The Competition Complexity of Dynamic Pricing

Abstract

We study the competition complexity of dynamic pricing relative to the optimal auction in the fundamental single-item setting. In prophet inequality terminology, we compare the expected reward Am(F) achievable by the optimal online policy on m independent and identically distributed (i.i.d.) random variables distributed according to F to the expected maximum Mn(F)of n i.i.d. draws from F. We ask how big m has to be to ensure that (1 + epsilon)Am(F) >_ Mn(F) for all F. We resolve this question and characterize the competition complexity as a function of epsilon. When epsilon = 0, the competition complexity is unbounded. That is, for any n and m there is a distribution F such that Am(F) < Mn(F). In contrast, for any epsilon > 0, it is sufficient and necessary to have m = phi(epsilon)n, where phi(epsilon) = Theta(log log 1=epsilon). Therefore, the competition complexity not only drops from unbounded to linear, it is actually linear with a very small constant. The technical core of our analysis is a lossless reduction to an infinite dimensional and nonlinear optimization problem that we solve optimally. A corollary of this reduction is a novel proof of the factor approximate to 0:745 i.i.d. prophet inequality, which simultaneously establishes matching upper and lower bounds.