Two-tier equilibria for supply chains with a single manufacturer and multiple asymmetric retailers

We derive sufficient conditions for the existence and uniqueness of the Stackelberg–Nash–Cournot equilibria for a supply chain problem with a single manufacturer and multiple asymmetric retailers and characterize the first and second order derivatives of the total equilibrium quantities. The Stackelberg manufacturer is assumed to supply a homogeneous product to all retailers with the retail price determined by a general nonlinear inverse demand function. We provide several extensions of our previous results [G.J. Kyparisis, C. Koulamas, A note on equilibria for two-tier supply chains with a single manufacturer and multiple retailers, Operations Research Letters 39 (2011) 471–474] obtained for a similar supply chain with symmetric retailers.