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! (ABSOLUTE VALUE). 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 ?B U = ?!0.7 ? 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) that
Manager 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 recommendation
from both managers, as in the previous question. Now, however, assume that both managers
strategically make their recommendations. What are the recommendations rA and rB made by
the Managers A and B, respectively, in a Nash equilibrium?
(c) What is the principal?s expected utility (or loss) under centralised decision making (as in
(d) Can you design a contract for both of the managers that can help the principal implement
their preferred option? Why might this contract be problematic in the real world?