War with Crazy Types

Abstract

This article introduces a model of war and peace in which leaders believe that their adversaries might be crazy types who always behave aggressively, regardless of whether it is strategically optimal to do so. In the model, two countries are involved in a dispute that can either end in a peaceful settlement or escalate into ‘limited war’ or ‘total war.’ If it is common knowledge that the leaders of the countries are strategically rational, then the only equilibrium outcome of the model is peace. Yet if a leader believes that there is a chance that her 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 strategically rational. The article shows that despite having two-sided incomplete information, the 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 pushes for more aggressive behavior. These two forces underlay several comparative statics results. For example, the study analyzes the effect of increasing the prior probability that a leader is crazy, and the effect of changing the relative military strengths of the countries, on the equilibrium behavior of both leaders. The analysis also characterizes the conditions under which the madman strategy is profitable (or not), which contributes to the debate in the literature about its effectiveness.