Abstract
We give recursive definitions for the Banzhaf Value and the Semivalues of cooperative TU games. These definitions were suggested by the concept of potential for the Shapley Value due to Hart and Mas-Colell and by some results of the author who introduced the potentials of these values and the Power Game of a given game.
Highlights
After some notations and concepts needed in this paper, as well as some references including earlier results, we give a new proof for the recursive definition of the Shapley Value in the second section
× [V (S) − V (S − {i})], ∀i ∈ T, for all V ∈ G(N), T ⊆ N, where s = |S|, t = |T|. Another axiomatization has been given by Hart and Mas-Colell [6], based upon the concept of potential of the Shapley Value they have introduced
We conclude that the characterization offered by the last Theorem is allowing us to consider that (7) gives a recursive definition of the Shapley Value
Summary
After some notations and concepts needed in this paper, as well as some references including earlier results, we give a new proof for the recursive definition of the Shapley Value in the second section. Let S ⊆ N be any coalition in V ∈ G(N) and denote by G(S) the space of games with the set of players S. The main tools in the present work are some results of Linear Algebra, a concept of potential of the Shapley Value, due to Hart and Mas-Colell, [6], as well as earlier results of the author [7,8,9], that will be individually mentioned in connection with the new results
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