developments and deployment challenges. IEEE Netw. 20, 20–25 (2006)CrossRefGoogle Scholar
  23. 23.
    Fitzek, F.H.P., Katz, M.D. (eds.): Cooperation in Wireless Networks: Principles and Applications-Real Egoistic Behavior is to Cooperate!. Springer, Berlin (2006)Google Scholar
  24. 24.
    Saad, W., Han, Z., Başar, T., Debbah, M., Hjørungnes, A.: Coalition formation games for collaborative spectrum sensing. IEEE Trans. Veh. Technol. 60, 276–297 (2011)CrossRefGoogle Scholar
  25. 25.
    Saad, W., Han, Z., Zheng, R., Hjørungnes, A., Başar, T., Poor, H.V.: Coalitional games in partition form for joint spectrum sensing and access in cognitive radio networks. IEEE J. Sel. Top. Sig. Proc. 6, 195–209 (2012)CrossRefGoogle Scholar
  26. 26.
    Borm, P., Owen, G., Tijs, S.: On the position value for communication situations. SIAM J. Discrete Math. 5, 305–320 (1992)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Gilboa, I., Lehrer, E.: Global games. Int. J. Game Theory 20, 120–147 (1990)MathSciNetzbMATHGoogle Scholar
  28. 28.
    Rossi, G.: Worth-sharing through Möbius inversion. Homo Economicus 24, 411–433 (2007)Google Scholar
  29. 29.
    Thrall, R.M., Lucas, W.F.: \(n\)-person games in partition function form. Naval Res. Logistic Q. 10, 281–298 (1963)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Grabisch, M., Funaki, Y.: A coalition formation value for games in partition function form. Eur. J. Oper. Res. 221(1), 175–185 (2012)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Myerson, R.: Values of games in partition function form. Int. J. Game Theory 6, 23–31 (1977)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Rossi, G.: The geometric lattice of embedded subsets, pp. 1–17. arXiv: 1612.05814 (2017)
  33. 33.
    Whitney, H.: On the abstract properties of linear dependence. Am. J. Math. 57, 509–533 (1935)MathSciNetCrossRefGoogle Scholar
  34. 34.
    Rosas, M.H., Sagan, B.E.: Symmetric functions in noncommuting variables. Trans. AMS 358, 215–232 (2006)MathSciNetCrossRefGoogle Scholar
  35. 35.
    Conitzer, V., Sandholm, T.: Complexity of constructing solutions in the core based on synergies among coalitions. Artif. Intell. 170, 607–619 (2006)MathSciNetCrossRefGoogle Scholar
  36. 36.
    Shapley, L.S.: Cores of convex games. Int. J. Game Theory 1(1), 11–26 (1971)MathSciNetCrossRefGoogle Scholar
  37. 37.
    Fortunato, S.: Community detection in graphs. Phys. Rep. 486(3–5), 75–174 (2010)MathSciNetCrossRefGoogle Scholar
  38. 38.
    Gilboa, I., Lehrer, E.: The value of information—an axiomatic approach. J. Math. Econ. 20(5), 443–459 (1991)MathSciNetCrossRefGoogle Scholar
  39. 39.
    Rota, G.C.: On the foundations of combinatorial theory I: theory of Möbius functions. Z. Wahrscheinlichkeitsrechnung Verw. Geb. 2, 340–368 (1964)Google Scholar
  40. 40.
    Weber, R.J.: Probabilistic values for games. In: A.E. Roth (ed.) The Shapley Value—Essays in Honor of Lloyd S. Shapley, pp. 101–119. Cambridge University Press, Cambridge (1988)Google Scholar
  41. 41.
    Grabisch, M.: The lattice of embedded subsets. Discrete Appl. Math. 158, 479–488 (2010)MathSciNetCrossRefGoogle Scholar
  42. 42.
    Shapley, L.S.: A value for \(n\)-person games. In: Kuhn, H., Tucker, A.W. (eds.) Contributions to the Theory of Games, pp. 307–317. Princeton University Press, Princeton (1953)Google Scholar
  43. 43.
    Stanley, R.: Enumerative Combinatorics, 2nd edn. Cambridge University Press, Cambridge (2012)zbMATHGoogle Scholar
  44. 44.
    Knuth, D.E.: Generating all combinations and partitions. The Art of Computer Programming, vol. 4, no. 3, pp. 1–150. Addison-Wesley (2005)Google Scholar
  45. 45.
    Kung, J.P.S., Rota, G.C., Yan, C.H. (eds.): Combinatorics: The Rota Way. Cambridge University Press, Cambridge (2009)zbMATHGoogle Scholar
  46. 46.
    Rahwan, T., Jenning, N.: An algorithm for distributing coalitional value calculations among cooperating agents. Artif. Intell. 171, 535–567 (2007)MathSciNetCrossRefGoogle Scholar
  47. 47.
    Korte, B., Vygen, J.: Combinatorial Optimization-Theory and Algorithms. Springer, Berlin (2002)zbMATHGoogle Scholar
  48. 48.
    Myerson, R.: Graphs and cooperation in games. Math. Oper. Res. 2, 225–229 (1977)MathSciNetCrossRefGoogle Scholar
  49. 49.
    Owen, G.: Values of graph-restricted games. SIAM J. ADM 7(2), 210–220 (1986)MathSciNetzbMATHGoogle Scholar
  50. 50.
    Diestel, R.: Graph Theory. Springer, Berlin (2010)CrossRefGoogle Scholar
  51. 51.
    Azrieli, Y., Lehrer, E.: Concavification and convex games. Working Paper, Tel Aviv University, pp. 0–18 (2005)Google Scholar
  52. 52.
    Holzman, R., Lehrer, E., Linial, N.: Some bounds for the Banzhaf index and other semivalues. Math. Oper. Res. 13, 358–363 (1988)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  1. 1.Department of Computer Science and Engineering DISIUniversity of BolognaBolognaItaly

Personalised recommendations