Efficient extensions of communication values

We study values for transferable utility games enriched by a communication graph. The most well-known such values are component-efficient and characterized by some deletion link property. We study efficient extensions of such values: for a given component-efficient value, we look for a value that (i) satisfies efficiency, (ii) satisfies the link-deletion property underlying the original component-efficient value, and (iii) coincides with the original component-efficient value whenever the underlying graph is connected. Béal et al. (Soc Choice Welf 45:819–827, 2015) prove that the Myerson value (Myerson in Math Oper Res 2:225–229, 1977) admits a unique efficient extension, which has been introduced by van den Brink et al. (Econ Lett 117:786–789, 2012). We pursue this line of research by showing that the average tree solution (Herings et al. in Games Econ Behav 62:77–92, 2008) and the compensation solution (Béal et al. in Int J Game Theory 41:157–178, 2012b) admit similar unique efficient extensions, and that there exists no efficient extension of the position value (Meessen in Communication games, 1988; Borm et al. in SIAM J Discrete Math 5:305–320, 1992). As byproducts, we obtain new characterizations of the average tree solution and the compensation solution, and of their efficient extensions.