please solve problem 1 " regret" and problem 2 "value of information"

1. REGRET (10 POINTS) Consider a decisionmaker whose choices are determined in the following way: Given a choice set A, the agent chooses: are min{ max #06, ail}- .z-EA yEA\\x The story here is that the agent knows that if they choose cc, they will face some regret for not having picked y, where this regret is given by (:L', 3;). Their goal is to minimize the most regret they could face. 1. (3 points) Suppose (1501;, y) is constant in 3;. Is this model equivalent to utility maxi mization? 2. (3 points) Suppose A = {01, b}. Is this model equivalent to utility maximization? 3. (4 points) Construct a violation of transitivity when there are 3 or more choices. 2. VALUE OF INFORMATION (10 POINTS) 1. (7 points) Let: V(p) = maxlEg~p[v(a, 9)] (EA Show that V(p) is convex. (To consider a convex combination of beliefs, it may be helpful to write out the expectation as a sum) 2. (3 points) Suppose the state space is O = {G, B}, and consider the following information structures: For both, the signal space is T = {g, b}, and for Ti (with i E {1, 2} ) we have: P[t = g | 0 = G] = r;, P[t = b | 0 = B] =1 With r1 > r2. Is there a decisionmaker who would prefer T2 to I1? Prove that your answer is correct