I have a problem with finding nucleolus and prenuclelous for a given weighted majority game. I know all the definitions, but I cannot really grasp the concepts, so I would also appreciate some explanations as well.
Let N = {1,...,5};w = (2; 2; 1; 1; 1), and q = 4. Consider the weighted majority game (N; v) represented by (q;w), i.e., for S ⊆ N, v(S) = 1 if w(S) = ∑w > q and v(S) = 0 if w(S) < q.
The coalitions of the weighted majority game (wmg) whose value is one are:
{{1,2},{1,2,3},{1,2,4},{1,3,4},{2,3,4},{1,2,5},{1,3,5},{2,3,5},{1,4,5},{2,4,5},{1,2,3,4},{1,2,3,5},{1,2,4,5},{1,3,4,5},{2,3,4,5},N }
otherwise zero. Then the nucleolus/pre-nucleolus of the game is given by
$\nu(wmg)=(2,2,1,1,1)/7.$
while solving a sequence of linear programs. For more information how doing that, I recommend to read Chapter 20 of the book M. Maschler, E. Solan, and Zamir Sh. Game Theory. Cambridge University Press, Cambridge, 1 edition, 2013.
However, you should try to become more familiar with this solution concept while studying first a simpler game at
Nucleolus of a Three Person TU Game
Accompanying your progress in understanding this solution concept, you may want to rely on the following software packages:
Software for Computing the (Pre-)Nucleolus