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Let N be a finite set of players, |N| = n; a cooperative TU game in coalitional form is a function v : P(N) -> R, with v(ø) = 0. It is well known that the set of all games with the set of players N, denoted below G(N), is a space of dimension 2n - 1. Let S be any coalition in v E G(N) and denote by G(S) the space of games with the set of players S. If v E G(N), then the restriction of v to S is a game in G(S). To avoid a notation like vs, we shall denote the game v by (N, v), and its restriction to S by (S, v). Denote by GN the union of all spaces G(S), for all [see pdf for notation]. Then, a value on GN is a functional ^ on GN with values in R8 for all w E G(S) and all S C N. In particular, for v E G(N) the value ^ gives s-vectors ^(S, v) for all subgames of v. Obviously, for i E S we have in general ^i (5, v) ^ ^i (N, v) when S N. This agrees with the game theoretic meaning of the value as a payoff: the win of player i in the subgame (S, v) is, in general, different of the win of the same player in the game (N, v), when S N. A value ^ on GN is a linear value if for any game v E G(N) which is a linear combination v = av1 + bv2, with v1, v2 E G(N) and a, b E R, we have for all [see pdf for notation], the equality [see pdf for notation]. We intend to give recursive definitions for the Shapley value (see [13]), the Banzhaf value (see [1] and [10]), the Least Square values (see [12]) and the Semivalues (see [8]). As it will be shown below, the proofs for these characterizations are using different tools, and auxiliary results interesting by themselves. |
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