funetion u (a, to} be 'What. is an optimal decision as the function of the probability r = Pr (to = all}? What is the expected utility as the function of this probability? Illustrate. Answer the same questions for the following matrix: Problem 3 Let the set of states be 9 ={wmwl} with the uniform prior belief, nu = (If-1, 3E4}, so that IP [to = tun]- = Il'i P [to = col]- = . Consider the following experiment E: Calculate the probability of observing each signal? [3}, the posterior beliefs after observ- ing them {43, and conrm that the belief evolution forms a martingale. Problem 4 Your volleyball team can either be good quality or bad quality (your prior belief is that the two qualities are equally likely}. Your team plays once a week. You do not know the quality of the opponents, but if your team is strong1 you win with probability; each week {independently of the other weeks}. If your team is weak1 then you win with probabilitg each week {independently of the other weeks}. There are no draws in volleyball. l. 1iii-"hat is your belief about the quality of your team after lweek, depending on your performance? 2. 1What is your belief about the quality of your team aer 1'! weeks, depending on your performance? 3. What is your belief about the quality of your team afterT' weeks, depending on your performance