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

Funding: This work was supported by ANID (Anillo ICMD) [Grant ACT210005] and the Center for Mathematical Modeling [Grant FB210005].