, Volume 26, Issue 1, pp 146–163 | Cite as

The precore: converse consistent enlargements and alternative axiomatic results

Original Paper


Since the precore violates (weak) converse consistency, two converse consistent enlargements are proposed. These two converse consistent enlargements are the smallest (weak) converse consistent solutions that contain the precore. On the other hand, we turn to a different notion of the reduction by considering the players and the activity levels simultaneously. Based on such revised reductions, we offer several axiomatizations of the precore.


The precore Converse consistency Converse consistent enlargement Axiomatization 

Mathematics Subject Classification

91A 91B 


  1. Davis M, Maschler M (1965) The kernel of a cooperative game. Naval Res Logist Q 12:223–259CrossRefGoogle Scholar
  2. Faigle U, Kern W (1992) The Shapley value for cooperative games under precedence constraints. Int J Game Theory 21:249266CrossRefGoogle Scholar
  3. Grabisch M, Xie L (2007) A new approach to the core and Weber set of multichoice games. Math Methods Oper Res 66:491–512CrossRefGoogle Scholar
  4. Harsanyi JC (1959) A bargaining model for the cooperative N-person game. In: Tucker AW, Luce RD (eds) Contributions to the theory of games IV. Annals of Mathematics Studies 40. Princeton University Press, Princeton, pp 325–355Google Scholar
  5. Hwang YA, Liao YH (2013) A note on the core: Minimal conversely consistent enlargement. Inf Sci 243:100–105CrossRefGoogle Scholar
  6. Hwang YA, Liao YH, Yeh CH (2015) Consistent extensions and subsolutions of the core for the multi-choice transferable-utility games. Optimization 64:913–928CrossRefGoogle Scholar
  7. Hwang YA, Sudhölter P (2001) Axiomatizations of the core on the universal domain and other natural domains. Int J Game Theory 29:597–623CrossRefGoogle Scholar
  8. Liao YH (2012) Converse consistent enlargements of the unit-level-core of the multi-choice games. Cent Eur J Oper Res 20:743–753CrossRefGoogle Scholar
  9. Moulin H (1985) The separability axiom and equal sharing methods. J Econ Theory 36:120–148CrossRefGoogle Scholar
  10. van den Nouweland A, Potters J, Tijs S, Zarzuelo J (1995) Core and related solution concepts for multi-choice games. ZOR-Math Methods Oper Res 41:289–311CrossRefGoogle Scholar
  11. Peleg B (1985) An axiomatization of the core of cooperative games without side payments. J Math Econ 14:203–214CrossRefGoogle Scholar
  12. Peleg B (1986) On the reduced game property and its converse. Int J Game Theory 15:187–200CrossRefGoogle Scholar
  13. Peleg B (1989) An axiomatization of the core of market games. Math Oper Res 14:448–456CrossRefGoogle Scholar
  14. Serrano R, Volij O (1998) Axiomatizations of neoclassical concepts for economies. J Math Econ 30:87–108CrossRefGoogle Scholar
  15. Sobolev AI (1975) The characterization of optimality principles in cooperative games by functional equations. Math Methods Soc Sci 6:150–165Google Scholar
  16. Tadenuma K (1992) Reduced games, consistency, and the core. Int J Game Theory 20:325–334CrossRefGoogle Scholar
  17. Thomson W (1994) Consistent extensions. Math Soc Sci 28:219–245CrossRefGoogle Scholar
  18. Thomson W (2005) Consistent allocation rules. University of Rochester, MimeoGoogle Scholar
  19. Voorneveld M, van den Nouweland A (1998) A new axiomatization of the core of games with transferable utility. Econ Lett 60:151–155CrossRefGoogle Scholar

Copyright information

© Sociedad de Estadística e Investigación Operativa 2017

Authors and Affiliations

  1. 1.Department of Applied MathematicsNational Pingtung UniversityPingtungTaiwan

Personalised recommendations