One of the properties characterizing cooperative game solutions is consistency connecting solution vectors of a cooperative game with finite set of players and its reduced game defined by removing one or more players and by assigning them the payoffs according to some specific principle (e.g., a proposed payoff vector). Consistency of a solution means that any part (defined by a coalition of the original game) of a solution payoff vector belongs to the solution set of the corresponding reduced game. In the paper the proportional solutions for TU-games are defined as those depending only on the proportional excess vectors in the same manner as translation covariant solutions depend on the usual Davis–Maschler excess vectors. The general form of the reduced games defining consistent proportional solutions is given. The efficient, anonymous, proportional TU cooperative game solutions meeting the consistency property with respect to any reduced game are described.
Consistency and Potentials in Cooperative TU-Games: Sobolev’s Reduced Game Revived
