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 1-convexity under this framework. We are interested in two heavily intertwined questions. First, given an incomplete game, how can we fill in the missing values to obtain a complete 1-convex game? Second, how to determine in a rational, fair, and efficient way the payoffs of players based only on the known values of coalitions? We illustrate the analysis with two classes of incomplete games—minimal incomplete games and incomplete games with defined upper vector. To answer the first question, for both classes, we provide a description of the set of 1-convex extensions in terms of its extreme points and extreme rays. Based on the description of the set of 1-convex extensions, we introduce generalisations of three solution concepts for complete games, namely the \(\tau \)-value, the Shapley value and the nucleolus. For minimal incomplete games, we show that all of the generalised values coincide. We call it the average value and provide different axiomatisations. For incomplete games with defined upper vector, we show that the generalised values do not coincide in general. This highlights the importance and also the difficulty of considering more general classes of incomplete games.

不完全合作博弈模型通过考虑部分特征函数，将不确定性纳入经典合作博弈模型中。因此，某些联盟的价值是未知的。本文的主要关注点是该框架下的1-凸性。我们对两个紧密交织的问题感兴趣。首先，给定一个不完全博弈，我们如何填充缺失值以获得完整的1-凸博弈？其次，如何仅根据已知的联盟价值，合理、公平、高效地确定参与者的收益？我们用两类不完全博弈来说明分析——最小不完全博弈和具有定义的上向量的不完全博弈。为了回答第一个问题，对于这两个类，我们根据其极值点和极值射线提供了 1-凸扩展集的描述。基于 1-凸扩展集的描述，我们引入了完整博弈的三个解概念的概括，即 \ (\tau \)值、Shapley 值和核仁。对于最小的不完全游戏，我们证明所有的广义值都是一致的。我们将其称为平均值并提供不同的公理化。对于具有定义的上向量的不完全博弈，我们表明广义值通常并不重合。这凸显了考虑更一般的不完全博弈类别的重要性和难度。