We consider the problem of selling a single item to one of n bidders who arrive sequentially with values drawn independently from identical distributions, and ask how much more revenue can be obtained by posting discriminatory prices to individual bidders rather than the same anonymous price to all of them. The ratio between the maximum revenue from discriminatory pricing and that from anonymous pricing is at most 2 − 1/n for arbitrary distributions and at most 1/(1 − (1 − 1/n) n ) ≤ e/(e − 1) ≈ 1.582 for regular distributions, and these bounds can in fact be obtained by using one of the discriminatory prices as an anonymous one. The bounds are shown via a relaxation of the discriminatory pricing problem rather than virtual values and thus apply to distributions without a density, and they are tight for all values of n. For a class of distributions that includes the uniform and the exponential distribution we show the maximization of revenue to be equivalent to the maximization of welfare with an additional bidder, in the sense that both use the same discriminatory prices. The problem of welfare maximization is the well-known Cayley-Moser problem, and this connection can be used to establish that the revenue gap between discriminatory and anonymous pricing is approximately 1.037 for the uniform distribution and approximately 1.073 for the exponential distribution. * Helpful discussions with Rad Niazadeh and Frieder Smolny are gratefully acknowledged. The second and third author are supported by the Einstein Foundation Berlin.
Closer to our problem, REF explicitly study the revenue gap between discriminatory and anonymous sequential posted pricing.
Revenue Gaps for Discriminatory and Anonymous Sequential Posted Pricing
ArXiv
ArXiv
Mathematics Computer Science