War with Crazy Types∗

We introduce a model of war and peace in which leaders believe that their adversaries might be crazy types that always behave aggressively, regardless of whether it is optimal to do so. In the model, two countries are involved in a dispute. The dispute can end in a peaceful settlement, or it can escalate to “limited war” or “total war.” If it is common knowledge that the leaders of countries are strategically rational, then the only equilibrium outcome of the model is peace. If, on the other hand, a leader believes that there is some chance that his adversary is a crazy type, then even a strategically rational adversary may have an incentive to adopt a “madman strategy” in which he pretends to be crazy. This leads to limited war with positive probability, even when both leaders are actually strategically rational. We show that despite having two-sided incomplete information, our model has a generically unique equilibrium. Moreover, the model identifies two countervailing forces that drive equilibrium behavior: a reputation motive and a defense motive. When the prior probability that a leader is crazy decreases, the reputation motive promotes less aggressive behavior by that leader, while the defense motive promotes more aggressive behavior by her adversary. These two forces underly several comparative statics results. For example, we analyze the effect of increasing the prior probability that a leader is crazy, as well as the effect of changing the relative military strengths of the countries, on the equilibrium behavior of both leaders. Our analysis also characterizes conditions under which the madman strategy is profitable, as well as conditions under which it is not, thereby contributing to a debate in the literature about the effectiveness of the madman strategy. JEL Codes: C7, F5, N4