The Value of Knowing a Demand Curve: Bounds on Regret for On-line Posted-Price Auctions PRELIMINARY VERSION – DO NOT DISTRIBUTE

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. ∗Department of Mathematics, MIT, Cambridge MA 02139, and Akamai Technologies, 8 Cambridge Center, Cambridge, MA 02142. Email: [email protected]. Supported by a Fannie and John Hertz Foundation Fellowship. †Department of Mathematics, MIT, Cambridge MA 02139, and Akamai Technologies, 8 Cambridge Center, Cambridge, MA 02142. Email: [email protected]. PRELIMINARY VERSION – DO NOT DISTRIBUTE 1