Values of games with probabilistic graphs

Calvo, Emílio; Lasaga, J.; Nouweland, Anne van den

doi:10.1016/s0165-4896(98)00013-4

Cited by 47 publications

(51 citation statements)

References 16 publications

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“…Now, we are ready to define the Weighted Attachment Centrality for node-and edge-weighted graphs. For a graph G = (V, E, 1, ρ) with unweighted nodes, the Weighted Attachment Centrality is equal to the Probabilistic Myerson value, defined by Calvo et al [1999] with characteristic function f * G . Based on Theorem 8, we get the following two axiomatizations of the Weighted Degree Centrality and the Weighted Attachment Centrality.…”

Section: Node-weighted Graphsmentioning

confidence: 99%

Attachment Centrality for Weighted Graphs

Sosnowska

Skibski

2017

Proceedings of the Twenty-Sixth International Joint Conference on Artificial Intelligence

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Measuring how central nodes are in terms of connecting a network has recently received increasing attention in the literature. While a few dedicated centrality measures have been proposed, Skibski et al. [2016] showed that the Attachment Centrality is the only one that satisfies certain natural axioms desirable for connectivity. Unfortunately, the Attachment Centrality is defined only for unweighted graphs which makes this measure ill-fitted for various applications. For instance, covert networks are typically weighted, where the weights carry additional intelligence available about criminals or terrorists and the links between them. To analyse such settings, in this paper we extend the Attachment Centrality to node-weighted and edgeweighted graphs. By an axiomatic analysis, we show that the Attachment Centrality is closely related to the Degree Centrality in weighted graphs.

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Section: Node-weighted Graphsmentioning

confidence: 99%

Attachment Centrality for Weighted Graphs

Sosnowska

Skibski

2017

Proceedings of the Twenty-Sixth International Joint Conference on Artificial Intelligence

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“…In this last case, w l ∈ [0, ∞), for all l ∈ γ. Calvo et al (1999) propose to interpret each weight as the probability of the corresponding relation and to assume the links independence.…”

Section: Weighted Graphsmentioning

confidence: 99%

“…Remark 3.4 If (N, γ w ) is a weighted network in which, for each l ∈ γ, w l is interpreted as the probability of a direct relation between its two incident nodes, and these probabilities are considered as independent (as in Calvo et al, 1999) then, for R connected in γ and T ⊆ R:…”

Section: Weighted-graph Restricted Gamesmentioning

confidence: 99%

“…A weighted graph consists of a set of nodes and a set of links, each link having an associated weight, a non-negative real number that can be interpreted in different ways: the capacity or the capability of the communication channel, the flow across it, the degree of intimacy, intensity or frequency if the link represents a social relation, the distance between both incident nodes, or even the cost of building or maintaining the communication link. Calvo et al (1999) introduced a probabilistic model in which the weight of a link is the probability of to establish the relation, these probabilities being independent. Later, Gómez et al (2008) generalized this probabilistic model dropping out the independence hypothesis.…”

Section: Introductionmentioning

confidence: 99%

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Values of games with weighted graphs

González–Arangüena

Manuel

Pozo

2015

European Journal of Operational Research

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“…In other words, these coalitions are those corresponding to the communication situations between the players: A ⊆ N is feasible if and only if there is a communication chain for any pair of players of A. More details about games with communication graphs may be found in Calvo, Lasage, and Nouweland (1999) and van den Brink, van der Laan, and Pruzhansky (2004).…”

Section: Regular Gamesmentioning

confidence: 99%

Values on regular games under Kirchhoff’s laws

Lange¹,

Grabisch

2009

Mathematical Social Sciences

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The Shapley value is a central notion defining a rational way to share the total worth of a cooperative game among players. We address a general framework leading to applications to games with communication graphs, where the feasible coalitions form a poset whose all maximal chains have the same length. Considering a new way to define the symmetry among players, we propose an axiomatization of the Shapley value of these games. Borrowing ideas from electric networks theory, we show that our symmetry axiom and the efficiency axiom correspond to the two Kirchhoff's laws in the circuit associated to the Hasse diagram of feasible coalitions. We thank both reviewers for their work and valuable remarks, as well as the Associate Editor. Below is the list of changes and argued rebuttals. Fabien Reviewer #2• Another definition of regular set systems is proposed: N is regular iff there is a set O of strict orders of N s.t. N contains precisely all initial segments of orderings in O. The "if part" is wrong: for instance, if N = {1, . . . , 4} and O := {(1, 2, 3, 4); (3, 4, 1, 2)}, the resulting N does not satisfy the definition ({3} ≺ {1, 2, 3} but |{1, 2, 3} \ {3}| = 1).• We quite disagree with the idea that the statement of the regularity axiom is too long, and the proposed restatement does not enable to understand its intuition. As specified, this axiom means that for an equidistributed game v, the sum of marginal contributions of players should not depend on the considered maximal chain. For such a game, the path taken from coalition A to B has no effect on the successive increasing worth v(C) (A ⊆ C ⊆ B), so that our statement of the axiom is defended.• It is true that the "Shapley-Kirchhoff value" is not explicitely presented.Actually, it appears that the coefficients are computable only for a given regular set sytem, since a linear system has to be solved, where equations depend on the structure of the system. By the way, it is false to say that this value is the arithmetic average of the order values of Weber over all orderings in O. This is actually a compatible-order value, and consequently an average of the order values, but the mean is not arithmetic (coefficients are not the same, depending on the ordering, and may be negative as a corollary of the counterexample of Annex B).• Theorem 12 does not show that the SK value is monotonic (which is false, according to the aforementionned counterexample), it shows its aggregate monotonicity. 1 * Response to ReviewersReviewer #1• About the fact that we do not give very much support to the claim that the class of regular set systems is natural: in addition of the example of communication situations, we added after Proposition 1 a paragraph with another approach in the process of building regular set systems : by considering permitted/forbidden permutations. The introduction is also amended (p. 2, paragraph 1).• Technical points p. 8 and p. 12: these are now corrected. Abstract. The Shapley value is a central notion defining a rational way to share the total wor...

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Values of games with probabilistic graphs

Cited by 47 publications

References 16 publications

Attachment Centrality for Weighted Graphs

Attachment Centrality for Weighted Graphs

Values of games with weighted graphs

Values on regular games under Kirchhoff’s laws

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