Incomplete third-degree price discrimination, and market partition problem

We introduce in this paper the "incomplete" third-degree price discrimination, which is the situation where a monopolist must charge at most k different prices while the total market is composed of n markets, with n>k. We thus study the optimal partition problem of the n markets in k groups. As a byproduct, when k=2, we are able to reconsider the so-called (Robinsonian) weak-strong partition.