Strategic pricing and production planning using a Stackelberg differential game with unknown demand parameters

We consider a dynamic pricing and production planning problem for make-to-stock firms in an uncooperative competitive market. We model the market as a one-leader-multiple-follower Stackelberg differential game, and express the demand dynamics as a differential equation with unknown demand parameters based on evolutionary game theory. A demand learning approach based on Markov chain Monte Carlo method is applied to predict the unknown demand parameters. First, we assume backorder is not allowed, and formulate the followers' optimal control problem as a differential variational inequality (DVI) subproblem, and the leader's problem as a mathematical program with equilibrium constraints (MPECs). A fixed point algorithm is employed to solve the DVI subproblem, and a simulated annealing algorithm is used to solve the MPECs. A Stackelberg- generalized-differential-Nash equilibrium is achieved when the leader solves his optimal control problem, where the followers reach a generalized-differential- Nash equilibrium. Through numerical examples, we conclude that the leader outperforms the followers, and each firm can increase his profit by using the demand learning method. We explain in which case a firm can obtain his highest profit. Finally, the pricing and production problem is extended to allow backorder, and the result shows that allowing backorder can improve a firm's performance, compared with no backorder.