I have a problem with the concept of the bargaining set which is given below in some detail. Let $N=\{1,\ldots,n\}$ be a set of players and $v:2^N\to \mathbb{R}$ a superadditive game (meaning $S,T \subseteq N ,\,S \cap T=\emptyset \Longrightarrow v(S)+v(T)\le v(S \cup T).$) in coalitional function form. For $y\in\mathbb{R}^C$ and $S\subseteq N$ write $y(S)$ for $\sum_{i\in C}y_i.$ So $y(\emptyset)=0.$ Write ${\cal S_{kl}}=\{T \subseteq N\,|\,k \in T,\, l \notin T\}.$ Let $x\in \mathbb{R}^D$ be an individually rational imputation with $x(D)=v(D)$. What is the relationship between the following two formulas-I'd like them to be equivalent $\forall y\in \mathbb{R}^C$:
$(\exists D \in {\cal S_{lk}})\big(D \cap C=\emptyset \;\wedge\; x(D)\leq v(D)\big) \quad\vee$ $\Big((\forall i \in C)(y_i > x_i)\quad\wedge$ $(\forall D \in {\cal S_{lk}})\big(x(D \setminus C)+y(D \cap C) > v(D)\big)$ $\ \Longrightarrow \ y(C)>v(C)\Big).$
and
$(\exists D \in {\cal S_{lk}})\big(D \cap C=\emptyset \;\wedge\; x(D)\leq v(D)\big) \quad\vee$ $\Big((\forall i \in C)(y_i \geq x_i)\quad\wedge$ $(\forall D \in {\cal S_{lk}})\big(x(D \setminus C)+y(D \cap C) \geq v(D)\big)$ $\ \Longrightarrow \ y(C)\geq v(C)\Big).$
We may assume that the first part before $\vee$ which is the same in both is not satisfied. The only difference is by replacing some $>$ with $\geq.$ The problem arises when $D\cap C=\emptyset,$ otherwise they are equivalent by increasing coordinates of $y$ and slight considertaions. Perhaps here may be used the superadditivity of $v$?
I believe that the second is equivalent to this
$\Big(k,l\in N \;\wedge\; k \neq l \;\wedge\; C \in {\cal S_{kl}} \quad \Longrightarrow$
$(\exists D \in {\cal S_{lk}})\big(D \cap C=\emptyset \;\wedge\; x(D)\leq v(D)\big)\quad\vee$
$ \min\big\{y(C)\,|\, x(D \setminus C) +y(D \cap C) \geq v(D),\, D \in {\cal S_{lk}},\, y_i \geq x_i,\, i \in C \big\} \geq v(C)\Big),$ which is my aim to obtain from the first displayed formula with strict inequalities which are problematic to obtain these weak inequalities and $\min$. The entire thing here is to obtain weak linear inequalities connected by Boolean connectives that determine the Maschler bargaining set which is a quite interesting concept in cooperative game theory. The formulas are supposed to describe the bargaining set $\cal M$, which is such that there is no justified objection of a player $k$ to $l$. And it is of the form finite union of polytopes in $\mathbb{R}^N.$
