A cowboy rides into a new town and stops at the local hotel for breakfast. He knows that there is a local outlaw who enjoys killing people in gunfights, but who is (naturally) averse to being killed. The cowboy may be of two types: a coward or a gunslinger. If the outlaw attacks the coward, the outlaw wins. If the outlaw attacks the gunslinger, the gunslinger wins. The hotel serves two things at breakfast: quiche and beer. The coward prefers quiche and the gunslinger prefers beer. The cowboy must decide what breakfast to have, and the outlaw observes this choice, and must decide whether to start a gunfight with the cowboy.
A) Is there a separating Nash equilibrium in which the coward orders quiche for breakfast and the gunslinger orders beer?
Now consider a slightly modiﬁed version of this game, in which the coward has a much strongerpreference for quiche: