1-convex extensions of partially defined cooperative games and the average value

Partially defined cooperative games are a generalisation of classical cooperative games in which the worth of some of the coalitions is not known. Therefore, they are one of the possible approaches to uncertainty in cooperative game theory. The main focus of this paper is the class of 1-convex cooperative games under this framework. For incomplete cooperative games with minimal information, we present a compact description of the set of 1-convex extensions employing its extreme points and its extreme rays. Then we investigate generalisations of three solution concepts for complete games, namely the $\tau$-value, the Shapley value and the nucleolus. We consider two variants where we compute the centre of gravity of either extreme games or of a combination of extreme games and extreme rays. We show that all of the generalised values coincide for games with minimal information and we call this solution concept the \emph{average value}. Further, we provide three different axiomatisations of the average value and outline a method to generalise several axiomatisations of the $\tau$-value and the Shapley value into an axiomatisation of the average value. We also briefly mention a similar derivation for incomplete games with defined upper vector and indicate several open questions.