Can you provide a TU game $[N,\nu]$ that satisfies $\mathcal{PK}(N,\nu)\neq\mathcal{PN}(N,\nu)$? Following this question I posted some days ago, I have been suggested to explore several TU solution concepts, in particular the Prekernel ($\mathcal{PK}$). Unfortunately, this is a TU solution concept I'm not familiar with. For what I've read, the Prekernel coincides with the Prenucleolus ($\mathcal{PN}$) whenever the Prekernel consists of a single imputation, but it is different from the Prenucleolus whenever the Prekernel contains more than one imputation (see Peleg, 1986). I've been trying to come up with a game $[N,\nu]$ in which $\mathcal{PK}\neq\mathcal{PN}$, but I haven't been able to do so.
Thus, could anyone please provide a super additive game $[N,\nu]$ (ideally, with $|N|=4$) in which $\mathcal{PK}\neq\mathcal{PN}$? (please, make explicit the sets $\mathcal{PN}$ and $\mathcal{PK}$).
Thank you all very much for your time and effort.
 A: Consider a four person game which we quantify through
$v=[0,0,0,0,0,1,1/2,1,1,0,1,1,1,1,1,2]$
while taking the usual lexicographic order of coalitions. The pre-nucleolus of the game is given by
$\mathcal{PN}(N,v)=\{1,1,1,1\}/2,$
whereas the pre-kernel is a line segment that coincides with the core, and it is given by
$\mathcal{PK}(N,v)=conv(\{\{1,0,1,0\},\{1,3,1,3\}/4\})$.
We get $\mathcal{PK}(N,v)\neq \mathcal{PN}(N,v)$. The solutions are visualized by a small graphic

The green enlarged point is the pre-nucleolus, the blue dot is the Shapley value, the red dot is the pre-kernel element $\{1,3,1,3\}/4$, and the yellow line segment represents the pre-kernel solution as well as the core of the game.
The details of how to compute a pre-kernel element would be beyond the scope of our discussion. Nevertheless, a procedure as well as the theoretical foundation of calculating a pre-kernel element is explained in more details within my book that can be found at
The Pre-Kernel as a Tractable Solution for Cooperative Games
or you should have a look at the following link
Determining the Pre-Nucleolus
that gives at least some idea of how to proceed.
