We consider a network revenue management problem and advance its dual formulation. The dual formulation reveals that the (optimal) shadow price of capacity forms a nonnegative martingale. This result is proved under minimal assumptions on network topology and stochastic nature of demand, allowing an arbitrary statistical dependence structure across time and products. Next, we consider a quadratic perturbation of the network revenue management problem and show that a simple (perturbed) bid-price control is optimal for the perturbed problem; and it is $\varepsilon$-optimal for the original network revenue management problem. Finally, we consider a predictable version of this control, where the bid prices used in the current period are last updated in the previous period, and provide an upper bound on its optimality gap in terms of the (quadratic) variation of demand. Using this upper bound we show that there exists a near-optimal such control in the usual case when periods are small compared to the planning horizon provided that either demand or the incremental information arriving during each period is small. We establish the martingale property of the (near) optimal bid prices in both settings. The martingale property can have important implications in practice as it may offer a tool for monitoring the revenue management systems.
"On bid-price controls for network revenue management." Stoch. Syst. 5 (2) 268 - 323, 2015. https://doi.org/10.1214/12-SSY081