War with Crazy Types

We model a situation in which two countries are involved in a dispute. The dispute can end in a peaceful settlement, or it can escalate to war. If it is common knowledge that the countries are strategically rational, then the only equilibrium outcome of the model is peace. If, on the other hand, each country believes that there is some chance that its adversary is a crazy type that always behaves aggressively, then even a strategically rational country may have an incentive to pretend to be crazy. This leads to war with positive probability. In addition to being qualitatively different from the existing literature, our model (i) enables a more tractable analysis of two-sided incomplete informa- tion, (ii) has a generically unique equilibrium prediction, and (iii) yields several new comparative statics results. For example, we analyze the effect of increas- ing the prior probability that the countries are crazy types, as well as the effect of changing the relative military strengths of the countries, on equilib- rium behavior. In studying these comparative statics, our model identifies two countervailing forces that arise when the prior probability that a country is crazy decreases: a reputation motive that promotes less aggressive behavior by that country, and a defense motive that promotes more aggressive behavior by the other country.