, Volume 20, Issue 3, pp 578–591 | Cite as

1-concave basis for TU games and the library game

  • Theo S. H. Driessen
  • Anna B. KhmelnitskayaEmail author
  • Jordi Sales
Original Paper


The study of 1-convex/1-concave TU games possessing a nonempty core and for which the nucleolus is linear was initiated by Driessen and Tijs (Methods Oper. Res. 46:395–406, 1983) and Driessen (OR Spectrum 7:19–26, 1985). However, until recently appealing abstract and practical examples of these classes of games were missing. The paper solves these drawbacks. We introduce a 1-concave basis for the entire space of all TU games wherefrom it follows that every TU game is either 1-convex/1-concave or is a sum of 1-convex and 1-concave games. Thus we may conclude that the classes of 1-convex/1-concave games constitute rather considerable subsets in the entire game space. On the other hand, an appealing practical example of 1-concave game has cropped up in Sales’s study (Ph. D. thesis, 2002) of Catalan university library consortium for subscription to journals issued by Kluwer publishing house. The so-called library game turns out to be decomposable into suitably chosen 1-concave games of the basis mentioned above.


Cooperative TU game 1-concavity Library cost game Shapley value Nucleolus 

Mathematics Subject Classification (2000)

91A12 91A40 91A43 


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Copyright information

© Sociedad de Estadística e Investigación Operativa 2010

Authors and Affiliations

  • Theo S. H. Driessen
    • 1
  • Anna B. Khmelnitskaya
    • 2
    Email author
  • Jordi Sales
    • 3
  1. 1.Department of Applied MathematicsUniversity of TwenteAE EnschedeThe Netherlands
  2. 2.SPb Institute for Economics and Mathematics, Russian Academy of SciencesSt. PetersburgRussia
  3. 3.Departament de Matemática Económica, Financera i ActuarialUniversitat de BarcelonaBarcelonaSpain

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