This report uses a generic two-stage escalation model to ask whether and when limited conflicts can occur. There are two players in the model: Challenger and Defender. Challenger can either initiate a conflict or not. If Challenger initiates, Defender can concede, respond-in-kind, or escalate. If Defender does not concede, Challenger can escalate. The process continues until one side concedes or both escalate. Limited conflicts do not occur in our model when information is complete or when Defender's threat to respond-in-kind is seen to be completely noncredible. They are also extremely unlikely when Defender is seen strictly to prefer a response-in-kind to immediate capitulation when challenged. Limited conflicts are most probable under a Constrained Limited-Response Equilibrium (CLRE). Constrained Limited-Response Equilibria only occur when there is uncertainty about Defender's willingness to respond-in-kind to an initiation. The conditions associated with the existence of a CLRE and the other equilibria of the model are illustrated, both graphically and via a numerical example. Typically, under a CLRE, Challenger initiates and Defender concedes. From time to time, however, Challenger misjudges Defender's intentions and is surprised by a limited response. At this point, Defender chooses not to escalate the conflict because it concludes that Defender will counter-escalate and an all-out conflict will occur. Real-life examples of this process include the Gulf War, the Cuban Missile Crisis, the Fashoda Crisis of 1898, and the Korean crisis of 1950.
