Characterization of the Average Tree solution and its kernel

Abstract

In this article, we study cooperative games with limited cooperation possibilities, represented by a tree on the set of agents. Agents in the game can cooperate if they are connected in the tree. We first derive direct-sum decompositions of the space of TU-games on a fixed tree, and two new basis for these spaces of TU-games. We then focus our attention on the Average (rooted)-Tree solution (see Herings, P., van der Laan, G., Talman, D., 2008. The Average Tree Solution for Cycle-free Games. Games and Economic Behavior 62, 77-92). We provide a basis for its kernel and a new axiomatic characterization by using the classical axiom for inessential games, and two new axioms of invariance, namely Invariance with respect to irrelevant coalitions and Weighted addition invariance on bi-partitions.

Discounted tree solutions

“…There is a major difference between (5) and 4: the linear combination of the hierarchical outcomes is specified by the node-weighted tree TU-game (N, v, L, p, δ). From an axiomatic point of view, this difference has some consequences.…”

“…Node-weighted efficiency. The fact that the solution Ψ satisfies node-weighted efficiency follows from (3) and (5). Invariance with respect to cone amalgamation.…”

“…Béal et al (2015b) use the equivalent axiom of Invariance to irrelevant coalitions in order to characterize the average tree solution.…”