Dynamic Pricing and Learning with Long-term Reference Effects
Shipra Agrawal Wei Tang
We consider a dynamic pricing problem where customer response to the current
price is impacted by the customer price expectation, aka reference price. We
study a simple and novel reference price mechanism where reference price is the
average of the past prices offered by the seller. As opposed to the more
commonly studied exponential smoothing mechanism, in our reference price
mechanism the prices offered by seller have a longer term effect on the future
customer expectations.
  We show that under this mechanism, a markdown policy is near-optimal
irrespective of the parameters of the model. This matches the common intuition
that a seller may be better off by starting with a higher price and then
decreasing it, as the customers feel like they are getting bargains on items
that are ordinarily more expensive. For linear demand models, we also provide a
detailed characterization of the near-optimal markdown policy along with an
efficient way of computing it.
  We then consider a more challenging dynamic pricing and learning problem,
where the demand model parameters are apriori unknown, and the seller needs to
learn them online from the customers' responses to the offered prices while
simultaneously optimizing revenue. The objective is to minimize regret, i.e.,
the $T$-round revenue loss compared to a clairvoyant optimal policy. This task
essentially amounts to learning a non-stationary optimal policy in a
time-variant Markov Decision Process (MDP). For linear demand models, we
provide an efficient learning algorithm with an optimal $\tilde{O}(\sqrt{T})$
regret upper bound.