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

We consider such a problem: Multiple sellers sell a family ofsubstitutable perishable products in a common market withcorrelative random demands. Due to the perishable nature of aproduct, delivery time directly affects its freshness level, whichin turn affects its demand. Each seller's demand is formulated asa function of all sellers' prices and delivery-times. To maximizehis own expected profit, each seller needs to set his sellingprice and delivery time simultaneously, by taking into account hiscompetitors' 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 assumedelivery-time dependent operating and purchasing costs withoutservice reliability constraints in Model II and Model IIIrespectively. We establish the existence of a price anddelivery-time equilibrium, under mild conditions in Model I and II,and restrictive conditions in Model III. A novel method is adoptedto establish the uniqueness of the equilibrium in Model I, and aniterative procedure is designed to compute the equilibrium pricesand delivery-times in Model III.