Consider any cooperative TU game $[N,\nu]$ in characteristic function form, where $N$ is a finite set of Players $i\in N$ and $\nu:2^N\to\mathbb{R}$ is a characteristic function assigning some real worth to every coalition $L\subseteq N$. Then, let's define some concepts (all definitions are from Maschler, Solan and Zamir, Chapter 16)
SET OF IMPUTATIONS
Given any game $[N,\nu]$, the set of vectors $\mathbf{x}\in\mathbb{R}^N$ that satisfy $\sum_{i\in N}x_i=\nu(N)$ and $x_i\geqslant\nu(\{i\})$ for all Players $i\in N$ is the set of imputations of $[N,\nu]$, denoted $X^*(N,\nu)$.
SUPER ADDITIVE GAME
A game $[N,\nu]$ in characteristic function form is super additive if for every pair of coalitions $L,L^{\prime}\subset N$ such that $L\cap L^{\prime}=\emptyset$, $\nu(L\cup L^{\prime})\geqslant \nu(L)+\nu(L^{\prime})$.
Clearly, if a game is super additive, the grand coalition is the efficient partition of $N$ and the set of imputations is non-empty. Now, consider a super additive game $[N,\nu]$, fix a Player $i\in N$ and consider all non-identical couples that include Player $i$ (each couple has Player $i$ and some other Player $m\in N\backslash\{i\}$; these add up to $|N|-1$ couples). I am looking for the entire set of imputations $\mathbf{x}\in X^*$ that, for every such couple $\{i,j\}$, satisfy:
\begin{gather*}
x_i=\nu^{\prime}(\{i\})+\frac{1}{2}\left[\nu^{\prime}(i,j)-\nu^{\prime}(\{i\})-\nu^{\prime}(\{j\})\right]\\
x_j=\nu^{\prime}(\{j\})+\frac{1}{2}\left[\nu^{\prime}(i,j)-\nu^{\prime}(\{j\})-\nu^{\prime}(\{i\})\right]
\end{gather*}
Let $Q=N\backslash\{i,j\}$. Then, $\nu^{\prime}(\{i,j\})$, $\nu^{\prime}(\{i\})$ and $\nu^{\prime}(\{j\})$ are defined as follows:
\begin{gather*}
\nu^{\prime}(\{i,j\})=\nu(N)-\sum_{k\in Q}x_k\\
\nu^{\prime}(\{i\})=\max_{R_i\subseteq Q}\left\{\nu(\{i\}\cup R_i)-\sum_{k\in R_i}x_k\right\}\\
\nu^{\prime}(\{j\})=\max_{R_j\subseteq Q}\left\{\nu(\{j\}\cup R_j)-\sum_{k\in R_j}x_k\right\}
\end{gather*}
I am kind of sure that the Prenucleolus is one such vector, but I am not sure it is the only one. Intuitively, the reason is that the Prenucleolus is both standard for two and Davis and Maschler consistent (as well as individually rational for all Players $i\in N$ whenever $[N,\nu]$ is super additive). However, I can't find an intuitive reason why it should be the only one such vector satisfying the condition defined above.
Thank you all very much in advance for your time and effort.
EDIT 1 (from the first comment): if the original game $[N,\nu]$ is such that $N=\{i,j\}$, then the Shapley Value also satisfies the desired criterion, but as long as $|N|>2$, it fails to do so. The reason is simple: if $[N,\nu]$ is a two-Player game, then $[\{i,j\},\nu^{\prime}]=[N,\nu]$. But if $[N,\nu]$ is such that $|N|>2$, then $[\{i,j\},\nu^{\prime}]\neq [N,\nu]$ (at least, not necessarily). I hope this clarification, along with the comments, help to clarify the question.
EDIT 2 (from the second comment): no solution concept that can prescribe more than a single imputation for a $2$-Player game can satisfy the above criterion because it may not be standard for two in all DM reduced games $[\{i,j\},\nu^{\prime}]$. Thus, the Core does not satisfy the desired criterion. On the other hand, the Prenucleolus satisfies the criterion because it is Davis and Maschler consistent and standard for two. Thus, in any DM-reduced game $[S,\nu^{\prime}]$ at the Prenucleolus with $|S|=2$, the standard for two split is the Prenucleolus again! Hence, I am trying to figure out if the Prenucleolus is the only solution concept with this property (notice that this weak form of the RGP is implied by the standard RGP).
EDIT 3 (from the last comment): Because the phrasing of the original question made it almost impossible to answer, I re-phrase it in the following way: given some super additive game $[N,\nu]$, is there any imputation $\mathbf{x}\in X^*$ different from the Prenucleolus that satisfies the previous condition?
Best Answer
I am still not sure if I really got your point, however, I shall try to provide at least a small example -- but not a proof -- to give some evidence that every element of the pre-kernel also satisfies the standard solution for every two person DM-reduced game. For a proof of RGP and the pre-kernel -- that includes this special case -- have a look at Peleg and Sudhoelter (2007).
For this purpose, let us resume the game I have presented at
Four Person Game
I consider here only the reduced game of $T:=\{2,4\}$ w.r.t. the pre-kernel element $\mathbf{x}:=\{1,3,1,3\}/4$. This game has coalitional values of
$$v_{\mathbf{x},T}(\{2\})= 3/4 \qquad v_{\mathbf{x},T}(\{4\}) = 3/4 \qquad v_{\mathbf{x},T}(\{2,4\}) = 3/2, $$
which is a two person DM-reduced game. Applying now the standard solution, this gives $\vec{y}=\{3,3\}/4$. However, one can check out that the pre-kernel is given by $\mathcal{PK}(T,v_{\mathbf{x},T})=\{3,3\}/4$, which coincides with the pre-nucleolus. Since up to three person games both solution concepts are identical. Thus, we get $$\{\mathbf{x}_{T}\} = \mathcal{PK}(T,v_{\mathbf{x},T}) = \mathcal{PN}(T,v_{\mathbf{x},T}).$$
This examples reproduces the theoretical expected result that whenever $\langle N,v \rangle \in \Gamma, \emptyset \neq S \subseteq N$, and $\mathbf{x} \in \mathcal{PK}(N,v)$, then $\langle N,v_{\mathbf{x},S} \rangle \in \Gamma$ and $\mathbf{x}_{S} \in \mathcal{PK}(S,v_{\mathbf{x},S})$ must hold. This also includes the standard solution for every $\arrowvert S \arrowvert = 2$.