A rough set approach to estimating the game value and the Shapley value from data

A value of a game v is a function which to each coalition S assigns the value v(S) of this coalition, meaning the expected pay-off for players in that coalition. A classical approach of von Neumann and Morgenstern [6] had set some formal requirements on v which contemporary theories of value adhere to. A Shapley value of the game with a value nu [14] is a functional Phi giving for each player p the value Phi(p) (v) estimating the expected pay-off of the player p in the game. Game as well as conflict theory have been given recently much attention on the part of rough and fuzzy set communities [11, 8, 1, 4, 7, 2]. In particular, problems of plausible strategies [1] in conflicts as well as problems related to Shapley's value [3, 2] have been addressed. We confront here the problem of estimating a value as well as Shapley's value of a game from a partial data about the game. We apply to this end the rough set ideas of approximations, defining the lower and the upper value of the game and, respectively, the lower and the upper Shapley value. We also define a notion of an exact coalition, on which both values coincide giving the true value of the game; we investigate the structure of the family of exact sets showing its closeness on complements, disjoint sums, and intersections of coalitions covering the set of players. This work sets open a new area of rough set applications in mining constructs from data. The constructs mined in this case are values as well as Shapley values of games.