Values on regular games under Kirchhoff’s laws

Abstract

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...

Power Measures and Solutions for Games Under Precedence Constraints

“…For example, Algaba et al [24] consider it in games on antimatroids, Bilbao and Edelman [5] studied it on convex geometries, and Bilbao and Ordoñez [25] and Algaba et al [26] on augmenting systems. Convex geometries and augmenting systems have been shown to be contained in the class of so-called regular set systems considered by Honda and Grabisch [7] and Lange and Grabisch [8], who also consider an extension of the precedence Shapley value to games on regular set systems. We can use the 'maximal chains' in the definition of a regular set system to extend the hierarchical measure to this class.…”

“…In many applications, there is some structure on the player set. In the literature, such structures can be modeled by various combinatorial structures, such as 'classic' undirected graphs in Myerson [1] or directed graphs in Gilles et al [2] or Faigle and Kern [3], but also more general structures such as antimatroids in Algaba et al [4], convex geometries in Bilbao and Edelman [5], augmenting systems in Bilbao [6], regular set systems in Honda and Grabisch [7] and Lange and Grabisch [8] or union stable systems in Algaba et al [9,10]. In the underlying paper, we focus on games, where there is a hierarchical structure on the set of players.…”

Monge extensions of cooperation and communication structures

“…This value, however, does not coincide with the Shapley value (3) that arises naturally from the Monge algorithm for this class. The notion of games on regular set systems introduced by Lange and Grabisch (see [22], where a Shapley-like value is proposed) is also closely related to Myerson games. (F, v) is an augmenting system in the sense of Bilbao [5] if it satisfies for all F, G ∈ F with F ⊆ G,…”

“…Other examples arise from models where N is (partially) ordered by some dominance or preference relation (e.g., Derks and Gilles [8], Faigle and Kern [14,15], Gilles et al [17], Grabisch and Lange [18], Hsiao and Raghavan [19]). The latter model was further relaxed and studied by Algaba et al [3], Bilbao et al [2,5] to combinatorial coalition structures of so-called antimatroids, convex geometries and augmenting systems, and by Lange and Grabisch to regular set systems [22]. All these generalized models for cooperation rely on their particular combinatorial structure for the definition of Shapley-type values, Weber sets and cores.…”

Average tree solutions and the distribution of Harsanyi dividends

“…If one discards the assumption that individuals are completely free in forming coalitions, one arrives at refinements which incorporate certain constraints in coalition formation. Examples are games with permission structures (Gilles et al [6]), games with precedence constraints (Faigle and Kern [5]) and games on regular systems (Lange and Grabisch [12]). Such constraints can also be motivated by restrictions on communication.…”