Regime Dependent Approximations for the Single-Item Dynamic Pricing Problem

We study the single-item dynamic pricing problem in three separate asymptotic regimes characterized by their ratio of inventory to market size. We first consider the case of customer item valuations following an exponential distribution where we are able to derive a sharp characterization of the boundaries between each of the regimes. We then proceed to the case of customer item valuations following a general distribution. In this case, we derive for each regime approximations to the optimal value function, optimal pricing policy and optimal purchasing probability policy for both the offline and the online setting. We also provide for each regime an approximation to the regret of the optimal online value function relative to the optimal offline value function. In addition to these results, we show that in the regime when the inventory-to-market size ratio is low, a static run-out rate policy asymptotically fails to be first-order optimal. However, a dynamic run-out rate policy is shown to achieve both first and second-order optimality. Finally, in the numerics section, we test the quality of our approximations and determine the boundaries between each of the three asymptotic regimes in the case of general customer item valuation distributions.