Price and delivery-time competition of perishable products: Existence and uniqueness of Nash equilibrium

We consider such a problem: Multiple sellers sell a family of substitutable perishable products in a common market with correlative random demands. Due to the perishable nature of a product, delivery time directly affects its freshness level, which in turn affects its demand. Each seller's demand is formulated as a function of all sellers' prices and delivery-times. To maximize his own expected profit, each seller needs to set his selling price and delivery time simultaneously, by taking into account his competitors' reactions. We present three models to address different practical situations. In Model I, we assume that each seller faces constant operating and purchasing costs, but is subject to a given service reliability constraint. In contrast, we assume delivery-time dependent operating and purchasing costs without service reliability constraints in Model II and Model III respectively. We establish the existence of a price and delivery-time equilibrium, under mild conditions in Model I and II, and restrictive conditions in Model III. A novel method is adopted to establish the uniqueness of the equilibrium in Model I, and an iterative procedure is designed to compute the equilibrium prices and delivery-times in Model III.