Abstract
The model of incomplete cooperative games incorporates uncer7 tainty into the classical model of cooperative games by considering a partial 8 characteristic function. Thus the values for some of the coalitions are not 9 known.
The main focus of this paper is 1-convexity under this framework. 10 We are interested in two heavily intertwined questions. First, given an 11 incomplete game, how can we ll in the missing values to obtain a complete 12 1-convex game? Second, how to determine in a rational, fair, and ecient way 13 the payos of players based only on the known values of coalitions? 14 We illustrate the analysis with two classes of incomplete games - minimal 15 incomplete games and incomplete games with dened upper vector.
To answer 16 the rst question, for both classes, we provide a description of the set of 1- 17 convex extensions in terms of its extreme points and extreme rays. Based on 18 the description of the set of 1-convex extensions, we introduce generalisations 19 of three solution concepts for complete games, namely the -value, the Shapley 20 value and the nucleolus.
For minimal incomplete games, we show that all 21 of the generalised values coincide. We call it the average value and provide 22 dierent axiomatisations.
For incomplete games with dened upper vector, we 23 show that the generalised values do not coincide in general. This highlights 24 the importance and also the diculty of considering more general classes of 25 incomplete games.