A general characterization for non-balanced games in terms of U-cycles

In a paper by Cesco [Cesco, J.C., 2003. Fundamental cycles of pre-imputations in non-balanced TU-games. International Journal of Game Theory 32, 211–222], it was proven that the existence of a certain type of cycles of pre-imputations, fundamental cycles, is equivalent to the non-balancedness of a TU-game, i.e., the emptiness of the core of the game. There are two characteristic sub-classes related to fundamental cycles: U-cycles and maximal U-cycles. In this note we show that it is enough to consider U-cycles in obtaining a similar characterization for non-balanced TU-games.