This chapter presents the formal details of the mathematical model of the laws of war discussed in Chapter 3. It is intended for those interested in those details. Other readers may wish to skip ahead to Chapter 4, which presents the patterns of compliance with the laws of war. I present each of the three models separately and examine their equilibria. The first focuses on soldier-to-soldier interaction on the battlefield as a matching model between the two infinite sets of soldiers. The second models military discipline within a state’s military as a principal-agent model. The third model builds on Smith’s (1998a) model of war as a gambler’s ruin problem where the parties can influence their chance of winning by having their soldiers commit violations. The sides’ interests in limiting violence divide into three cases: (1) neither side wishes to commit violations, (2) both sides would like to commit violations on their own, and (3) one side would like to commit violations and the other does not. An agreement to limit violence is self-enforcing in the first case, and reciprocity could be used to enforce such an agreement in the other two cases.

After presenting the three models separately, I examine their interaction. The battlefield model has multiple equilibria with some having higher levels of violence than others. Additionally, states that wish to control the violations of their own soldiers can use military discipline to do so. Because the battlefield model has multiple equilibria with different levels of violations, states may wish to signal their preference for a restrained or an unrestrained battlefield. They lack the incentive to reveal this information on their own, but public acceptance of a treaty standard through ratification can screen out the parties that want an unrestrained battlefield. Those parties reject the treaty standard to induce the unrestrained battlefield equilibrium.