Cops and Robbers. A police officer must decide whether to patrol the streets or hang out at the coffee shop. Simultaneously, a robber must decide whether to prowl the streets or stay in his hideaway. If the police officer patrols and the robber prowls, then the officer will surely make an arrest with a payoff of 10 for the officer's reputation. If the officer is in the coffee shop while the robber prowls, the payoff (in the form of damage to the officer's reputation) is -1. If the robber stays hidden, then the payoff to the officer is zero to patrol and 5 to hang out in the coffee shop.

(a) Formulate this decision problem as a two-person zero-sum game and determine the optimal strategies for the officer and robber.

(b) Compute the value of the game.

(c) Suppose that the robber, who did not take a course in game theory, has a police radio. He overhears the officer telling her partner of her mixed strategy. That is to say, the robber knows the probabilities with which the officer will patrol or get coffee (but he does not know which pure strategy the officer will actually choose). The robber thinks "Hmmm. That officer is more likely to get coffee than to patrol. Since I know this, I will prowl the streets and my chances of not getting caught are improved." Is the robber's argument valid? In other words, does the value of the game change if the robber knows the officer's mixing probabilities (and the officer sticks to them)?