Consistency and potentials in cooperative TU-games: Sobolev's reduced game revived

Theo Driessen

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It was a quarter of a century ago that Sobolev proved the reduced game (otherwise called consistency) property for the much-discussed Shapley value of cooperative TU-games. The purpose of this paper is to extend Sobolev's result in two ways. On the one hand the unified approach applies to the enlarged class consisting of game-theoretic solutions that possess a so-called potential representation; on the other Sobolev's reduced game is strongly adapted in order to establish the consistency property for solutions that admit a potential. Actually, Sobolev's explicit description of the reduced game is now replaced by a similar, but implicit definition of the modified reduced game; the characteristic function of which is implicitly determined by a bijective mapping on the universal game space (induced by the solution in question). The resulting consistency property solves an outstanding open problem for a wide class of game-theoretic solutions. As usual, the consistency together with some kind of standardness for two-person games fully characterize the solution. A detailed exposition of the developed theory is given in the event of dealing with so-called semivalues of cooperative TU-games and the Shapley and Banzhaf values in particular.
Original languageEnglish
Place of PublicationEnschede
PublisherUniversity of Twente
Number of pages16
Publication statusPublished - 2000

Publication series

PublisherDepartment of Applied Mathematics, University of Twente
ISSN (Print)0169-2690


  • MSC-91D12
  • MSC-91D40


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