Convexity and positivity in partially defined cooperative games

Partially defined cooperative games are a generalisation of classical cooperative games in which payoffs for some of the coalitions are not known. In this paper we perform a systematic study of partially defined games, focusing on two important classes of cooperative games: convex games and positive games. In the first part, we focus on convexity and give a polynomially decidable condition for extendability and a full description of the set of symmetric convex extensions. The extreme games of this set, together with the lower game and the upper game, are also described. In the second part, we study positivity. We characterise the non-extendability to a positive game by existence of a certificate and provide a characterisation for the extreme games of the set of positive extensions. We use both characterisations to describe the positive extensions of several classes of incomplete games with special structures. Our results complement and extend the existing theory of partially defined cooperative games. We provide context to the problem of completing partial functions and, finally, we outline an entirely new perspective on a connection between partially defined cooperative games and cooperative interval games.