Average tree solutions and the distribution of Harsanyi dividends

Abstract

We consider communication situations games being the combination of a TU-game and a communication graph. We study the average tree (AT) solutions introduced by Herings et al. [9] and [10]. The AT solutions are defined with respect to a set, say T , of rooted spanning trees of the communication graph. We characterize these solutions by efficiency, linearity and an axiom of T -hierarchy. Then we prove the following results. Firstly, the AT solution with respect to T is a Harsanyi solution if and only if T is a subset of the set of trees introduced in [10]. Secondly, the latter set is constructed by the classical DFS algorithm and the associated AT solution coincides with the Shapley value when the communication graph is complete. Thirdly, the AT solution with respect to trees constructed by the other classical algorithm BFS yields the equal surplus division when the communication graph is complete.

The Communication Tree Value for TU-Games with Graph Communication

“…The property in condition (2) says that if in a communication tree K 1 and K 2 are two different components in the set of subordinates of some player, then there is no link in the graph between any player of K 1 and any player of K 2 . This means that if in the tree two players are not subordinate of each other, they are not able to communicate directly with each other.…”

“…Take any i ∈ N and K ∈ C L (B i (T )). Then there exists j ∈ N such that B j (T ) = K and since T ∈ ST L , {i, j} ∈ L, so that condition (2) in the definition of the average tree solution is also satisfied. Hence, the mapping f is well-defined.…”

“…In [7], also a characterization of the average tree solution for the class of games with cycle-free graph communication is given by means of component efficiency and component fairness. The average tree solution is the average of those marginal contribution vectors that correspond to a specific collection of spanning trees for the underlying graph, see also [2]. In Koshevoy and Talman [9] the average of all different marginal contribution vectors is taken as solution concept.…”

Characterization of the Average Tree solution and its kernel

“…Other characterizations on the class of cycle-free graph games have been provided by van den Brink (2009), Mishra and Talman (2010), Béal et al (2010Béal et al ( , 2012b and Ju and Park (2012). Generalizations of the AT solution to the class of all graph games have been examined by Herings et al (2010) and Baron et al (2011). The average tree solution has also been implemented by van den Brink et al (2013), and applied to and characterized in the richer frameworks of multichoice communication games by Béal et al (2012a) and of games with a permission tree by van den Brink et al (2014).…”

“…The axiom of Invariance to irrelevant coalitions requires that the solution should prescribe the same payoff vector in two tree games where the worths of all cones are the same. 1 The axiom of Weighted addition invariance on bi-partitions requires that the solution is not affected if the worths of each of the two coalitions of a bi-partition of the agent set changes in proportion to their respective size. On the class of tree games, we show that the AT solution is characterized by Invariance to irrelevant coalitions, Weighted addition invariance on bi-partitions and the Inessential game axiom (Proposition 4).…”