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Some Recursive Definitions of the Shapley Value and Other Linear Values of Cooperative TU Games

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Some Recursive Definitions of the Shapley Value and Other Linear Values of Cooperative TU Games

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dc.contributor.author Dragan, Irinel C. en
dc.date.accessioned 2010-06-03T18:21:37Z en
dc.date.available 2010-06-03T18:21:37Z en
dc.date.issued 1997 en
dc.identifier.uri http://hdl.handle.net/10106/2348 en
dc.description.abstract 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. en
dc.language.iso en_US en
dc.publisher University of Texas at Arlington en
dc.relation.ispartofseries Technical Report;328 en
dc.subject Shapley value en
dc.subject Banzhaf value en
dc.subject Least Square values en
dc.subject TU games en
dc.subject Cooperative game en
dc.subject.lcsh Game theory en
dc.subject.lcsh Mathematics Research en
dc.title Some Recursive Definitions of the Shapley Value and Other Linear Values of Cooperative TU Games en
dc.type Technical Report en
dc.publisher.department Department of Mathematics en

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