A Matrix Approach to TU Games with Coalition and Communication Structures

The aim of this article is to present a technique to construct extensions of the Shapley value. Only basic matrix algebra is used. We concentrate on TU games with coalition structures and with communication structures. We define an efficient Aumann–Drèze value and an efficient Myerson value. We also define two families of values for TU games, the first being a convex combination of the efficient Aumann–Drèze value and of the Shapley value and the second a convex combination of the efficient Myerson value and of the Shapley value. We show that the Myerson value, the Aumann–Drèze value, the Shapley value and the four new solutions above are linked by a relationship of “similarity”.