Think twice before attacking: Effort, restraint, and sanctions in war conflicts

There are contrasting empirical findings about how third parties mitigate conflict and few theoretical results about war efforts and restraint. Applying a new approach that distinguishes between quantity and effectiveness of effort, we explain countries’ restraint incentives. In our setting, restraint is manifested in reducing the effectiveness of military efforts when third parties impose costs (sanctions) on unrestrained behavior. We show that intermediate sanctions may lead to a mixed-strategy equilibrium or pure strategy equilibria in which only one country reduces its effectiveness. We present the conditions that yield pure-strategy (in which, surprisingly, restraint is a corner solution) and mixed-strategy (MS) equilibria in our multi-stage model. Interestingly, a country's effort decreases with its or its rival's self-restraint, allowing a third party to reduce efforts by sanctioning only one country. Our analysis enriches the classical results, particularly those obtained in the context of war conflicts, which focus on the extent of effort expended and on means of mitigating the conflict. The theoretical study lays the groundwork for future research into war contests and suggests various avenues for further exploration.