Consider the cooperative game $[N,\nu]$ given by a player set $N=\{i,m,n\}$ and the characteristic function $\nu:2^N\to\mathbb{R}$ given by $\nu(N)=8$, $\nu(\{i\})=3$, $\nu(\{m\})=2$, $\nu(\{n\})=1$ and $0$ else. I want to find the Prenucleolus of the game $[N,\nu]$ defined above, but I’m having some serious trouble.
At the present time, I do not have access to MatLab or any other software of that sort, and I don’t seem to be able to figure it out myself from the axioms that the Prenucleolus satisfies. To find the Prenucleolus from its defining axioms, I generally rely on the null player and the symmetry axioms, but neither of these axioms is helpful here (because there are no symmetric nor null players). Also, I generally check whether the Core contains a single element, case in which the Prenucleolus coincides with it (because it’s a core-selector). However, the Core of the game $[N,\nu]$ I defined above is very large, so the axiom of core selection is not helpful at all. Moreover, this game does not belong to the class of games for which the Shapley Value and the Prenucleolus coincide, so computing the Shapley Value (which is a trivial task) is not helpful, either.
Could someone help me compute the Prenucleolus of the game $[N,\nu]$ defined above?
EDIT: I made a profoundly dumb mistake when typesetting the original question. It turns out that the game whose Prenucleolus I need to find is the following one:
$N=\{i,m,n\}$, $\nu(N)=8$, $\nu(\{i,m\})=3$, $\nu(\{i,n\})=2$, $\nu(\{m,n\})=1$ and $0$ else.