Average-case interactive communication

X and Y are random variables. Person P_x knows X, Person P_y knows Y, and both know the joint probability distribution of the pair (X,Y). Using a predetermined protocol, they communicate over a binary error-free channel in order for P_y to learn X. P_x may or may not learn Y. It is determined how many information bits must be transmitted (by both persons) on the average. The results show that, when the arithmetic average number of bits is considered, there is no asymptotic advantage to P_x knowing Y in advance and four messages are asymptotically optimum. By contrast, for the worst-case number of bits, communication can be significantly reduced if P_x knows Y in advance, and it is not known whether a constant number of messages is asymptotically optimum