Moving Forward vs. Inflicting Costs in a Random-Walk Model of War

How do belligerents choose and change their military strategies during war? How do these strategies shape war? To address these questions, we develop a random-walk model of war, where two belligerents fight over "forts" across periods. The random walk represents a battlefront, which moves as the war evolves, resulting in the occupation of more forts for the winning side and less forts for the losing side. Unlike existing models, ours allows the belligerents to choose an action out of moving forward, inflicting costs, and surrender in every battle. We found that equilibrium strategies are monotonic with respect to the walk---a belligerent will punish its opponent if it is sufficiently advantageous and surrender if it is too disadvantageous. Accordingly, the punishment strategy can function to shorten the war. Moreover, a severer punishment tends to make the war even shorter.