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, 16:1 | Cite as

Transversality of the Shapley value

  • Stefano MorettiEmail author
  • Fioravante Patrone
Invited Paper
First Online:

Abstract

A few applications of the Shapley value are described. The main choice criterion is to look at quite diversified fields, to appreciate how wide is the terrain that has been explored and colonized using this and related tools.

Keywords

Coalitional game Shapley value Applied game theory Axiomatizations Game practice 

Mathematics Subject Classification (2000)

91-02 91A12 91A80 
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References

  1. Arrow KJ (1951) Social choice and individual values. Wiley, New York. 2nd edn: 1963 Google Scholar
  2. Aubin JP (1981) Cooperative fuzzy games. Math Oper Res 6:1–13 Google Scholar
  3. Aumann RJ, Dréze JH (1974) Cooperative games with coalition structure. Int J Game Theory 3:217–237 CrossRefGoogle Scholar
  4. Aumann RJ, Hart S (eds) (2002) Handbook of game theory, with economic applications, vol 3. North-Holland, Amsterdam Google Scholar
  5. Aumann RJ, Shapley LS (1974) Values of non-atomic games. Princeton University Press, Princeton Google Scholar
  6. Baker M, Associates (1965) Runaway cost impact study. Report presented to the Association of Local Transport Airlines, Jackson, MI Google Scholar
  7. Banzhaf JF III (1965) Weighted voting doesn’t work: a game theoretic approach. Rutgers Law Rev 19:317–343 Google Scholar
  8. Barlow RE, Proschan F (1975) Importance of system components and fault tree event. Stoch Process Appl 3:153–172 CrossRefGoogle Scholar
  9. Baumgartner JP (1997) Ordine di grandezza di alcuni costi delle ferrovie. Ing Ferrov 7:459–469 Google Scholar
  10. Birnbaum ZW (1969) On the importance of different components in a multicomponent system. In: Krishnaiah PR (ed) Multivariate analysis II. Academic, New York Google Scholar
  11. Branzei R, Dimitrov D, Tijs S (2005) Models in cooperative game theory: crisp, fuzzy, and multi-choice games. Springer, Berlin Google Scholar
  12. Cohen S, Dror G, Ruppin E (2005) Feature selection based on the Shapley value. In: International joint conference on artificial intelligence, pp 665–670 Google Scholar
  13. Cox LA Jr (1985) A new measure of attributable risk for public health applications. Manag Sci 31:800–813 Google Scholar
  14. Dinar A, Kannai Y, Yaron D (1986) Sharing regional cooperative gains from reusing effluent for irrigation. Water Resour Res 22:339–344 CrossRefGoogle Scholar
  15. Dubey P (1975) On the uniqueness of the Shapley value. Int J Game Theory 4:131–139 CrossRefGoogle Scholar
  16. Dubey P (1982) The Shapley value as aircraft landing fees–revisited. Manag Sci 20:869–874 CrossRefGoogle Scholar
  17. Dubois D, Prade H (1982) On several representations of an uncertain body of evidence. In: Gupta MM, Sanchez E (eds) Fuzzy information and decision processes. North-Holland, Amsterdam, pp 167–181 Google Scholar
  18. Dubois D, Prade H (2002) Quantitative possibility theory and its probabilistic connections. In: Grzegorzewski P et al. (eds) Soft methods in probability, statistics and data analysis. Physica Verlag, Heidelberg, pp 3–26 Google Scholar
  19. Fragnelli V, García-Jurado I, Norde H, Patrone F, Tijs S (1999) How to share railways infrastructure costs? In: Patrone F, García-Jurado I, Tijs S (eds) Game practice: contributions from applied game theory. Kluwer Academic, Dordrecht, pp 91–101 Google Scholar
  20. Freixas J, Puente MA (2002) Reliability importance measures of the components in a system based on semivalues and probabilistic values. Ann Oper Res 109:331–342 CrossRefGoogle Scholar
  21. Gefeller O, Land M, Eide GE (1998) Averaging attributable fractions in the multifactorial situation: assumptions and interpretation. J Clin Epidemiol 51:437–441 CrossRefGoogle Scholar
  22. Gillies DB (1953) Some theorems on n-person games. PhD thesis, Department of Mathematics, Princeton University, Princeton Google Scholar
  23. Gómez D, González-Arangüena E, Manuel C, Owen G, del Pozo M, Tejada J (2003) Centrality and power in social networks: a game theoretic approach. Math Soc Sci 46:27–54 CrossRefGoogle Scholar
  24. González P, Herrero C (2004) Optimal sharing of surgical costs in the presence of queues. Math Methods Oper Res 59:435–446 Google Scholar
  25. Grabisch M (2000) Fuzzy integral for classification and feature extraction. In: Grabisch M, Murofushi T, Sugeno M (eds) Fuzzy measures and integrals: theory and applications. Physica-Verlag, New York, pp 415–434 Google Scholar
  26. Grabisch M, Roubens M (1999) An axiomatic approach to the concept of interaction among players in cooperative games. Int J Game Theory 28:547–565 CrossRefGoogle Scholar
  27. Haake CJ, Kashiwada A, Su FE (2005) The Shapley value of phylogenetic trees. J Math Biol 56:479–497, 2008 CrossRefGoogle Scholar
  28. Harsanyi JC (1959) A bargaining model for cooperative n-person games. In: Tucker AW, Luce RD (eds) Contributions to the theory of games IV. Princeton UP, Princeton, pp 325–355 Google Scholar
  29. Harsanyi JC (1963) A simplified bargaining model for the n-person cooperative game. Int Econ Rev 4:194–220 CrossRefGoogle Scholar
  30. Hart S (2002) Values of perfectly competitive economies. In: Aumann RJ, Hart S (eds) Handbook of game theory, with economic applications, vol 3. North-Holland, Amsterdam, pp 2169–2184, Chap 57 Google Scholar
  31. Hart S (2006) Cooperative games: value theory. On web page: http://www.ma.huji.ac.il/~hart/value.html. Last updated 05/07/2006
  32. Hart S, Mas-Colell A (1987) Potential, value and consistency. Econometrica 57:589–614 CrossRefGoogle Scholar
  33. Hartmann K, Steel M (2006) Maximizing phylogenetic diversity in biodiversity conservation: greedy solutions to the Noah’s Ark problem. Syst Biol 55:644–651 CrossRefGoogle Scholar
  34. Kalai E, Samet D (1988) Weighted Shapley values. In: Roth AE (ed) The Shapley value. Cambridge University Press, Cambridge, pp 83–100 Google Scholar
  35. Kargin V (2005) Uncertainty of the Shapley value. Int Game Theory Rev 33:959–976 Google Scholar
  36. Kaufman A, Kupiec M, Ruppin E (2004a) Multi-knockout genetic network analysis: the rad6 example. In: Proceedings of the 2004 IEEE computational systems bioinformatics conference (CSB’04), August 16–19, 2004, Standford, California Google Scholar
  37. Kaufman A, Keinan A, Meilijson I, Kupiec M, Ruppin E (2004b) Quantitative analysis of genetic and neuronal multi-perturbation experiments. PLoS Comput Biol 1(6):e64 CrossRefGoogle Scholar
  38. Keinan A, Sandbank B, Hilgetag CC, Meilijson I, Ruppin E (2004) Fair attribution of functional contribution in artificial and biological networks. Neural Comput 16:1887–1915 CrossRefGoogle Scholar
  39. Land M, Gefeller O (1997) A game-theoretic approach to partitioning attributable risks in epidemiology. Biom J 39(7):777–792 CrossRefGoogle Scholar
  40. Land M, Gefeller O (2000) A multiplicative variant of the Shapley value for factorizing the risk of disease. In: Patrone F, Garcia-Jurado I, Tijs S (eds) Game practice: contributions from applied game theory. Kluwer Academic, Dordrecht, pp 143–158 Google Scholar
  41. Littlechild S, Owen G (1973) A simple expression for the Shapley value in a special case. Manag Sci 2:370–372 Google Scholar
  42. Littlechild S, Thompson G (1977) Aircraft landing fees. A game theory approach. Bell J Econ 8:186–204 CrossRefGoogle Scholar
  43. Loehman E, Whinston A (1976) A generalized cost allocation scheme. In: Lin SAY (ed) Theory and measurement of economic externalities. Academic, New York, pp 87–101 Google Scholar
  44. Loehman E, Orlando J, Tschirhart J, Whinston A (1979) Cost allocation for a regional wastewater treatment system. Water Resour Res 15:193–202 CrossRefGoogle Scholar
  45. Luce RD, Raiffa H (1957) Games and decisions. Wiley, New York Google Scholar
  46. McLean RP (2002) Values of non-transferable utility games. In: Aumann RJ, Hart S (eds) Handbook of game theory, with economic applications, vol 3. North-Holland, Amsterdam, pp 2077–2120 Google Scholar
  47. Mertens J-F (2002) Some other economic applications of the value. In: Aumann RJ, Hart S (eds) Handbook of game theory, with economic applications, vol 3. North-Holland, Amsterdam, pp 2185–2201, Chap 58 Google Scholar
  48. Monderer D, Samet D (2002) Variations on the Shapley value. In: Aumann RJ, Hart S (eds) Handbook of game theory, with economic applications, vol 3. North-Holland, Amsterdam, pp 2055–2076, Chap 54 Google Scholar
  49. Moretti S, Patrone F (2004) Cost allocation games with information costs. Math Methods Oper Res 59:419–434 Google Scholar
  50. Moretti S, Patrone F, Bonassi S (2007) The class of microarray games and the relevance index for genes. Top 15:256–280 CrossRefGoogle Scholar
  51. Myerson RB (1977) Graphs and cooperation in games. Math Oper Res 2:225–229 Google Scholar
  52. Myerson RB (1980) Conference structures and fair allocation rules. Int J Game Theory 9:169–182 CrossRefGoogle Scholar
  53. Myerson RB (1991) Game theory: analysis of conflict. Harvard University Press, Cambridge Google Scholar
  54. Nash JF Jr (1950) The bargaining problem. Econometrica 18:155–162 CrossRefGoogle Scholar
  55. Neyman A (2002) Values of games with infinitely many players. In: Aumann RJ, Hart S (eds) Handbook of game theory, with economic applications, vol 3. North-Holland, Amsterdam, pp 2121–2168, Chap 56 Google Scholar
  56. Osborne M, Rubinstein A (1994) A course in game theory. MIT Press, Cambridge Google Scholar
  57. Owen G (1977) Values of games with a priori unions. In: Henn R, Moeschlin O (eds) Essays in mathematical economics and game theory. Springer, New York Google Scholar
  58. Owen G (1995) Game theory, 3rd edn. Academic, San Diego. 1st edn 1968 Google Scholar
  59. Ramamurthy KG (1990) Coherent structures and simple games. Kluwer Academic, Dordrecht Google Scholar
  60. Roth AE (1977) The Shapley value as a von Neumann–Morgenstern utility. Econometrica 45:657–664 CrossRefGoogle Scholar
  61. Roth AE (ed) (1988a) The Shapley value, essays in honor of Lloyd S. Shapley. Cambridge University Press, Cambridge Google Scholar
  62. Roth AE (1988b) The expected value of playing a game. In: Roth AE (ed) The Shapley value. University Press, Cambridge, pp 51–70 Google Scholar
  63. Shafer G (1976) A mathematical theory of evidence. Princeton University Press, Princeton Google Scholar
  64. Shapley LS (1953) A value for n-person games. In: Kuhn HW, Tucker AW (eds) Contributions to the theory of games II. Annals of mathematics studies, vol 28. Princeton University Press, Princeton, pp 307–317. Reprinted in: Roth AE ed (1988a), pp 31–40 Google Scholar
  65. Shapley LS (1969) Utility comparison and the theory of games. In: Guilbaud G (ed) La décision: aggrégation et dynamique des ordres de préference. Editions du CNRS, Paris, pp 251–263. Reprinted in: Roth AE ed (1988a), pp 307–319 Google Scholar
  66. Shapley LS, Shubik M (1954) A method for evaluating the distribution of power in a committee system. Am Polit Sci Rev 48:787–792 CrossRefGoogle Scholar
  67. Smets P (1990) Constructing the pignistic probability function in a context of uncertainty. In: Henrion M et al. (eds) Uncertainty in artificial intelligence, vol 5. North-Holland, Amsterdam, pp 29–39 Google Scholar
  68. Smets P (1997) The normative representation of quantifield beliefs by belief functions. Artif Intell 92:229–242 CrossRefGoogle Scholar
  69. Smets P (2005) Decision making in the TBM: the necessity of the pignistic transformation. Int J Approx Reason 38:133–147 CrossRefGoogle Scholar
  70. Smets P, Kennes R (1994) The transferable belief model. Artif Intell 66:191–234 CrossRefGoogle Scholar
  71. Straffin PD (1976) Power indices in politics. MAA modules in applied mathematics. Cornell University, Ithaca Google Scholar
  72. Suijs J, Borm P (1999) Stochastic cooperative games: superadditivity, convexity, and certainty equivalents. Games Econ Behav 27:331–345 CrossRefGoogle Scholar
  73. Suijs J, Borm P, De Waegenaere A, Tijs S (1999) Cooperative games with stochastic payoffs. Eur J Oper Res 113:193–205 CrossRefGoogle Scholar
  74. Thompson G (1971) Airport costs and pricing. Unpublished PhD dissertation, University of Birmingham Google Scholar
  75. Timmer J, Borm P, Tijs S (2004) On three Shapley-like solutions for cooperative games with random payoffs. Int J Game Theory 32:595–613 CrossRefGoogle Scholar
  76. Vázquez-Brage M, van den Nouweland A, García-Jurado I (1997) Owen’s coalitional value and aircraft landing fees. Math Soc Sci 34:273–286 CrossRefGoogle Scholar
  77. von Neumann J (1928) Zur Theorie der Gesellschaftsspiele. Mat Ann 100:295–320 CrossRefGoogle Scholar
  78. Weitzman ML (1998) The Noah’s ark problem. Econometrica 66:1279–1298 CrossRefGoogle Scholar
  79. Winter E (2002) The Shapley value. In: Aumann RJ, Hart S (eds) Handbook of game theory, with economic applications, vol 3. North-Holland, Amsterdam, pp 2025–2054, Chap 53 Google Scholar
  80. Young HP (1985) Monotonic solutions of cooperative games. Int J Game Theory 14:65–72 CrossRefGoogle Scholar
  81. Young HP (1988) Individual contribution and just compensation. In: Roth AE (ed) The Shapley value. Cambridge University Press, Cambridge, pp 267–278 Google Scholar
  82. Young HP (1994) Cost allocation. In: Aumann RJ, Hart S (eds) Handbook of game theory, with economic applications, vol 2. North-Holland, Amsterdam, pp 1193–1235, Chap 34 Google Scholar
  83. Young HP, Okada N, Hashimoto T (1982) Cost allocation in water resources development. Water Resour Res 18:361–373 Google Scholar

Copyright information

© Sociedad de Estadística e Investigación Operativa 2008

Authors and Affiliations

  1. 1.Unit of Molecular EpidemiologyNational Cancer Research InstituteGenoaItaly
  2. 2.DIPTEMUniversity of GenovaGenoaItaly

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