Consistency and Potentials in Cooperative TU-Games: Sobolev’s Reduced Game Revived

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 TUgames. 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.