  Price-based revenue management is an important problem in operations
management with many practical applications. The problem considers a retailer
who sells a product (or multiple products) over $T$ consecutive time periods
and is subject to constraints on the initial inventory levels. While the
optimal pricing policy could be obtained via dynamic programming, such an
approach is sometimes undesirable because of high computational costs.
Approximate policies, such as the re-solving heuristics, are often applied as
computationally tractable alternatives. In this paper, we show the following
two results. First, we prove that a natural re-solving heuristic attains $O(1)$
regret compared to the value of the optimal policy. This improves the $O(\ln
T)$ regret upper bound established in the prior work of
\cite{jasin2014reoptimization}. Second, we prove that there is an $\Omega(\ln
T)$ gap between the value of the optimal policy and that of the fluid model.
This complements our upper bound result by showing that the fluid is not an
adequate information-relaxed benchmark when analyzing price-based revenue
management algorithms.
