Three messages are not optimal in worst case interactive communication

Let X and Y be two jointly distributed random variables. Suppose person P_X, the informant, knows X, and person P_Y, the recipient, 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, whereas P_X may or may not learn Y. Cˆm(X|Y) is the minimum number of bits required to be transmitted (by both persons) in the worst case when only m message exchanges are allowed. Cˆ∞(X|Y) is the number of bits required when P_X and P_Y can communicate back and forth an arbitrary number of times. Orlitsky proved that for all (X,Y) pairs, Cˆ₂(X|Y)⩽4Cˆ∞(X|Y)+3, and that for every positive c and ∈ with ∈<1, there exist (X,Y) pairs with Cˆ₂(X|Y)⩾(2-∈)Cˆ₃ (X|Y)⩾(2-∈)Cˆ-∞(X|Y)⩾c. These results show that two messages are almost optimal, but not optimal. A natural question, then, is whether three messages are asymptotically optimal. In this work, the authors prove that for any c and ∈ with 0<∈<1 and c>0, there exist some (X,Y) pairs for which Cˆ₃(X|Y)⩾(2-∈)Cˆ₄(X|Y)⩾c. That is, three messages are not optimal either