Document Type
Report
Source Publication Title
Technical Report 328
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.
Disciplines
Mathematics | Physical Sciences and Mathematics
Publication Date
1-1-1997
Language
English
License
This work is licensed under a Creative Commons Attribution-NonCommercial-Share Alike 4.0 International License.
Recommended Citation
Dragan, Irinel C., "Some Recursive Definitions of the Shapley Value and Other Linear Values of Cooperative TU Games" (1997). Mathematics Technical Papers. 39.
https://mavmatrix.uta.edu/math_technicalpapers/39