I am a bit stuck on this problem:
(a) $\quad $ Consider a jury system with $12$ jurors in which a defendant is found guilty if voted guilty by $10$ or more of the jurors. We represent this jury system as a coalitional game where $v(S)=1$, if the defendant is found guilty if voted guilty by all members of $S$, and $v(S)=0$ otherwise.
Find the nucleolus of the game. (I know how to find the core of the game easily.)
(b) $\quad$ Now assume that there is a judge in addition to the jurors ,and that to be found guilty the defendant in particular has to be voted guilty by the judge.
Find the nucleolus of the new game.
My main problem is that finding the nucleolus in this $12$ (resp. $13$)-player game relies on solving $12$ (resp. $13$) linear programs, which is practically way too tedious. Any reasonable methods to find this?
I would propose to have a look on my Matlab Game Theory Toolbox (MatTuGames) that can be found at the following URL
http://www.mathworks.com/matlabcentral/fileexchange/35933-mattugames
You can compute the (pre-)nucleolus using Matlab's Optimization Toolbox or, a CPLEX, GUROBI, MOSEK, CDDMEX, GLPK interface. The expected computation time for a 12-person game on a system with two octa-core Intel Xeon processors E5-2670 is about 15-60 seconds depending on the game class and commercial solvers you are calling. On such a system you can compute the (pre-)nucleolus in a reasonable time up to 15 players. However, much faster is the search process for a (pre-)kernel element, it is less than 1/10 second. So if you know that the (pre-)kernel coincides with the (pre-)nucleolus use this function. The implemented algorithm to evaluate a pre-kernel point by this toolbox is described in my book that can be found at
http://www.springer.com/economics/game+theory/book/978-3-642-39548-2
Finding a (pre-)kernel element for a 34-person TU game lasts about 20 minutes.
Update: In the meantime I had some time to look closer on your problem. The simple games are symmetric, therefore the nucleolus distributes 1/n to each player. You can verify these results with the above toolbox in less than a second.
Update 2 For completeness I will give now the answer to question b. The judge is here the veto player, and all jurors must be treated equally in accordance with the equal treatment property (ETP). This implies that the nucleolus distributes zero to the jurors, and one to the judge.