Strategy and numerical analysis for sellers based on Priceline

Based on the famous reverse auction website Priceline, the following two situations are modeled for sellers: one is that customers bid and then sellers decide to accept or reject, and the other is that sellers set a price-limit before customers' bidding. Theoretical analysis proves that both the models are equivalent, which means the expressions of the optimal value function and the optimal price for sellers are the same separately. Three demand functions are considered of the second situation in numerical analysis with the following properties: 1) the marginal revenue increases in the remaining time, 2) the price-limit is monotone in the remaining time and items, 3) and the optimal value is the concave function of the remaining time and items separately.