A note on the paradox of smaller coalitions

We consider hedonic coalition formation games that are induced by a simple TU-game and a cooperative solution. For such models, Shenoy's (1979) absence of the paradox of smaller coalitions provides a sufficient condition for core existence. We present three different versions of his condition in order to compare it to the top coalition property of Banerjee et al. (2001) that guarantees nonemptiness of the core in more general models. As it turns out, the top coalition property implies a condition in which Shenoy's paradox is not present for at least one minimal winning coalition. Conversely, if for each non-null player Shenoy's paradox is not present for at least one minimal winnig coalition containing that player, then the induced hedonic game satisfies the top coalition property.