This paper first establishes a new core theorem using the concept of generated payoffs: the TU (transferable utility) core is empty if and only if the maximum of generated payoffs (mgp) is greater than the grand coalition's payoff v(N), or if and only if it is irrational to split v(N). It then provides answers to the questions of what payoffs to split, how to split the payoff, what coalitions to form, and how long each of the coalitions will be formed by rational players in coalitional TU games. Finally, it obtains analogous results in coalitional NTU (non-transferable utility) games.