Component-wise proportional solutions for communication graph games

• The axiom of fairness for degrees is proposed by considering still connected players.
• The graph structure is highly emphasized in the proposed values.
• The proposed rules are novel and simple.

We introduce a class of solutions for graph games by considering the cooperation capacity which represented by bilateral agreements, i.e. degree of nodes in graphs. We replace the axiom of fairness for neighbors proposed by Béal et al. (2012a) by axioms of fairness for degree in order to characterize the component-wise proportional solutions. When a link of a graph is removed, fairness for neighbors states that a player incident to the link and any of his other neighbors should be affected similarly while fairness for degree states that a player incident to the link and all players connected to the player should be affected proportionally to their degree. We first characterize the component-wise proportional solution and the component-wise proportional surplus solution in terms of component efficiency, some kind of fairness for degree and equal treatment or fairness for two-player components. Secondly, we obtain a characterization of the two-step component-wise proportional surplus solution in terms of efficiency, fairness for degree with degree bi-surplus worth, fairness for two-player components and proportional distribution of the surplus.