Can the inequality below be satisfied for an imputation $\mathbf{x}\in\mathcal{PK}(N,\nu)$?

Question

Consider any $0$-monotonic TU game $[N,\nu]$.

A game $[N,\nu]$ is $0$-monotonic if $\nu(L\cup\{i\})\geqslant\nu(L)+\nu(\{i\})$ for all Players $i\in N$ and all coalitions $L\subseteq N\backslash\{i\}$.

Further, consider the Kernel of such a game, denoted $\mathcal{K}(N,\nu)$. By Maschler, Peleg & Shapley (1971), the Kernel and the Prekernel, denoted $\mathcal{PK}(N,\nu)$, coincide whenever $[N,\nu]$ is $0$-monotonic.

The Prekernel is defined as the set of vectors $\mathbf{x}\in\mathbb{R}^N$ that satisfy $\sum_{i\in N}x_i=\nu(N)$ and \begin{gather*} \max_{S\subseteq N}\left\{\nu(S)-\sum_{k\in S}x_k\mid i\in S,j\notin S\right\}=\max_{S^{\prime}\subseteq N}\left\{\nu(S^{\prime})-\sum_{k\in S^{\prime}}x_k\mid j\in S^{\prime},i\notin S^{\prime}\right\} \end{gather*}

Let me now phrase my question: given a $0$-monotonic game $[N,\nu]$, is it possible to find a vector $\mathbf{x}\in\mathcal{K}(N,\nu)$ and a pair of Players $i,j\in N$ for whom the inequality below is satisfied? \begin{gather}\label{eq1} \nu(N)-\sum_{k\in N\backslash\{i,j\}}x_k<\max_{S\subseteq N\backslash\{i,j\}}\left\{\nu(\{i\}\cup S)-\sum_{k\in S}x_k\right\}+\max_{S^{\prime}\subseteq N\backslash\{i,j\}}\left\{\nu(\{j\}\cup S^{\prime})-\sum_{k\in S^{\prime}}x_k\right\} \end{gather}

Either provide an example showing that it is possible, or provide a solid argument showing it is not possible.

Thank you all very much in advanced for your time.

EDIT: following the first comment, I provide here some background. I am basically studying the set of SPE outcomes of a bargaining game. So far, I have been able to prove that the if $\mathbf{x}\notin\mathcal{PK}(N,\nu)$, then $\mathbf{x}$ is not a SPE outcome. However, I am still not sure for which family of games $[N,\nu]$ a SPE exists, nor which Prekernel elements can be sustained as an SPE. I know that if $\mathbf{x}$ satisfies the above inequality, then $\mathbf{x}$ is not a SPE outcome. Thus, I was curious to know if such an inequality could be satisfied for some $0$-monotonic game $[N,\nu]$ and some $\mathbf{x}\in\mathcal{PK}(N,\nu)$.

Answer 1

Let us consider the following six-person weighted majority game that we obtain from the following parameters:

$$[16;2,4,4,5,6,7],$$

which we have borrowed from Kopelowitz (1967). This game has a disconnected pre-kernel given by the two points

$$\mathcal{PK}(N,v)=\{\{0,0,0,0,1,1\}/2,\{0,1,1,1,1,1\}/5\}.$$

Notice, that this game is zero-monotonic and has an empty core. If we now select the second point of the pre-kernel solution, we get the following matrix of maximum surpluses

     0    0.4000    0.4000    0.4000    0.4000    0.4000
0.4000         0    0.4000    0.4000    0.4000    0.4000
0.4000    0.4000         0    0.4000    0.4000    0.4000
0.4000    0.4000    0.4000         0    0.4000    0.4000
0.4000    0.4000    0.4000    0.4000         0    0.4000
0.4000    0.4000    0.4000    0.4000    0.4000         0

Let us denote this matrix as $smat$. Similar, we get a matrix of surpluses over efficiency for all coalitions of $N\backslash \{i,j\}$ with $i,j \in N, i \neq j$, which is quantified through

     0    0.2000    0.2000    0.2000    0.2000    0.2000
0.2000         0    0.4000    0.4000    0.4000    0.4000
0.2000    0.4000         0    0.4000    0.4000    0.4000
0.2000    0.4000    0.4000         0    0.4000    0.4000
0.2000    0.4000    0.4000    0.4000         0    0.4000
0.2000    0.4000    0.4000    0.4000    0.4000         0

and denoting this matrix as $B$. Set then $A=smat+smat'$ and take the difference $A-B$, and we observe that the above condition is satisfied through

     0    0.6000    0.6000    0.6000    0.6000    0.6000
0.6000         0    0.4000    0.4000    0.4000    0.4000
0.6000    0.4000         0    0.4000    0.4000    0.4000
0.6000    0.4000    0.4000         0    0.4000    0.4000
0.6000    0.4000    0.4000    0.4000         0    0.4000
0.6000    0.4000    0.4000    0.4000    0.4000         0

Update: Weighted Majority Game:

To avoid to supply the whole game with $64$ coalitions, I just concentrate on the definition of a weighted majority game, and giving a short explanation of it.

Let the total number of votes are denoted as $w(N) \in \mathbb{N}$. For passing a bill at least $0 < th \le w(N)$ votes are needed. A simple game is referred to a weighted majority game, if there exists a quota/threshold $ th > 0$ and weights $w_{k} \ge 0$ for all $k \in N$ such that for all $S \subseteq N$ it holds either $v(S) = 1$ if $w(S) \ge th$ or $v(S) = 0$ otherwise. Such a game is generically represented as $[th; w_{1}, \ldots, w_{n}]$.

I hope this clarifies your question.