Two messages are almost optimal for conveying information

Abstract

X and Y are random variables. Person Px knows X, Person Py 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 Py to learn X. Px may or may not learn Y. How many information bits must be transmitted (by both persons) in the worst case if only m messages are allowed? C1(X|Y) is the number of bits required when at most one message is allowed, necessarily from Px to Py. C2(X|Y) is the number of bits required when at most two messages are permitted: Py transmits a message to Px, then Px responds with a message to Py. C∞(X|Y) is the number of bits required when communication is unrestricted: Px and Py can communicate back and forth. This article discusses the reduction in communication achievable via interaction and estimates its complexity.