This question is a follow-up from this other question I recently posted on this site about the nucleolus, a well-known vectorial solution concept in cooperative Game Theory. In the first question, I asked if the nucleolus could have a negative entry. I was told it could; and I was provided with a correct example of such situation. Let me reproduce the example:
Let $(N,v)$ be a game with $N=4$ and $v(\cdot)$ defined as follows (in the usual lexicographic order): $v(\cdot) = [−4,−2,−4,−3,1,1,4,−2,−1,−1,3,2,5−6,3]$. The nucleolus of this game is just:
Nc(N,v) = [25/6,−5/3,1/6,1/3]
So far, so good. My question, though, is: how should one interpret a nucleolus with a negative entry?
Let me further clarify my doubt. The grand coalition in the example gets $3$. If you add up all four elements of the nucleolus, you get what the grand coalition can provide: $3$. But it seems to me that the Players getting a positive amount of utility in the nucleolus are getting it just because there is a guy getting negative which makes the total amount feasible. In other words, if it wasn't for the guy getting a negative utility, the total amount of those getting a positive utility would not be feasible (it exceeds $3$, which is the maximum the grand coalition can provide). In a way, there are three guys that, in total, are getting more than the grand coalition can provide; and then, this amount turns out to be feasible because someone else is getting negative. This is pretty much equivalent to saying that a subset of Players can get more than the grand coalition can provide as long as another subset of Players are getting negative utility. This, in turn, is kind of equivalent to saying that negative utility for some Players magically transforms into positive utility for some others.
Then, my question becomes three-fold:
How can a subset of Players get more than the grand coalition can provide as long as this extra amount is negative utility for some other Players?
What is the sense / meaning of this?
Would not it make much more sense that the amount that any subset of Players get is always feasible, independently of what other Players are getting?
This is not strictly a mathematical question; but rather, how to properly understand a concept in Game Theory, which is a mathematical field. Thank you all very much for your time and patience.