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In cooperative game theory, a well-known solution concept is the so-called nucleolus. Following Maschler et altri (2013), the nucleolus is defined as follows:

Let $(N;v)$ be a coalitional game and let $K \subseteq \mathbb{R}^N$. The nucleolus of the game $(N;v)$ relative to $K$ is the set

$\mathcal{N}(N;v;K):= \{ x \in K : \theta(x) \succsim_L \theta(y), \forall y \in K \}$

Notice that $\succsim_L$ indicates lexicographic preference. In short, the nucleolus is a vector that assigns an amount of utility to every player. In case someone wants more information about the nucleolus, you can visit: Wikipedia. However, I cannot find anywhere if this payoff vector can be negative for some (or all) Players when the characteristic function $v(\cdot)$ is always non-negative.

Any help will be appreciated. Thank you all very much in advance.

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Note that Maschler et. al (2013, p. 804) introduce a general definition of the nucleolus over an arbitrary set $K \subseteq \mathbb{R}^{N}$. This set $K$ can be, for instance, the imputation set defined by $$X(N;v):=\{\mathbf{x} \in \mathbb{R}^{N}\; \big\arrowvert\; v(N) = x(N), \;\; x_{i} \ge v(\{i\})\;\;\forall i \in N\} $$ or the pre-imputation set given by $$X^{0}(N;v):=\{\mathbf{x} \in \mathbb{R}^{N}\; \big\arrowvert\; v(N) = x(N)\}, $$ in this case it is called the pre-nucleolus. If $K:=X^{0}(N;v)$, then it is permissible that the pre-nucleolus distributes negative payoffs. However, if $K:=X(N;v)$, then it depends whether $v(\{i\})$ is negative or not. For most economical relevant (non-cost) situations the outside option $v(\{i\})$ of all players $i \in N$ is set to a non-negative value, then, of course, it is not permissible that negative payoffs are distributed by the nucleolus. In case that it is allowed that $v(\{i\})$ is negative for at least one player $i$, a negative payoff is permissible.

Update:

To answer your request, consider, for example, the following non zero-monotonic four person game given by

$$v =[-4,-2,-4,-3,1,1,4,-2,-1,-1,3,2,5-6,3],$$

in the usual lexicographical order. We see that all players have negative outside options, and some coalitions $|S|>1$ have negative benefits too. Then the nucleolus of the game is given by

$$ nuc=[25/6,-5/3,1/6,1/3]. $$

We observe that player $2$ has a negative payoff under the distribution of the nucleolus. Notice in this context that even though the game is not zero-monotonic, the pre-nucleolus coincides in this case with the nucleolus of the game $v$.

If you want to play around a little bit with some examples by yourself, then you might be interested in my Matlab toolbox that can be found at

Matlab Toolbox MatTuGames

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  • $\begingroup$ Thank you very much. You perfectly replied to my question. I understand, then, that unless $v(\{ i \})$ for some $i \in N$, the Nucleolus is always non-negative. Correct? $\endgroup$ – Héctor Jun 9 '17 at 12:31
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    $\begingroup$ @Héctor Yes, if for all players $i \in N$ the outside option $v(\{i\})$ is non-negative, then the nucleolus distributes non-negative payoffs. $\endgroup$ – Holger I. Meinhardt Jun 9 '17 at 12:39
  • $\begingroup$ Since you posted the answer, I've been looking for a game with a nucleolus with a negative entry and I haven't been able to find one yet. Could you provide one such example, please? $\endgroup$ – Héctor Jun 9 '17 at 17:17
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    $\begingroup$ @Héctor I have updated my answer to satisfy your request. $\endgroup$ – Holger I. Meinhardt Jun 10 '17 at 11:54
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    $\begingroup$ Yes, this is correct. $\endgroup$ – Holger I. Meinhardt Mar 22 '18 at 8:34

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