  The model of incomplete cooperative games incorporates uncertainty into the
classical model of cooperative games by considering a partial characteristic
function. Thus the values for some of the coalitions are not known. The main
focus of this paper is the class of 1-convex cooperative games under this
framework.
  We are interested in two heavily intertwined questions. First, given an
incomplete game, in which ways can we fill in the missing values to obtain a
classical 1-convex game? Such complete games are called \emph{1-convex
extensions}. For the class of minimal incomplete games (in which precisely the
values of singletons and grand coalitions are known), we provide an answer in
terms of a description of the set of 1-convex extensions. The description
employs extreme points and extreme rays of the set.
  Second, how to determine in a rational, fair, and efficient way the payoffs
of players based only on the known values of coalitions? Based on the
description of the set of 1-convex extensions, we introduce generalisations of
three solution concepts (values) 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
minimal incomplete games which allows to introduce the \emph{average value}.
For this value, we provide three different axiomatisations based on axiomatic
characterisations of the $\tau$-value and the Shapley value for classical
cooperative games.
  Finally, we turn our attention to \emph{incomplete games with defined upper
vector}, asking the same questions and this time arriving to different
conclusions. This provides a benchmark to test our tools and knowledge of the
average value.
