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The purpose of this paper is to compare and contrast two formal models of escalation and de-escalation: the attractor landscape model and the S-shaped reaction function model. Also, the paper aims to enumerate conditions that affect the shape and location of reaction functions and, hence, the stability of less and more escalated states. Both models are presented together with geometric proofs of the main assertions of the second model. Overlap and comparative strengths of the models are reviewed. Parts of the social science literature are synthesized in a discussion of the antecedents of stability. Though derived from totally different traditions, these models are similar in their basic assumptions and predictions. Each model has value. The attractor landscape model is easier to grasp and contains a concept of resistance to escalation that is not found in the S-shaped reaction function model. The latter model looks at individual parties rather than the dyad as a whole and, thus, offers an explanation for most of the phenomena described by the former model. It also allows identification of many variables that affect the shape and location of reaction functions and, hence, can be viewed as antecedents of escalation and de-escalation. Seven testable hypotheses are presented in the Conclusions section. Laboratory tasks for testing such hypotheses have yet to be developed and there is only one study employing real-life measures. However, it is clear that once research on these phenomena really begins, new variables will be found that moderate the strength of the effects hypothesized. The models provide concepts for thinking about how to avoid runaway escalation and promote runaway de-escalation. The variables mentioned in the hypotheses suggest ways to diminish the likelihood of runaway escalation and can also be used for constructing measures of the likelihood of that phenomenon. The theories also imply that when the likelihood of runaway escalation increases, disputants should be doubly careful to avoid initiating escalative behavior. The article is original in that the S-shaped reaction function model is refined and further developed and the proofs are new. The comparison between the models is also new, as is most of the enumeration of conditions affecting the stability of low and high escalation. The value of the article is to provide concepts and theory for thinking about escalation and de-escalation, and testable hypotheses for studying these phenomena.
