Three results on interactive communication

X and Y are random variables. Person P_x knows X, Person P_y knows Y, and both know the underlying probability distribution of the random pair (X, Y). Using a predetermined protocol, they exchange messages over a binary, error-free, channel in order for P _y to learn X. P_x may or may not learn Y. Cˆ _m is the number of information bits that must be transmitted (by both persons) in the worst case if only m messages are allowed. Cˆ_∞ is the corresponding number of bits when there is no restriction on the number of messages exchanged. We consider three aspects of this problem. Cˆ₄. It is known that one-message communication may require exponentially more bits than the minimum possible: for some random pairs, Cˆ₁=2^Cˆ∞-1. Yet just two messages suffice to reduce communication to almost the minimum: for all random pairs, Cˆ₂⩽4Cˆ_∞+3. We show that, asymptotically, four messages require at most three times the minimum number of bits: for all random pairs, Cˆ₄⩽3Cˆ_∞+o(Cˆ_∞ ). Balanced pairs. Let μˆ be the maximum number of X values possible with a given Y value, and let ηˆ be the maximum number of Y values possible with a given X value. A random pair is balanced if μˆ=ηˆ. It is known that for all balanced pairs, three messages require at most log μˆ+o(log μˆ) bits, hence are asymptotically optimum. It was not known whether two messages are asymptotically optimum. We show that for every c and positive E there is a balanced pair such that Cˆ₂⩾(2-∈)Cˆ_∞⩾c. Asymptotically, this is the largest possible discrepancy. Amortized complexity. The amortized complexity of (X,Y) is the limit, as k grows, of the number of bits required in the worst case for L independent repetitions of (X, Y), normalized by k. We show that the four-message amortized complexity of all random pairs is exactly log μˆ. Hence, when a random pair is repeated many times, no bits can be saved if P_x knows Y in advance