The Power of Two Choices in Dynamic Pricing: As Good As Full Feedback, Instance-Wise

Yan Dai, Tao Lin, Aleksandrs Slivkins

We study the dynamic pricing problem, where a firm sells a product to $T$ myopic customers and their values are independent samples from an unknown value distribution $F$. With posted prices, the firm only sees whether each customer buys, and the minimax regret is $\widetilde Θ(T^{2/3})$ [KL03]. A menu instead offers several options, each selling the product with some probability at some expected payment. Infinitely many menu options allow the seller to fully reveal customer values, reducing the regret to $Θ(\sqrt T)$. How many options are needed? Perhaps surprisingly, a second menu option suffices, achieving the same regret as infinite menus (up to $\log$ factors). This holds broadly: without knowing any property of $F$, our algorithm with two menu options attains $\tilde O(\sqrt T)$ regret in general, $\tilde O(T^{1/3})$ regret for monotone-hazard-rate distributions, and $\tilde O(T^{d/(d+1)})$ instance-dependent regret when $d\in(0,1)$ is a suitable notion of "dimension". All are optimal even with infinite menus. Moreover, we argue, via a novel lower-bound construction, that our algorithm is near-optimal for each problem instance. Along the way, we study the query complexity…