Affiliations: Polish-Japanese Institute of Information Technology,
Koszykowa 86, 02-008 Warsaw, Poland | Department of Mathematics and Computer Science,
University Warmia and Mazury, Żołnierska 14a, 10-650 Olsztyn,
Poland
Abstract: 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 v [14] is a
functional Φ giving for each player p the value Φ_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 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.
Keywords: rough sets, the lower and the upper value (Shapley value) of a game, coalition, exact coalition
