A note on the paradox of smaller coalitions

Abstract

We consider hedonic coalition formation games that are induced by a simple TUgame 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 winning coalition containing that player, then the induced hedonic game satisfies the top coalition property.

A note on monotonic power indices, smaller coalitions, and new members

“…Dimitrov and Haake (2008) commented on this paradox as it plays a role when looking for stable coalitions. They introduced a weaker condition on a power index and a simple game by requiring that the following stronger paradox is not exhibited.…”

“…Shenoy (1979) showed that if a power index does not exhibit the paradox of smaller coalitions on a simple game, then there is a minimal winning coalition which is core stable-that is, which will not be left by any group of players. Dimitrov and Haake (2008) relaxed this condition and showed that it is sufficient that any coalition has at least one subcoalition in which each player is not worse off. This seems like a fair generalization of Shenoy's result; however, I will show that in the case of monotonic power indices, it is still quite restricting.…”