The finite stage dynamic n-person games with transferable payoffs are considered. The cooperative version of the game is defined, and a new approach for constructing characteristic functions in multistage games based on characteristic functions defined in stage games is proposed. It is proved that the values of this new characteristic function dominate the values of characteristic function constructed using the min-max approach. This allows constructing the subcore of the classical core in the multistage game under consideration and guarantees that this new approach leads to time-consistent (works L. Petrosyan, G. Zaccour, 2003; L. Petrosyan, 1991) and in some cases strongly time-consistent solutions (paper L. Petrosyan, 1993). The example is provided showing the construction of this newly defined characteristic function and the time-consistency and strong time-consistency of the core.