In the thesis two distinct topics in cooperative game theory are investigated. The first problem analyzed is one of the oldest unsolved problems in cooperative game theory. The question asks, under what conditions does an n-person, cooperative, TU game have a stable core? This problem is fundamental for n-person, cooperative, TU game theory as the solution of this problem would provide vital insights into certain properties of the core as well as revealing certain aspects of von Neumann-Morgernstern stable sets. In the thesis new sufficient conditions for core stability are found that turn out to also be necessary for certain classes of games.

In the second chapter of the dissertation the question of core stability is analyzed from a different perspective using the concept of a fuzzy game. This style of game is used to provide new necessary and sufficient conditions for core stability in terms of properties of two correspondences.

The second topic examined in this PhD, in the third chapter, concerns what is known as the apportionment problem. The problem in question is how one can apportion seats, power, etc., in a parliament, committee, etc., corresponding to the size, power, etc., of certain states or parties within a country, company, etc. One is confronted with this problem as soon as one wishes to represent the interests of certain groups in some sort of committee. Hence, this problem is age old but has only recently received a proper mathematical treatment in the twentieth century. In this thesis, a new apportionment method based on game theoretical concepts is investigated for its suitability as an apportionment method to be applied in reality. It is shown that the new apportionment does not fulfill certain desirable criteria. In addition, variations of the new apportionment methods are considered.