A Consistent Value for NTU Games with Coalition Structure

Resumo

We introduce a new value for NTU games with coalition structure. This value coincides with the consistent Shapley value for trivial coalition structures, and with the Owen coalitional value for TU games with coalition structure. Furthermore, we propose two axiomatizations in terms of balanced contributions and by a consistency property. The consistent value is presented by Maschler and Owen (1989, 1992) as a generalization of the Shapley value for hyperplane games. The main idea behind this generalization is to maintain (as far as possible) the consistency property from the Shapley value. It is remarkable that their value arises as the mean of random order marginal contributions of the players. Maschler and Owen (1992) even suggest the name �random order value� instead of �consistent value�. Later, Hart and Mas-Colell (1996) develop a bargaining mechanism which implements the coalitional value. They also characterize this value by means of balanced contributions. On the other hand, NTU games with coalition structure are studied by Winter (1992). Winter axiomatizes the Game Coalition Structure (GCS) value, which is a generalization of the Harsanyi value for trivial coalition structures and the Owen value for TU games. It was of our interest to know whether the consistent value could be generalized the same way to games with coalition structure. Both the Shapley value and the consistent value are obtained as an average of marginal contributions depending on equal-likely orders. Furthermore, the Owen value can be obtained by restricting these orders to compatibility with the coalition structure. Thus, it seems reasonable to generalize the consistent value using the same restrictions. Remarkably, the arising value, however generalizes both consistent and Owen value, misses most of their nice properties; namely, it is not consistent, nor satisfies a �balanced contributions� property. We call this value the �random order coalitional value�. We use the random order coalitional value as a first step in defining our value. We show that the suggested value is consistent (in the sense of Maschler and Owen (1989) and Winter (1992)) and it satisfies a balanced contributions property. We believe these properties make the value a proper generalization of the Owen value for NTU games with coalition structure