In order to find an optimal and time consistent cooperative path in multicriteria multistage game the minimal sum of relative deviations rule is introduced. Using this rule one can construct a vector-valued characteristic function that is weakly superadditive. The sustainability of the cooperative agreement is ensured by using an imputation distribution procedure (IDP) based approach. We formulate the conditions an IDP should satisfy to guarantee that the core is strongly time consistent (STC). Namely, if the imputation distribution procedure for the Shapley value satisfies the efficiency condition, the strict balance condition and the strong irrational-behavior-proof condition, given that the Shapley value belongs to the core of each subgame along the cooperative path, it can be used as a “supporting imputation” which guarantees that the whole core is STC. We discuss three payment schedules and check whether they can be used as supporting imputation distribution procedures for the considered multicriteria game.