  A ubiquitous learning problem in today's digital market is, during repeated
interactions between a seller and a buyer, how a seller can gradually learn
optimal pricing decisions based on the buyer's past purchase responses. A
fundamental challenge of learning in such a strategic setup is that the buyer
will naturally have incentives to manipulate his responses in order to induce
more favorable learning outcomes for him. To understand the limits of the
seller's learning when facing such a strategic and possibly manipulative buyer,
we study a natural yet powerful buyer manipulation strategy. That is, before
the pricing game starts, the buyer simply commits to "imitate" a different
value function by pretending to always react optimally according to this
imitative value function.
  We fully characterize the optimal imitative value function that the buyer
should imitate as well as the resultant seller revenue and buyer surplus under
this optimal buyer manipulation. Our characterizations reveal many useful
insights about what happens at equilibrium. For example, a seller with concave
production cost will obtain essentially 0 revenue at equilibrium whereas the
revenue for a seller with convex production cost is the Bregman divergence of
her cost function between no production and certain production. Finally, and
importantly, we show that a more powerful class of pricing schemes does not
necessarily increase, in fact, may be harmful to, the seller's revenue. Our
results not only lead to an effective prescriptive way for buyers to manipulate
learning algorithms but also shed lights on the limits of what a seller can
really achieve when pricing in the dark.
