Power Measures and Solutions for Games Under Precedence Constraints

Abstract

Games under precedence constraints model situations, where players in a cooperative transferable utility game belong to some hierarchical structure, which is represented by an acyclic digraph (partial order). In this paper, we introduce the class of precedence power solutions for games under precedence constraints. These solutions are obtained by allocating the dividends in the game proportional to some power measure for acyclic digraphs. We show that all these solutions satisfy the desirable axiom of irrelevant player independence, which establishes that the payoffs assigned to relevant players are not affected by the presence of irrelevant players. We axiomatize these precedence power solutions using irrelevant player independence and an axiom that uses a digraph power measure. We give special attention to the hierarchical solution, which applies the hierarchical measure. We argue how this solution is related to the known precedence Shapley value, which does not satisfy irrelevant player independence, and thus is not a precedence power solution. We also axiomatize the hierarchical measure as a digraph power measure.

Harsanyi power solutions for cooperative games on voting structures

“…In the property of equality is revealed the strong connection between economic influence and political power. It is also worth mentioning that the weak null agent out evokes the irrelevant agent independence which is introduced for games under precedence constraints to characterize the hierarchical solution (Algaba et al, 2017). The main distinction between the latter two axioms can be found in the structural differences of the system of feasible coalitions of each approach.…”

The Shapley Value and Games with Hierarchies

“…Alternatively, this precedence Shapley value can be written as an allocation of the Harsanyi dividends, where the dividend of every feasible coalition is allocated proportional to the so-called hierarchical strength being a power measure for digraphs that assigns to every player the number of admissible permutations where it is the last to enter. Algaba et al (2017) showed that in this solution the payoff allocation is influenced by the presence of irrelevant players. These are players who do not generate worth in the game and, moreover, also all the players who depend on their presence do not generate worth in the game.…”

“…Power or centrality measures for digraphs are applied to define solutions for games under precedence constraints by Algaba et al (2017). In this section, we first review their result, and then apply this approach to games with a permission structure.…”

“…Algaba et al, 2017) A solution for games under precedence constraints is equal to the p-hierarchical solution H p if and only if it satisfies efficiency, linearity, the irrelevant player property, irrelevant player independence and p-strength.Note that this gives Theorem 3.4 as a corollary by taking the hierarchical strength as power measure. The axioms of Theorem 4.2 are not logically independent.…”

Network Structures with Hierarchy and Communication

“…Union stable network structures are closely related to hypergraphs, see Algaba et al[6] 2. Another model dealing with hierarchies, defined by digraphs, is that of games under predecence constraints, introduced by Faigle and Kern[12], and recently studied in Algaba et al[13].…”