Algorithmic aspects of core nonemptiness and core stability

In 1944, von Neumann and Morgenstern developed the concept of stable sets as a solution for coalitional games. Fifteen years later, Gillies popularized the concept of the core, which is a convex polytope when nonempty. In the next decade, Bondareva and Shapley formulated independently a theorem describing a necessary and sufficient condition for the non emptiness of the core, using the mathematical objects of minimal balanced collections. We start our investigations of the core by implementing Peleg's (1965) inductive method for generating the minimal balanced collections as a computer program and then, an algorithm that checks if a game admits a nonemptiy core or not. In 2020, Grabisch and Sudhölter formulated a theorem describing a necessary and sufficient condition for a game to admit a stable core, using several mathematical objects and concepts such as nested balancedness, balanced subsets which generalized balanced collections, exact and vital coalitions, etc. In order to reformulate the aforementioned theorem as an algorithm, a set coalitions has to be found that, among other conditions, determines the core of the game. We study core stability, geometric properties of the core and in particular, such core determining sets of coalitions. Furthermore, we describe a procedure for checking whether a subset of a given set is balanced. Finally, we implement the algorithm as a computer program that allows to check if an arbitrary balanced game admits a stable core or not.