Game Theory pp 210-216 | Cite as

Shapley Value

  • Sergiu Hart
Part of the The New Palgrave book series


The value of an uncertain outcome (a ‘gamble’, ‘lottery’, etc.) to a participant is an evaluation, in the participant’s utility scale, of the prospective outcomes: It is an a priori measure of what he expects to obtain (this is the subject of ‘utility theory’). In a similar way, one is interested in evaluating a game; that is, measuring the value of each player in the game.


Game Theory Coalition Structure Marginal Contribution Grand Coalition Cost Allocation 
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  1. Artstein, Z. 1971. Values of games with denumerably many players. International Journal of Game Theory 1, 27–37.CrossRefGoogle Scholar
  2. Aumann, R.J. 1975. Values of markets with a continuum of traders. Econometrica 43, 611–46.CrossRefGoogle Scholar
  3. Aumann, R.J. 1978. Recent developments in the theory of the Shapley Value. Proceedings of the International Congress of Mathematicians, Helsinki, 995–1003.Google Scholar
  4. Aumann, R.J. 1985. On the non-transferable utility value: a comment on the Roth—Shafer examples. Econometrica 53, 667–78.CrossRefGoogle Scholar
  5. Aumann, R.J. and Drize, J.H. 1974. Cooperative games with coalition structures. International Journal of Game Theory 3, 217–37.CrossRefGoogle Scholar
  6. Aumann, R.J. and Kurz, M. 1977. Power and taxes. Econometrica 45, 1137–61.CrossRefGoogle Scholar
  7. Aumann, R.J. and Shapley, L.S. 1974. Values of Non-atomic Games, Princeton: Princeton University Press.Google Scholar
  8. Banzhaf, J.F. 1965. Weighted voting doesn’t work: a mathematical analysis. Rutgers Law Review 19, 317–43.Google Scholar
  9. Berbee, H. 1981. On covering single points by randomly ordered intervals. Annals of Probability 9, 520–28.CrossRefGoogle Scholar
  10. Billera, L.J. and Heath, D.C. 1982. Allocation of shared costs: a set of axioms yielding a unique procedure. Mathematics of Operations Research 7, 32–9.CrossRefGoogle Scholar
  11. Billera, L.J., Heath, D.C. and Raanan, J. 1978. Internal telephone billing rates: a novel application of non-atomic game theory. Operations Research 26, 956–65.CrossRefGoogle Scholar
  12. Champsaur, P. 1975. Cooperation versus competition. Journal of Economic Theory 11, 393–417.CrossRefGoogle Scholar
  13. Dubey, P. and Shapley, L.S. 1979. Mathematical properties of the Banzhaf Power Index. Mathematics of Operations Research 4, 99–131.CrossRefGoogle Scholar
  14. Dubey, P., Neyman, A. and Weber, R.J. 1981. Value theory without efficiency. Mathematics of Operations Research 6, 122–8.CrossRefGoogle Scholar
  15. Fogelman, F. and Quinzii, M. 1980. Asymptotic value of mixed games. Mathematics of Operations Research 5, 86–93.CrossRefGoogle Scholar
  16. Harsanyi, J.C. 1959. A bargaining model for the cooperative n-person game. In Contributions to the Theory of GamesVol.4, ed. A.W. Tucker and D.R. Luce, Princeton: Princeton University Press, 324–56.Google Scholar
  17. Harsanyi, J.C. 1963. A simplified bargaining model for the n-person cooperative game. International Economic Review 4, 194–220.CrossRefGoogle Scholar
  18. Hart, S. 1973. Values of mixed games. International Journal of Game Theory 2, 69–85.CrossRefGoogle Scholar
  19. Hart, S. 1977a. Asymptotic value of games with a continuum of players. Journal of Mathematical Economics 4, 57–80.CrossRefGoogle Scholar
  20. Hart, S. 1977b. Values of non-differentiable markets with a continuum of traders. Journal of Mathematical Economics 4, 103–16.CrossRefGoogle Scholar
  21. Hart, S. and Kurz, M. 1983. Endogenous formation of coalitions. Econometrica 51, 1047–64.CrossRefGoogle Scholar
  22. Hart, S. and Mas-Colell, A. 1989. Potential, value and consistency. Econometrica 57 (forthcoming).Google Scholar
  23. Kannai, Y. 1966. Values of games with a continuum of players. Israel Journal of Mathematics 4, 54–8.CrossRefGoogle Scholar
  24. Littlechild, S.C. and Owen, G. 1973. A simple expression for the Shapley value in a special case. Management Science 20, 370–72.CrossRefGoogle Scholar
  25. Mas-Colell, A. 1977. Competitive and value allocations of large exchange economies. Journal of Economic Theory 14, 419–38.CrossRefGoogle Scholar
  26. Mertens, J.-F. 1980. Values and derivatives. Mathematics of Operations Research 5, 523–52.CrossRefGoogle Scholar
  27. Milnor, J.W. and Shapley, L.S. 1961. Values of large games II: oceanic games. RAND RM 2649. Also in Mathematics of Operations Research 3, 1978, 290–307.CrossRefGoogle Scholar
  28. Mirman, L.J. and Tauman, Y. 1982. Demand compatible equitable cost-sharing prices. Mathematics of Operations Research 7, 40–56.CrossRefGoogle Scholar
  29. Myerson, R.B. 1977. Graphs and cooperation in games. Mathematics of Operations Research 2, 225–9.CrossRefGoogle Scholar
  30. Nash, J.F. 1950. The bargaining problem. Econometrica 18, 155–62.CrossRefGoogle Scholar
  31. Neyman, A. 1977. Continuous values are diagonal. Mathematics of Operations Research 2, 338–42.CrossRefGoogle Scholar
  32. Neyman, A. 1981. Singular games have asymptotic values. Mathematics of Operations Research 6, 205–12.CrossRefGoogle Scholar
  33. Neyman, A. 1986. Weighted majority games have asymptotic values. The Hebrew University, Jerusalem, RM 69.Google Scholar
  34. Neyman, A. and Tauman, Y. 1976. The existence of non-diagonal axiomatic values. Mathematics of Operations Research 1, 246–50.CrossRefGoogle Scholar
  35. Owen, G. 1971. Political games. Naval Research Logistics Quarterly 18, 345–55.CrossRefGoogle Scholar
  36. Owen, G. 1977. Values of games with a priori unions. In Essays in Mathematical Economics and Game Theory, ed. R. Henn and O. Moeschlin, New York: Springer-Verlag, 76–88.CrossRefGoogle Scholar
  37. Roth, A.E. 1977. The Shapley value as a von Neumann-Morgenstern utility. Econometrica 45, 657–64.CrossRefGoogle Scholar
  38. Selten, R. 1964. Valuation of n-person games. In Advances in Game Theory, ed. M. Dresher, L.S. Shapley and A.W. Tucker, Princeton: Princeton University Press, 577–626.Google Scholar
  39. Shapiro, N.Z. and Shapley, L.S. 1960. Values of large games I: a limit theorem. RAND RM 2648. Also in Mathematics of Operations Research 3, 1978, 1–9.Google Scholar
  40. Shapley, L.S. 1951. Notes on the n-person game II: the value of an n-person game. RAND RM 670.Google Scholar
  41. Shapley, L.S. 1953a. A value for n-person games. In Contributions to the Theory of Games, Vol. II, ed. H.W. Kuhn and KW. Tucker, Princeton: Princeton University Press, 307–17.Google Scholar
  42. Shapley, L.S. 1953b. Additive and non-additive set functions. PhD thesis, Princeton University.Google Scholar
  43. Shapley, L.S. 1962. Values of games with infinitely many players. In Recent Advances in Game Theory, Princeton University Conferences, 113–18.Google Scholar
  44. Shapley, L.S. 1964. Values of large games VII: a general exchange economy with money. RAND RM 4248-PR.Google Scholar
  45. Shapley, L. S. 1969. Utility comparison and the theory of games. In La Décision: agrégation et dynamique des ordres de préférence, Paris: Editions du CNRS, 251–63.Google Scholar
  46. Shapley, L.S. 1977. A comparison of power indices and a nonsymmetric generalization. RAND P-5872.Google Scholar
  47. Shapley, L.S. 1981a. Comments on R.D. Banker’s ‘Equity considerations in traditional full cost allocation practices: an axiomatic perspective’. In Joint Cost Allocations, ed. S. Moriarity: University of Oklahoma, 131–6.Google Scholar
  48. Shapley, L.S. 1981b. Measurement of power in political systems. Game Theory and its Applications, Proceedings of Symposia in Applied Mathematics, Vol. 24, American Mathematical Society, 69–81.CrossRefGoogle Scholar
  49. Shapley, L.S. and Shubik, M. 1954. A method for evaluating the distribution of power in a committee system. American Political Science Review 48, 787–92.CrossRefGoogle Scholar
  50. Shapley, L.S. and Shubik, M. 1969. Pure competition, coalitional power, and fair division. International Economic Review 10, 337–62.CrossRefGoogle Scholar
  51. Shubik, M. 1962. Incentives, decentralized control, the assignment of joint costs and internal pricing. Management Science 8, 325–43.CrossRefGoogle Scholar
  52. Tauman, Y. 1977. A non-diagonal value on a reproducing space. Mathematics of Operations Research 2, 331–7.CrossRefGoogle Scholar
  53. Young, H.P. (ed.) 1985. Cost Allocation: Methods, Principles, Applications. New York: Elsevier Science.Google Scholar

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© Palgrave Macmillan, a division of Macmillan Publishers Limited 1989

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  • Sergiu Hart

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