Between the prekernel and the prenucleolus

A collection of $TU$ games solutions intermediate between the prekernel and the prenucleolus is considered. All these solutions are Davis-Maschler consistent, symmetric and covariant. Each solution from the collection is parametrized by a positive integer $k$ such that for all games with the number of players not greater than $k$, the solution for parameter $k$ coincides with the prenucleolus, and for games with more than $k$ players it is maximal, i.e. it satisfies the "$k$-converse consistency". The properties of solutions are described and their characterization in terms of balancedness is given.