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. How many information bits must be transmitted (by both persons) on the average? At least H (XjY) bits must be transmitted and H (X) + 1 bits always suuce 1. If the support set of (X; Y) is a Cartesian product of two sets, then H (X) bits must be transmitted. If the random pair (X; Y) is uniformly distributed over its support set, then H (XjY) + 3 log (H(XjY) + 1) + 17 bits suuce. Furthermore, this number of bits is achieved when P X and P Y exchange four messages (sequences of binary bits). The last two results show that when the arithmetic average number of bits is considered: (1) there is no asymptotic advantage to P X knowing Y in advance; (2) four messages are asymptotically optimum. By contrast, for the worst-case number of bits: (1) communication can be signiicantly reduced if P X knows Y in advance; (2) it is not known whether a constant number of messages is asymptotically optimum. These results extend the work of El Gamal and Orlitsky 1] on the number of bits that must be transmitted when P X and P Y wish to exchange X and Y. They have several implications for a problem of Slepian and Wolf 2] where P X knows a random variable X , P Y knows a random variable Y , and each communicates with a third person P who wants to learn both X and Y .