Extensions of the shapley value and their applications

Abstract

Young [97] axiomatized the Shapley value on G by using efficiency, symmetry and the so-called strong monotonicity. Inspired by his approach, we modify strong monotonicity, such that it can be used together with efficiency and symmetry, to axiomatize the ELS values:• The ELS value is the unique value on G satisfying efficiency, symmetry and B-strong monotonicity. (Theorem 2.7).In the uniqueness proof, we make use of a new basis of the game space G. The special feature of this new basis { N, u b T | T ⊆ N, T = ∅} is that, the ELS value for player i in N, u b T equals 1 if i ∈ T , otherwise it equals 0.results in the generalized game space. In this generalized game space, the worth of coalitions not only depend on the players in that coalition, but also on the orders of players entering into the game. Thus different permutations of a fixed set of players may admit different worths. This game model was introduced firstly by Nowak and Radzik [58], and then it was refined by Sanchez and Bergantinos [70]. In Chapter 3 the generalized Shapley value defined by Sanchez and Bergantinos [70] is studied. Evans [20] introduced a procedure on G, and proved that the unique value which is consistent with this procedure is the classical Shapley value. We modify Evans' procedure such that it can be used in the generalized model. More precisely, for any generalized game, firstly, we choose one permutation of the grand coalition; secondly, we select two subcoalitions according to this permutation; thirdly, we choose two leaders separately from the two subcoalitions; then the two leaders play a two-person bargaining game. The rule is that the two-person standard solution is used in the bargaining, and each leader gives the rest of players in his subcoalition an amount of payoff. We prove that if all chosen processes are subjected to uniform distribution, then the expected value of the procedure is just the generalized Shapley value:• The generalized Shapley value is the unique value on G satisfying Evans' consistency with respect to a chosen procedure and standardness on two-person games. (Theorem 3.1 and Corollary 3.1).It is shown by Hamiache [25] that the classical Shapley value is the unique value on G satisfying associated consistency, continuity and inessential game property. The proof of this axiomatization is very complicated, and later Xu et al.[93] used a matrix approach to simplify the proof. We modify the three properties used in the axiomatization from the classical game space to the generalized game space, and applying an analogous matrix approach to establish the proof:• The generalized Shapley value is the unique value on G satisfying generalized associated consistency, continuity, and generalized inessential game property. (Theorem 3.7).The main problem in the matrix approach is to prove that a certain matrix is diagonalizable. According to the Diagonalization Theory, a matrix is diagonalizable if and only if the sum of dimensions of the distinct eigenspaces equals the number of column vectors of this matrix, and th...