When a two-decision-maker conflict is generalized to a multi-decision-maker conflict, definitions of sequential stability must be refined because of ambiguities in meaning of sanctions. In an n-decision-maker conflict ( \(n>2)\) , when a focal decision maker/coalition has a unilateral improvement, its opponents will construct a sequence of movements to sanction the focal decision maker/coalition. A sanction is believable if it is preferred by all the sanctioners. After analyzing the problems of the existing definitions of non-cooperative and cooperative sequential stabilities within the paradigm of graph model, convincing definitions for sanctions of a focal decision maker/coalition’s initial unilateral improvements are proposed.