Balanced Sets

For each individual $i$ let $\nn(i)$ denote the set of all coalitions containing $i$. A weighting scheme is a collection of weights $\lambda(S)$, each between 0 and 1. A weighting scheme is balanced if for all $i$, $\sum_{S\in\nn(i)}\lambda(S)=1$. A characteristic function is balanced if for every balanced weighting scheme $\{\lambda_S\}$, \begin{equation*} \sum_{S\subset\nn}\lambda_Sv(S)\leq v(\nn). \end{equation*}

Examples.

• If $\ss$ is a partition of $\nn$, take $\lambda_S=1$ for all $S\in\ss$.
• The convex combination of any balanced weighting schemes is a balanced weighting scheme.
• For $\nn=\{1,2,3\}$ the collection of all three two-element sets can be balanced with weights $1/2$ on each. Thus not all weighting schemes are convex combinations of partition schemes.

Theorem. A TU game has a core iff its characteristic function is balanced.