Consider the following situation. Colin and Rowena are meeting for dinner, but they wonāt be able to communicate beforehand. Itās 2 pm now, and thereās a probability equal to 0.5 that, starting from 8 pm, it will rain for the whole night. If at 8 pm the sky is still clear, then it wonāt rain at all. Colin is leaving the house now while Rowena will leave after 8 pm. Before stepping out, they must both decide whether to take an umbrella with them or not. Hence, at the time of their decision, Colin is uncertain about the weather while Rowena is not. Bringing an umbrella always costs 1 utility point, whether it rains or not. If thereās rain and none of the two has an umbrella, then they will both suffer a cost of 4 utility points. One umbrella among the two is sufficient to avoid the penalty from the rain. If there is no rain, then thereās no utility loss due to the weather. Hence, if both are out without an umbrella and thereās no rain, then their utility is normalized equal to zero.
(i) Model the situation above as a Bayesian game with two players, two actions each and a binary state of the world. Be careful in defining all elements of a Bayesian game. (Hint: you should write down payoffs in the two states using two 2x2 matrices.) (6 marks)
(ii) Find the pure-strategy Bayes-Nash equilibria of this game. (6 marks)