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 Am(F) achievable by the optimal online policy on m i.i.d. random variables drawn from F to the expected maximum Mn(F) of n i.i.d. draws from the same distribution. We ask how big does m have to be to ensure that (1+ϵ) Am(F) ≥ Mn(F) for all F. We resolve this question and exhibit a stark phase transition: When ϵ = 0 the competition complexity is unbounded. That is, for any n and any m there is a distribution F such that Am(F) > Mn(F). In contrast, for any ϵ < 0, it is sufficient and necessary to have m = φ(ϵ)n where φ(ϵ) = φ(log log 1/ϵ). Therefore, the competition complexity not only drops from being unbounded to being linear, it is actually linear with a very small constant. The technical core of our analysis is a loss-less reduction to an infinite dimensional and non-linear optimization problem that we solve optimally. A corollary of this reduction, which may be of independent interest, is a novel proof of the factor ∼0.745 i.i.d. prophet inequality, which simultaneously establishes matching upper and lower bounds.