I present a 'cumulative assessment model', which describes dyadic antagonistic encounters in which a contestant's decision whether to persist or to flee is based upon a cumulative sum of its adversary's actions. It is particularly relevant to ritualized fights in which only a certain total of direct physical damage can be tolerated, but it can also be applied to displays without physical contact provided they are subject to external time costs (such as from predation risk). The cumulative assessment model provides an alternative to the sequential assessment model or the war of attrition as a description of temporally extended displays. I describe how the three may be distinguishable in real situations by consideration of escalatory properties and of characteristic intrapopulation variation. The model predicts that, under some circumstances, losers may start the encounter at a lower level of intensity but increase that level more rapidly than winners. Such behaviour has been observed in the cyprinodont fish Aphyosemion striatum. Copyright 1998 The Association for the Study of Animal Behaviour.