The Shapley value for directed graph games

Abstract

The Shapley value for directed graph (digraph) games, TU games with limited cooperation introduced by an arbitrary digraph prescribing the dominance relation among the players, is introduced. It is defined as the average of marginal contribution vectors corresponding to all permutations that do not violate the subordination of players. We assume that in order to cooperate players may join only coalitions containing no players dominating them. Properties of this solution are studied and a convexity type condition is provided that guarantees its stability with respect to an appropriately defined core concept. An axiomatization for cycle digraph games for which the digraphs are directed cycles is obtained.

Existence of Stable Coalition Structures in Three-player Games with Graph-constrained Solution

“…The set of permutations on N which are consistent with γ is denoted by Π γ . And in (Khmelnitskaya et al, 2016), it is stated that if π ∈ Π γ , then for any player i ∈ N , Pπ (i), P π (i) ∈ H(γ).…”

“…The Shapley value with γ as the exogenous directed graph constraint (see Khmelnitskaya et al, 2016) in cooperative game (N, v) is defined as…”

“…In the present paper, we examine the existence of a stable coalition structure in two-player as well as three-player games with respect to the solution concept based on the Shapley value with exogenous directed graph constraint (see Khmelnitskaya et al, 2016). We find there must be stable coalition structures in two-player games, while in three-player games, only when particular structures are applied as the exogenous directed graph constraints for such solution concept, there always exists a stable coalition structure.…”

El control coalicional en el marco de la teoría de juegos cooperativos

“…Alternativamente, es posible redefinir el juego (13) en cualquier subconjunto de T (Loehman and Whinston, 1976;Khmelnitskaya et al, 2016), o bien excluir coaliciones usando la propia regla de pagos (Hiller, 2016(Hiller, , 2018.…”