# Prenucleolus vs. nucleolus

I want to find a cooperative, characteristic function TU game $$v$$ (of at best of 3 or 4 players;2 players seem impossible to me) for which the prenucleolus is different from the nucleolus. I do not know if this is a research question or if it is more trivial as stated by the rules of this site. The prenucleolus is as usual defined by lexicographic minimization over ordered preimputations of excesses of all coalitions while the nucleolus is defined by lexicographic minimization over imputations. It seems that then the prenucleolus will be lexicographically strictly less than nucleolus. I just need an easy example of (pre)-nuclei which are not the same.

In the prenucleolus, symmetric players obtain the same payoff, hence the prenucleolus has the form $$(x,x,-2x)$$ for some real number $$x$$. The prenucleolus is the preimputation that minimizes the maximal excesses (in lexicographic order). Let us start by finding the $$x$$ that minimizes the maximal excess $$max_S E(S,x)$$. For each coalition $$S$$ draw the graph $$x \mapsto E(S,x) = v(S) - x(S)$$. This graph appears below. The maximal excess for a given $$x$$ is the upper contour $$\max\{ 1-2x,2x\}$$, whose minimum is attained at $$x=1/4$$. Hence the prenucleolus is $$(\frac{1}{4},\frac{1}{4},-\frac{1}{2})$$.
• How did you arrive at the maximal excess $-\frac{1}{2}$ ? Nov 10, 2019 at 16:29
• My guess was that $S=\{1,2\}$. But then the maximum is at $x=0$. Is there a better $S$ ? Nov 12, 2019 at 19:54
• I do not follow what if we take $S=\{1\}$ and $x_1=-1000 000 \ , x_2=x_3=500 000$. Then $x(\{1,2,3\})=0$, BUT $E(S,x)$ is virtually unbounded. What is wrong with this objection here? Nov 13, 2019 at 19:18