  We consider a generalization of the third degree price discrimination problem
studied in Bergemann et al. (2015), where an intermediary between the buyer and
the seller can design market segments to maximize any linear combination of
consumer surplus and seller revenue. Unlike in Bergemann et al. (2015), we
assume that the intermediary only has partial information about the buyer's
value. We consider three different models of information, with increasing order
of difficulty. In the first model, we assume that the intermediary's
information allows him to construct a probability distribution of the buyer's
value. Next we consider the sample complexity model, where we assume that the
intermediary only sees samples from this distribution. Finally, we consider a
bandit online learning model, where the intermediary can only observe past
purchasing decisions of the buyer, rather than her exact value. For each of
these models, we present algorithms to compute optimal or near optimal market
segmentation.
