Core extensions for non-balanced TU-games

Original Paper


A family of core extensions for cooperative TU-games is introduced. These solution concepts are non-empty when applied to non-balanced games yet coincide with the core whenever the core is non-empty. The extensions suggest how an exogenous regulator can sustain a stable and efficient outcome, financing a subsidy via individual taxes. Economic and geometric properties of the solution concepts are studied. When taxes are proportional, the proportional prenucleolus is proposed as a single-valued selection device. An application of these concepts to the decentralization of a public goods economy is discussed.


Core extensions Efficiency Taxation Public goods 

JEL Classification

C71 H21 H41 


  1. Aumann RJ, Maschler M (1964) The bargaining set for cooperative games. In: Dresher M, Shapley L, Tucker A (eds) Advances in game theory. Princeton University Press, Princeton, pp 442–476Google Scholar
  2. Border KC (1985) Fixed point theorems with applications to economics and game theory. Cambridge University Press, CambridgeGoogle Scholar
  3. Davis M, Maschler M (1965) The kernel of a cooperative game. Naval Res Logist Q 12: 223–259CrossRefGoogle Scholar
  4. Dréze J, LeBreton M, Savvateev A, Weber S (2007) ‘Almost’ subsidy-free spatial pricing in a multi-dimensional setting. CORE Working Paper 2007/30, Université catholique de LouvainGoogle Scholar
  5. Grotte JH (2000) Computation of and observations on the nucleolus, the normalized nucleolus and the central games. Master’s thesis, Cornell University, IthacaGoogle Scholar
  6. Hwang YA, Sudhölter P (2000) Axiomatizations of the core on the universal domain and other natural domains. Int J Game Theory 29(4): 597–623CrossRefGoogle Scholar
  7. Kannai Y (1992) The core and balancedness. In: Aumann RJ, Hart S (eds) Handbook of game theory with economic applications, vol 1. North-Holland, New York, pp 355–395Google Scholar
  8. Kwon Y, Yu P (1977) Stabilization through taxation in n-person games. J Optim Theory Appl 23(2): 277–284CrossRefGoogle Scholar
  9. Lucchetti R, Patrone F, Tijs SH, Torre A (1987) Continuity properties of solution concepts for cooperative games. OR Spektrum 9(2): 101–107CrossRefGoogle Scholar
  10. Mas-Colell A (1980) Efficiency and decentralization in the pure theory of public goods. Q J Econ 94(4): 625–641CrossRefGoogle Scholar
  11. Maschler M, Peleg B, Shapley L (1979) Geometric properties of the kernel, nucleolus, and related solution concepts. Math Oper Res 4(4): 303–338CrossRefGoogle Scholar
  12. Moulin H (1985) The separability axiom and equal-sharing methods. J Econ Theory 36(1): 120–148CrossRefGoogle Scholar
  13. Moulin H (1995) Une évaluation de la théorie des jeux coopératifs. Rev d’Economie Politique 105(4): 617–632Google Scholar
  14. Orshan G, Sudhölter P (2001) The positive core of a cooperative game. Discussion Paper 268, Center for Rationality and Interactive Decision Theory, Hebrew University of JerusalemGoogle Scholar
  15. Peleg B (1986) On the reduced game property and its converse. Int J Game Theory 15(3): 187–200CrossRefGoogle Scholar
  16. Schmeidler D (1969) The nucleolus of a characteristic function game. SIAM J Appl Math 17(6): 1163–1170CrossRefGoogle Scholar
  17. Shapley LS, Shubik M (1966) Quasi-cores in a monetary economy with nonconvex preferences. Econometrica 34(4): 805–827CrossRefGoogle Scholar
  18. Tadenuma K (1992) Reduced games, consistency, and the core. Int J Game Theory 20(4): 325–334CrossRefGoogle Scholar
  19. Thomson W (2003) Axiomatic and game-theoretic analysis of bankruptcy and taxation problems: a survey. Math Soc Sci 45(3): 249–297CrossRefGoogle Scholar
  20. Tijs SH, Driessen SH (1986) Extensions of solution concepts by means of multiplicative ε-tax games. Math Soc Sci 12(1): 9–20CrossRefGoogle Scholar
  21. Wooders MH, Zame WR (1984) Approximate cores of large games. Econometrica 52(6): 1327–1350CrossRefGoogle Scholar
  22. Young H (1985) Monotonic solutions of cooperative games. Int J Game Theory 14: 65–72CrossRefGoogle Scholar
  23. Young HP, Okada N, Hashimoto T (1982) Cost allocation in water resources development. Water Resour Res 18(3): 463–475CrossRefGoogle Scholar
  24. Zhao J (2000) The core in an oligopoly market with indivisibility. Econ Theory 16: 181–198CrossRefGoogle Scholar
  25. Zhao J (2001) The relative interior of the base polyhedron and the core. Econ Theory 18(3): 635–648CrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  1. 1.Department of EconomicsRice University MS-22HoustonUSA
  2. 2.Business Administration ProgramUniversity of Washington-BothellBothellUSA

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