Core, Least core and nucleolus

Solution 1:

I shall give you a new answer, since I suppose your edit requires a separate post.

In both cases, you are right. However, to compute the nucleolus under 2.) of your edit, it is sufficient to get the least-core at $\epsilon_{0}(v)=1/3$, since it consists of a single point.

Thus, the issue is more complex, if the least-core consists of infinite many points. For more complex games, the common method is to solve a sequence of LPs in order to find the nucleolus of a game. Fortunately, the nucleolus is part of the kernel, and if the kernel is single-valued, it is given by the nucleolus. This means, that for those classes of games, the nucleolus can be determined by methods that are designed to determine a kernel element, which are much easier to handle than solving sequences of LPs. Hence, you can apply the approach that was described in the following post

Finding the Nucleolus

to get the nucleolus for these classes of games. I suppose that this shall give you some insight where are the pitfalls in finding the nucleolus.