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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:

  1. 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?

  2. What is the sense / meaning of this?

  3. 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.

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I think I have to clarify some facts concerning a cooperative game before I can ask your question why the nucleolus can have negative outcomes when some players have negative outside options.

First of all let me mention that a cooperative game (Tu Game) can be considered as a stylized bargaining process. In this sense, the distribution of payoffs attributed by the nucleolus reflects the outcome of a bargaining process. If a player gets a negative payoff under the condition that she/he has a negative outside option, then this means that he had a weak negotiation position based under the rules of distributive justice related to the nucleolus, i.e., the set of axioms of the nucleolus. Here, you can replace the nucleolus with any other solution concept you know from cooperative game theory. Since, every solution concept has its own rules of distributive justice (set of axioms), and can therefore be interpreted as indicating an outcome of a bargaining process under these specific fairness rules.

Note that the rule of distributive justice needs not to be binding. Having made their choice, for instance, through a Rawlsian type of decision device about their preferred division scheme reflecting their objective norms of fairness, subjects must obey the underlying system of axioms by their self-interest. There is no possibility to renegotiate the outcome while formulating an objection. Such an objection must be based on a different set of unaccepted principles, and can therefore not be considered as valid in accordance with the underlying rule of distributive justice. In contrast, if the agreement is binding, the proposed solution can be enforced by an unbiased arbitrator or a legal system. This implies that an obstruction is impossible, even under the existence of strong incentives to deviate from it.

Finally, if you have problems with negative outside options, and therefore with negative outcomes for the nucleolus, then you have the possibility to transform this game into a zero-one normalized game that is equivalent to the default game.

I hope this helps to understand why distributed payoffs can be negative for cooperative solution concepts. If you want to study this topic in more details, then I can send you a treatise. In this case write me an e-mail that can be grasped from my Matlab toolbox.

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  • $\begingroup$ Thank you very much for your answer. It for sure helps, but I am still puzzled, to be honest. The whole point is that a subset of Players is getting more than the coalition can offer just because some other subset of Players is getting negative utility. This still looks weird to me. Suppose that a coalition of two Players can offer 0 total utility to its members. How could be reasonable to think that one Player gets 10 and the other one -10? I know this may the case, but I still find it too weird. I am busy now, but I'll definitely send you an email asking for that document later on. $\endgroup$ – Héctor Jun 12 '17 at 9:52
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    $\begingroup$ If the outside options gives $-10$ to this player, this means that the others can exploit some property assets from this player. It depends of how weak is her/his negotiation position whether the others can exploit the whole amount or only a part of it. $\endgroup$ – Holger I. Meinhardt Jun 12 '17 at 10:16
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    $\begingroup$ We can also put it differently. If the player stays alone, he loses the amount of $-10$ units instead of an amount $0>x >-10$, hence, it is more advantageous to join the grand coalition and to accept the proposed share. $\endgroup$ – Holger I. Meinhardt Jun 12 '17 at 10:24
  • $\begingroup$ Thank you for your comments and your help. I get the logic you are explaining me. Since it is how it is, I'll take it: there's not much to discuss, it's just my brain getting blocked. Anyway, all this feels absolutely unreal to me. If there are X units of anything to share, that someone gets less than 0 should not imply that someone else could get more than X. Say: "if you and I have 0 euros to share, how is it that you getting -5 implies me getting +5?" We had 0 at the beginning! Anyway, I'll keep forcing myself to believe that this is indeed the case. Once more, thank you for your patience. $\endgroup$ – Héctor Jun 12 '17 at 17:53

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