The value of knowing a demand curve: bounds on regret for online posted-price auctions

Abstract

We consider the revenue-maximization problem for a seller with an unlimited supply of identical goods, interacting sequentially with a population of n buyers through an on-line posted-price auction mechanism, a paradigm which is frequently available to vendors selling goods over the Internet. For each buyer, the seller names a price between 0 and 1; the buyer decides whether or not to buy the item at the specified price, based on her privately-held valuation. The price offered is allowed to vary as the auction proceeds, as the seller gains information from interactions with the earlier buyers.The additive regret of a pricing strategy is defined to be the difference between the strategy's expected revenue and the revenue derived from the optimal fixed-price strategy. In the case where buyers' valuations are independent samples from a fixed probability distribution (usually specified by a demand curve), one can interpret the regret as specifying how much the seller should be willing to pay for knowledge of the demand curve from which buyers' valuations are sampled.The answer to the problem depends on what assumptions one makes about the buyers' valuations. We consider three such assumptions: that the valuations are all equal to some unknown number p, that they are independent samples from an unknown probabilility distribution, or that they are chosen by an oblivious adversary. In each case, we derive upper and lower bounds on regret which match within a factor of log n; the bounds match up to a constant factor in the case of identical valuations.

Regret Minimization Under Partial Monitoring

“…We treat such limited-feedback (or partial monitoring) prediction problems in a more general framework which we describe next. The dynamic pricing problem described above, which is a special case of this more general framework, has been also investigated by Kleinberg and Leighton [27] in a simpler setting where the reward of the seller is defined as ρ(p t , y t ) = p t I pt≤yt . Note that, by using the feedback information (i.e., whether the customer bought the product or not), here the seller can compute the value of ρ(p t , y t ).…”

Adaptive limited-supply online auctions

“…Their model is an unlimited supply, unit-demand setting, with bidders' valuations constrained to the interval [1, h] and modeled adversarially. Kleinberg and Leighton [10] sharpen the Blum et al results by determining the additive regret: it is O(n 2/3 log(n) 1/3 ), with a lower bound of Ω(n 2/3 ), given n bidders. A revised lower bound of Ω(n 1/2 ) is stated for a model in which bids are distributed according to some fixed but unknown distribution, along with an upper bound of O((n log n) 1/2 ) with an additional technical hypothesis about the profit curve.…”

Online Auctions and Multi-scale Online Learning

“…Showing such bounds on the market share of the algorithm is an important avenue for future research. Kleinberg and Leighton (2003). Table 1: Number of rounds/samples needed to get a 1 − ǫ approximation to the best offline price/mechanism.…”

“…Online posted pricing Ω max{ Huang et al (2015); † Kleinberg and Leighton (2003). Table 2: Sample complexity & convergence rate w.r.t.…”