Optimal threshold in the group-buying auction with replenishment postponement

The group-buying auction is often used in the B2C market. The seller here is offered a quantity-discount-price contract and implements a replenishment postponement policy. We show that when customer's valuation follows a uniform distribution, the group-buying auction does not always outperform the fixed pricing. The optimal group-buying threshold is shown to depend on only purchasing costs, prices and the upper bound of customer's valuation.