Associated consistency and values for TU games

Driessen, Theo

doi:10.1007/s00182-010-0222-1

Cited by 19 publications

(15 citation statements)

References 6 publications

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“…Fourthly, a stronger version of uniform transfer invariance in which the two involved coalitions can have different sizes is used together with the dummy player property to characterize the Banzhaf value (Theorem 4). To the best of our knowledge, Hamiache [7] and Driessen [5] are the only other articles in which the classical axioms of efficiency, anonymity, symmetry and additivity are not incorporated to the characterizing set of axioms.…”

Section: Introductionmentioning

confidence: 99%

“…It should also be noted that the characterizations of the Shapley value in [24] and [4] incorporate at least one of the classical axioms of efficiency, symmetry and additivity. The axioms of associated consistency in Hamiache [7] and B-consistency in Driessen [5] require invariance of a rule if the worth of every coalition is adjusted in particularly specific ways, which allow much less freedom than our axiom of uniform addition invariance. Finally, our research can be connected to Kleinberg and Weiss [12] in which the set of TU-games for which the Shapley value recommends the null payoff vector is characterized.…”

Section: Introductionmentioning

confidence: 99%

“…Then, the axioms of invariance are employed to conclude on the payoff of the chosen player in the original TU-game. Apart from the present article, the idea to wend through the space of TU-games by constructing chains of TU-games connected through some axioms is also used by Hamiache [7] and Driessen [5], and by Pintér [15] for an alternative proof of Young's [24] result. A drawback of the approach in Hamiache [7] and Driessen [5] is that the chain of TU-games asymptotically converges, which necessitates to use a technical axiom of continuity.…”

Section: Introductionmentioning

confidence: 99%

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Axioms of invariance for TU-games

Béal

Rémila²,

Solal³

2014

Int J Game Theory

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We introduce new axioms for the class of all TU-games with a fixed but arbitrary player set, which require either invariance of an allocation rule or invariance of the payoff assigned by an allocation rule to a specified subset of players in two related TU-games. Comparisons with other axioms are provided. These new axioms are used to characterize the Shapley value, the equal division rule, the equal surplus division rule and the Banzhaf value. The classical axioms of efficiency, anonymity, symmetry and additivity are not used.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Axioms of invariance for TU-games

Béal

Rémila²,

Solal³

2014

Int J Game Theory

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“…Thus without either additivity or the efficiency axioms, the Shapley value is characterized by the inessential game property, associated consistency and continuity. Driessen [15] generalized this associated consistency to the ELS values. Notice that the uniqueness proof in Hamiache [25] as well as in Driessen [15] are very complicated and technical.…”

Section: Motivationmentioning

confidence: 91%

“…[17,15] A value Φ on G N satisfies the efficiency, linearity and symmetry if and only if there exists a (unique) collection of constants, B = {b n s | n ∈ N \ {0, 1}, s = 1, 2, . .…”

Section: Modified Potential Representation For the Els Valuementioning

confidence: 99%

Extensions of the shapley value and their applications

Feng

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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...

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A note on associated consistency and linear, symmetric values

Kleinberg

2017

Int J Game Theory

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Associated consistency and values for TU games

Cited by 19 publications

References 6 publications

Axioms of invariance for TU-games

Axioms of invariance for TU-games

Extensions of the shapley value and their applications

A note on associated consistency and linear, symmetric values

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