The content of this talk is two papers of ours. Two different problems in cooperative game theory, which are related indirectly, are studied. In the first one, different notions of merge proofness for allocation rules pertaining to transferable utility games are introduced. Relations between these merge proofness notions are studied, and some impossibility as well as possibility results for allocation rules are established. In the second one, a decomposition of transferable utility games is introduced. Based on that decomposition, the structure of the set of all transferable utility games is analyzed. Using the decomposition and the notion of minimal balanced collections, a set of necessary and sufficient conditions for a transferable utility game to have a singleton core is given. An ‘indirect’ relation between the two problems will be given.