2. Two mathematicians are held in separate cells. In the morning the jailer comes to each mathematician and flips a fair coin. Each mathematician observes (only) the outcome of his own coin flip. After observing the outcome of his own coin flip each mathematician predicts the outcome of the other's coin flip. If at least one of the mathematicians predicts correctly, their life is spared. Otherwise they are both executed. Suppose that each mathematician's payoff is the expected probability of survival. (a) Formalize this scenario as a Bayesian game. What are the players, what are their types, what are their information sets, what are their strategies, and what are their payoffs? (b) Find all pure strategy Bayesian Nash equilibria. (c) Find a mixed-strategy Bayesian Nash equilibrium. (d) Find all Pareto efficient Bayesian Nash equilibria. 2. Two mathematicians are held in separate cells. In the morning the jailer comes to each mathematician and flips a fair coin. Each mathematician observes (only) the outcome of his own coin flip. After observing the outcome of his own coin flip each mathematician predicts the outcome of the other's coin flip. If at least one of the mathematicians predicts correctly, their life is spared. Otherwise they are both executed. Suppose that each mathematician's payoff is the expected probability of survival. (a) Formalize this scenario as a Bayesian game. What are the players, what are their types, what are their information sets, what are their strategies, and what are their payoffs? (b) Find all pure strategy Bayesian Nash equilibria. (c) Find a mixed-strategy Bayesian Nash equilibrium. (d) Find all Pareto efficient Bayesian Nash equilibria