Pricing without Priors

Abstract

We consider the problem of pricing a single object when the seller has only minimal information about the true valuation of the buyer. Specifically, the seller only knows the support of the possible valuations and has no further distributional information. The seller is solving this choice problem under uncertainty by minimizing her regret. The pricing policy hedges against uncertainty by randomizing over a range of prices. The support of the pricing policy is bounded away from zero. Buyers with low valuations cannot generate substantial regret and are priced out of the market. We generalize the pricing policy without priors to encompass many buyers and many qualities.

Monopoly pricing with limited demand information

“…A similar mathematical programming proof of their result also appeared in (Lan et al, 2008). As mentioned earlier, yet another alternative interpretation is to treat the t j 's as probabilities and t as a randomized pricing policy; see (Bergemann and Schlag, 2008) for analogous results using the maximum regret criterion. We build on this result to explore issues of learning in upcoming sections.…”

“…They have been used extensively in the computer science literature, and have recently been applied in pricing and operations management problems. Specifically, Ball and Queyranne (2009) used a CR criterion for a singleresource capacity allocation problem, while Bergemann and Schlag (2008) and Perakis and Roels (2008) adopted the regret criterion to study the monopolist pricing and the newsvendor problems, respectively. Lan et al (2008) generalizes Ball and Queyranne's analysis and extends it to cover the regret criterion as well.…”

“…The firm has the ability to change its price over time, and its key decision is to figure out a pricing policy (how much to charge and for how long to stay at each price point) that would perform well even though the firm does not know the underlying consumers' WtP distribution. An alternate interpretation of this 'price-skimming' policy, in which different levels of prices are charged for varying amount of time, 2 is to treat the relative length of time over which a price is offered as a probability of that price point, thus interpreting the proposed price scheme as a randomizing pricing policy; this was done in (Bergemann and Schlag, 2008).…”

“…An alternative interpretation of our model is one where the firm sells one product through many potential stores, and where the firm controls how many stores will adopt any given price level in each period. (This is also an alternate interpretation for the randomized policy of Bergemann and Schlag (2008).) In that setting, the firm selects at how many stores to apply each potential price point, and then combines the information extracted from all these stores to update its demand information and optimize its downstream pricing decisions.…”

“…One exception is Perakis and Roels (2007), which incorporates partial demand information using the probabilistic tight bounds for mean and variance specifications derived by Bertsimas and Popescu (2005). Another is Bergemann and Schlag (2008), which allows for the unknown distribution to be within a distance of a nominal distribution that could encapsulate prior information. Both articles work with the regret criterion and do not generalize easily to the CR criterion, and both frameworks do not seem to allow for a tractable way in which to incorporate intuitive information extracted from past sales that usually translate to fractiles of the WtP distribution.…”