We propose an elementary theory of wars fought by fully rational contenders.
Two parties play a Markov game that combines stages of bargaining with stages
where one side has the ability to impose surrender on the other. Under uncertainty
and incomplete information, in the unique equilibrium of the game, long
confrontations occur: war arises when reality disappoints initial (rational) optimism,
and it persist longer 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