Cooperative Game as Non-Additive Measure

Part of the Studies in Fuzziness and Soft Computing book series (STUDFUZZ, volume 310)

Abstract

This chapter surveys cooperative game theory as an important application based on non-additive measures. In ordinary cooperative game theory, it is implicitly assumed that all coalitions of N can be formed; however, this is in general not the case. Let us elaborate on this, and distinguish several cases: 1) Some coalitions may not be meaningful. 2) Coalitions may not be “black and white”. In order to deal with such situations, various generalizations/extensions of the theory have been proposed, e.g., bi-cooperative games, games on networks, games on combinatorial structures. We give a survey on values and interaction indices for these extended cooperative game theory.

Keywords

cooperative game bi-cooperative game network combinatorial structure value interaction index 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Aubin, J.P.: Optima and equilibria: An Introduction to Nonlinear Analysis. Springer (1998)Google Scholar
  2. 2.
    Banzhaf, J.F.: Weighted voting doesn’t work: A mathematical analysis. Rutgers Law Review 19, 317–343 (1965)Google Scholar
  3. 3.
    Bilbao, J.M.: Cooperative games on combinatorial structures. Kluwer Acadmic Publ., Boston (2000)CrossRefMATHGoogle Scholar
  4. 4.
    Bilbao, J.M., Fernández, J.R., Jiménez, N., López, J.J.: The core and the Weber set for bicooperative games. International Journal of Game Theory 36(2), 209–222 (2007)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Bilbao, J.M., Fernandez, J.R., Losada, A.J., Lebron, E.: Bicooperative games. In: GAMES 2000, Bilbao, Spain (July 2000)Google Scholar
  6. 6.
    Birkhoff, G.: Lattice Theory, 3rd edn. American Mathematical Society (1967)Google Scholar
  7. 7.
    Borm, P., Owen, G., Tijs, S.: On the position value for communication situations. SIAM Journal on Discrete Mathematics 3, 305–320 (1992)MathSciNetCrossRefGoogle Scholar
  8. 8.
    van den Brink, R., Dietz, C.: Multi-player agents in cooperative TU-games. Tinbergen Institute Discussion Paper TI 2012-001/1 (2011)Google Scholar
  9. 9.
    van den Brink, R., van der Laan, G., Pruzhansky, V.: Harsanyi power solutions for graph-restricted games. International Journal of Game Theory 40, 87–110 (2011)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Chateauneuf, A., Jaffray, J.Y.: Some characterizations of lower probabilities and other monotone capacities through the use of Möbius inversion. Mathematical Social Sciences 17(3), 263–283 (1989)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Curiel, I.: Cooperative game theory and applications. Kluwer Acadmic Publ., Boston (1997)CrossRefGoogle Scholar
  12. 12.
    Derks, J., Haller, H., Peters, H.: The selectope for cooperative TU-games. International Journal of Game Theory 29, 23–38 (2000)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Dubey, P., Neyman, A., Weber, R.J.: Value theory without efficiency. Mathematics of Operations Research 6, 122–128 (1981)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Davey, B.A., Priestley, H.A.: Introduction to Lattices and orders. Cambridge University Press (1990)Google Scholar
  15. 15.
    Felsenthal, D., Machover, M.: Ternary voting games. International Journal of Game Theory 26, 335–351 (1997)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Freeman, L.C.: Centrality in network: conceptual classification. Social Network 1, 215–239 (1979)CrossRefGoogle Scholar
  17. 17.
    Fujimoto, K.: Network-formation and its stability based on peripherality of nodes in social networks. Journal of Japan Society for Fuzzy Theory and Intelligent Informatics 24(4), 901–908 (2012) (in Japanese)Google Scholar
  18. 18.
    Fujimoto, K., Kojadinovic, I., Marichal, J.L.: Axiomatic characterizations of probabilistic and cardinal-probabilistic interaction indices. Games and Economic Behavior 55, 72–99 (2006)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Fujimoto, K., Honda, A.: A value via posets induced by graph-restricted communication situation. In: Proc. 2009 IFSA World Congress/2009 EUSFLAT Conference, Lisbon, Portugal, pp. 636–641 (2009)Google Scholar
  20. 20.
    Fujimoto, K., Murofushi, T.: Some Characterizations of k-Monotonicity Through the Bipolar Möbius Transform in Bi-Capacities. Journal of Advanced Computational Intelligence and Intelligent Informatics 9(5), 484–495 (2005)Google Scholar
  21. 21.
    Grabisch, M.: Modeling data by the Choquet integral. Information Fusion in Data Mining, pp. 135–148. Physica-Verlag, Heidelberg (2003)Google Scholar
  22. 22.
    Grabisch, M.: Capacities and Games on Lattices: A Survey of Results. International Journal of Uncertainty, Fuzziness, and Knowledge-Based Systems 14(4), 371–392 (2006)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Grabisch, M., Labreuche, C.: Bi-capacities for decision making on bipolar scales. In: EUROFUSE Workshop on Informations Systems, Varenna, Italy (September 2002)Google Scholar
  24. 24.
    Grabisch, M., Labreuche, C.: Bi-capacities — I: definition, Möbius transform and interaction. Fuzzy Sets and Systems 151, 211–236 (2005)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Grabisch, M., Labreuche, C.: Bi-capacities — II: the Choquet integral. Fuzzy Sets and Systems 151, 237–259 (2005)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Grabisch, M., Labreuche, C.: Derivative of functions over lattices as a basis for the notion of interaction between attributes. Annals of Mathematics and Artificial Intelligence 49, 151–170 (2007)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Grabisch, M., Roubens, M.: An axiomatic approach to the concept of interaction among players in cooperative games. International Journal of Game Theory 28(4), 547–565 (1999)MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Grabisch, M., Marichal, J.-L., Roubens, M.: Equivalent representations of set functions. Mathematics of Operations Research 25(2), 157–178 (2000)MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Grabisch, M., Roubens, M.: Application of the Choquet integral in multicriteria decision making. In: Fuzzy Measures and Integrals, pp. 348–374. Physica Verlag, Heidelberg (2000)Google Scholar
  30. 30.
    Grabisch, M., Xie, L.: The core of games on distributive lattices: how to share benefits in a hierarchy. Document de Travail du Centre d’Economie de la Sorbonne (2008.77) (2008)Google Scholar
  31. 31.
    Hamiache, G.: A value with incomplete information. Games and Economic Behavior 26, 59–78 (1999)MathSciNetCrossRefMATHGoogle Scholar
  32. 32.
    Harsanyi, J.C.: A bargaining model for the cooperative n-person game. In: Contributions to the Theory of Games IV, pp. 325–355. Princeton University Press, Princeton (1959)Google Scholar
  33. 33.
    Harsanyi, J.C.: A simplified bargaining model for the n-person cooperative game. International Economic Review 4, 59–78 (1963)CrossRefGoogle Scholar
  34. 34.
    Honda, A., Fujimoto, K.: A generalization of cooperative games and its solution concept. Journal of Japan Society for Fuzzy Theory and Intelligent Informatics 21(4), 491–499 (2009) (in Japanese)Google Scholar
  35. 35.
    Honda, A., Grabish, M.: Entropy of capacities on lattices. Information Sciences 176, 3472–3489 (2006)MathSciNetCrossRefMATHGoogle Scholar
  36. 36.
    Hsiao, C.R., Raghavan, T.E.S.: Shapley value for multi-choice cooperative game. Games and Economic Behavior 5, 240–256 (1993)MathSciNetCrossRefMATHGoogle Scholar
  37. 37.
    Jackson, O.M.: Allocation rules for network games. Games and Economic Behavior 51, 128–154 (2005)MathSciNetCrossRefMATHGoogle Scholar
  38. 38.
    Kalai, E., Samet, D.: Weighted Shapley values. In: The Shapley Value. Essays in Honor of Lloyd S. Shapley, pp. 83–99. Cambridge Univ. Press (1988)Google Scholar
  39. 39.
    Kojadinovic, I.: Modeling interaction phenomena using non additive measures: Applications in data analysis. PhD thesis, Université de La Réunion, France (2002)Google Scholar
  40. 40.
    Kojadinovic, I.: Estimation of the weights of interacting criteria from the set of profiles by means of information-theoretic functionals. Europ. J. Operational Res. 155, 741–751 (2004)MathSciNetCrossRefMATHGoogle Scholar
  41. 41.
    Kojadinovic, I.: A weigh-based approach to the measurement of the interaction among criteria in the framework of aggregation by the bipolar Choquet integral. European Journal of Operational Research 179, 498–517 (2007)MathSciNetCrossRefMATHGoogle Scholar
  42. 42.
    Labreuche, C., Grabisch, M.: Modeling positive and negative pieces of evidence in uncertainty. In: Nielsen, T.D., Zhang, N.L. (eds.) ECSQARU 2003. LNCS (LNAI), vol. 2711, pp. 279–290. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  43. 43.
    Labreuche, C., Grabisch, M.: Axiomatisation of the Shapley value and power index for bi-cooperative games. Cahiers de la Maison des Sciences Économiques (2006.23) (2006)Google Scholar
  44. 44.
    Lange, F., Grabisch, M.: Games on distributive lattices and the Shapley interaction transform. In: Proc. of IPMU 2008, Torremplinos (Málaga), Spain, pp. 1462–1469 (2008)Google Scholar
  45. 45.
    Lange, F., Grabisch, M.: New axiomatizations of the Shapley interaction index for bi-capacities. Fuzzy Sets and Systems 176, 64–75 (2011)Google Scholar
  46. 46.
    Lange, F., Grabisch, M.: The interaction transform for functions on lattices. Discrete Mathematics 309, 4037–4048 (2009)MathSciNetCrossRefMATHGoogle Scholar
  47. 47.
    Lange, F., Grabisch, M.: Values on regular games under Kirchhoff’s laws. Mathematical Social Sciences 58, 322–340 (2009)MathSciNetCrossRefMATHGoogle Scholar
  48. 48.
    Marichal, J.-L.: Aggregation operators for multicriteria decision aid, Ph.D. thesis, University of Liège (1998)Google Scholar
  49. 49.
    Marichal, J.-L., Roubens, M.: The chaining interaction index among players in cooperative games. In: Meskens, N., Roubens, M. (eds.) Advances in Decision Analysis. Kluwer Acad. Publ., Dordrecht (1999)Google Scholar
  50. 50.
    Murofushi, T., Soneda, S.: Techniques for reading fuzzy measures (iii): interaction index. In: 9th Fuzzy System Symposium, Sapporo, Japan, pp. 693–696 (1993) (in Japanese)Google Scholar
  51. 51.
    Murofushi, T., Sugeno, M.: Fuzzy measures and fuzzy integrals. In: Grabisch, et al. (eds.) Fuzzy Measures and Integrals. Physica-Verlag, Heidelberg (2000)Google Scholar
  52. 52.
    Myerson, R.: Graphs and cooperation in games. Mathematics of Operations Research 2, 225–229 (1977)MathSciNetCrossRefMATHGoogle Scholar
  53. 53.
    Owen, G.: Multilinear extensions of games. Management Science 18, 64–79 (1972)MathSciNetCrossRefMATHGoogle Scholar
  54. 54.
    Peleg, B., Shudhölter, P.: Introduction to the theory of cooperative games, 2nd edn. Springer, Heidelberg (2007)MATHGoogle Scholar
  55. 55.
    Ramón, J., Mateo, S.C.: Multi Criteria Analysis in the Renewable Energy Industry. Springer, London (2012)Google Scholar
  56. 56.
    Rota, G.C.: On the foundations of combinatorial theory I. Theory of Möbius functions. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 2, 340–368 (1964)MathSciNetCrossRefMATHGoogle Scholar
  57. 57.
    Roubens, M.: Interaction between criteria and definition of weights in MCDA problems. In: Proc. 44th Meeting of the European Working Group “Multiple Criteria Decision Aiding” (1996)Google Scholar
  58. 58.
    Shafer, G.: A Mathematical Theory of Evidence. Princeton University Press (1976)Google Scholar
  59. 59.
    Shapley, L.: A value for n-person games. In: Tucker, A.W., Kuhn, H. (eds.) Contribution to the Theory of Games, II, Annals of Mathematics Studies, vol. 28, pp. 307–317. Princeton Univ. Press, Princeton (1953)Google Scholar
  60. 60.
    Shapley, L.S., Shubik, M.: A Method for Evaluating the Distribution of Power in a Committee System. American Political Science Review 48, 787–792 (1954)CrossRefGoogle Scholar
  61. 61.
    Slikker, M., van den Nouweland, A.: Social and economic networks in cooperative game theory. Kluwer Academic Publ., Boston (2001)CrossRefGoogle Scholar
  62. 62.
    Sugeno, M.: Theory of fuzzy integrals and its applications. Doctoral Thesis, Tokyo Institute of Technology (1974)Google Scholar
  63. 63.
    Vasil’ev, V.A.: On a class of operators in a space of regular set functions. Optimizacija 28, 102–111 (1982) (in Russian)Google Scholar
  64. 64.
    Vasil’ev, V.A.: Extreme points of the Weber polytope. Discretnyi Analiz i Issledonaviye Operatsyi Ser.1 10, 17–55 (2003) (in Russian)Google Scholar
  65. 65.
    Weber, R.J.: Probabilistic values for games. In: Roth, A.E. (ed.) The Shapley value. Essays in honor of Lloyd S. Shapley, pp. 101–120. Cambridge University Press, Cambridge (1988)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  1. 1.College of Symbiotic Systems ScienceFukushima UniversityFukushimaJapan

Personalised recommendations