Prosperity properties of TU-games

An important open problem in the theory of TU-games is to determine whether a game has a stable core (Von Neumann-Morgenstern solution (1944)). This s eems to be a rather difficult combinatorial problem. There are many suffici ent conditions for core-stability. Convexity is probably the best known of these properties. Other properties implying stability of the core are subco nvexity and largeness of the core (two properties introduced by Sharkey (19 82)) and a property that we have baptized extendability and is introduced b y Kikuta and Shapley (1986). These last three properties have a feature in common: if we start with an arbitrary TU-game and increase only the value o f the grand coalition, these properties arise at some moment and are kept i f we go on with increasing the value of the grand coalition. We call such p roperties prosperity properties. In this paper we investigate the relations between several prosperity properties and their relation with core-stabili ty. By counter examples we show that all the prosperity properties we consi der are different.