Compensations in the Shapley value and the compensation solutions for graph games

We consider an alternative expression of the Shapley value that reveals a system of compensations: each player receives an equal share of the worth of each coalition he belongs to, and has to compensate an equal share of the worth of any coalition he does not belong to. We give an interpretation in terms of formation of the grand coalition according to an ordering of the players and define the corresponding compensation vector. Then, we generalize this idea to cooperative games with a communication graph. Firstly, we consider cooperative games with a forest (cycle-free graph). We extend the compensation vector by considering all rooted spanning trees of the forest (see Demange 2004) instead of orderings of the players. The associated allocation rule, called the compensation solution, is characterized by component efficiency and relative fairness. The latter axiom takes into account the relative position of a player with respect to his component. Secondly, we consider cooperative games with arbitrary graphs and construct rooted spanning trees by using the classical algorithms DFS and BFS. If the graph is complete, we show that the compensation solutions associated with DFS and BFS coincide with the Shapley value and the equal surplus division respectively.

On the number of blocks required to access the coalition structure core

This article shows that, for any transferable utility game in coalitional form with nonempty coalition structure core, the number of steps required to switch from a payoff configuration out of the coalition structure core to a payoff configuration in the coalition structure core is less than or equal to (n*n+4n)/4, where n is the cardinality of the player set. This number considerably improves the upper bound found so far by Koczy and Lauwers (2004).coalition structure core; excess function; payoff configuration; outsider independent domination.

Average tree solutions and the distribution of Harsanyi dividends

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 \sl 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.

On the number of blocks required to access the core

For any transferable utility game in coalitional form with nonempty core, we show that that the number of blocks required to switch from an imputation out of the core to an imputation in the core is less than or equal to n(n-1)/2, where n is the cardinality of the player set. This number considerably improves the upper bounds found so far by Koczy (2006) and Yang (2010). Our result relies on an altered version of the procedure proposed by Sengupta and Sengupta (1996). The use of the Davis-Maschler reduced-games is also pointed out.Core; excess function; dominance path; Davis-Maschler reduced-game