Consider the following delegation versus centralisation model of decision making, loosely based on some of the discussion in class.

. Consider the following delegation versus centralisation model of decision making, loosely based on some of the discussion in class. A principal wishes to implement a decision that has to be a number between 0 and 1; that is, a decision d needs to be implemented where 0 ≤ d ≤1. The difficulty for the principal is that she does not know what decision is appropriate given the current state of the economy, but she would like to implement a decision that exactly equals what is required given the state of the economy. In other words, if the economy is in state s (where 0 ≤ s ≤1) the principal would like to implement a decision d = s as the principal’s utility Up (or loss from the maximum possible profit) is given by P U = − s − d . With such a utility function, maximising utility really means making the loss as small as possible. For simplicity, the two possible levels of s are 0.4 and 0.7, and each occurs with probability 0.5.There are two division managers A and B who each have their own biases. Manager A always wants a decision of 0.4 to be implemented, and incurs a disutility UA that is increasing thefurther from 0.4 the decision d that is actually implement, specifically, UA = − 0.4 − d . Similarly, Manager B always wants a decision of 0.7 to be implement, and incurs a disutility UB that is (linearly) increasing in the distance between 0.7 and the actually decision that is implemented – that is 0.7 B U = − − d . Each manager is completely informed, so that each of them knows exactly what the state of the economy s is.(a) The principal can opt to centralise the decision but before making her decision – given she does not know what the state of the economy is – she asks for recommendations from her two division managers. Centralisation means that the principal commits to implement a decision that is the average of the two recommendations she received from her managers. The recommendations are sent simultaneously and cannot be less than 0 or greater than 1.Assume that the state of the economy s = 0.7. What is the report (or recommendation) thatManager A will send if Manager B always truthfully reports s?(b) Again the principal is going to centralise the decision and will ask for a recommendationfrom both managers, as in the previous question. Now, however, assume that both managersstrategically make their recommendations. What are the recommendations rA and rB made bythe Managers A and B, respectively, in a Nash equilibrium?(c) What is the principal’s expected utility (or loss) under centralised decision making (as inpart b)?(d) Can you design a contract for both of the managers that can help the principal implementtheir preferred option? Why might this contract be problematic in the real world?