A Theoretical Investigation on the Consistency Property of Rank-Shapley Value for Super-Additive Games

Consistency property is an essential property of the solution concept of transferable utility games which preserves the same value for both the original game and a kind of modified games. In this work, the consistency property of Rank-Shapley value is explored through the concept of reduced game and associated game, respectively. A reduced game is a game that remains after some players have left and have been rewarded according to an established principle. Through the reduced game principle, we have demonstrated that the Rank-Shapley value admits consistency in a whole class of inessential games and null games. However, in other classes of games, it only preserves consistency for games with the number of players, . On the other hand, an associated game is a game generated by re-assessing (revaluing) the worth of each coalition (through a function of the worth of the coalition in the original game) such that the payoff of any player in the original game is the same as the payoff of the same player in the associated game. Here, we considered the modification of an associated game known as the Hamiache’s framework. The modification generates a unique and specific associated game for any given game by adapting the proportional share of the surpluses. Hence, the Rank-Shapley value admits consistency in all classes of game through the modification of the Hamiache’s framework. The consistency property of Rank-Shapley value espouses its flexibility in the theory of cooperative games involving dynamic number of players where interest is on maintaining fairness in the payoff of players. Thus, it supports the implementation of the value in a sequence (dynamic) of games.