The competition complexity of dynamic pricing

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 ⁡() 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 ⁡() of n i.i.d. draws from F. We ask how big m has to be to ensure that (1+)⁢⁡()≥⁡() for all F. We resolve this question and characterize the competition complexity as a function of ε. When =0, the competition complexity is unbounded. That is, for any n and m there is a distribution F such that ⁡()<⁡(). In contrast, for any >0, it is sufficient and necessary to have =⁡()⁢, where ⁡()=Θ⁡(log log 1/). 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 ≈0.745 i.i.d. prophet inequality, which simultaneously establishes matching upper and lower bounds.