Optimism and commitment: An elementary theory of bargaining and war

We propose an elementary theory of wars fought by fully rational contenders that features three of the main rationalist explanations for armed conflicts: uncertainty, commitment, and indivisibility. Two parties play a Markov game that combines stages of bargaining, where offers are made, with claim stages, where one side can commit to impose surrender on the other. Under uncertainty on the persistence of claims, long confrontations occur in the unique equilibrium of the game: war arises when reality disappoints initial (rational) optimism, and it persists when both agents are optimists but reality proves both wrong. Bargaining proposals that are rejected initially might eventually be accepted after several periods of confrontation. We provide an explicit computation of the equilibrium, evaluating the probability of war, and its expected losses as a function of (i) the costs of confrontation, (ii) the asymmetry of the split imposed under surrender, and (iii) the strengths of contenders at attack and defense. Changes in these parameters display non-monotonic effects.