A SURVEY OF STACKELBERG DIFFERENTIAL GAME MODELS IN SUPPLY AND MARKETING CHANNELS

Stackelberg differential game models have been used to study sequential decision making in noncooperative games in diverse fields. In this paper, we survey recent applications of Stackelberg differential game models to the supply chain management and marketing channels literatures. A common feature of these applications is the specification of the game structure： a decentralized channel composed of a manufacturer and independent retailers, and a sequential decision procedure with demand and supply dynamics and coordination issues. In supply chain management, Stackelberg differential games have been used to investigate inventory issues, wholesale and retail pricing strategies, and outsourcing in dynamic environments. The underlying demand typically has growth dynamics or seasonal variation. In marketing, Stackelberg differential games have been used to model cooperative advertising programs, store brand and national brand advertising strategies, shelf space allocation, and pricing and advertising decisions. The demand dynamics are usually extensions of the classical advertising capital models or sales-advertising response models. We begin by explaining the Stackelberg differential game solution methodology and then provide a description of the models and results reported in the literature.

A Mean-Field Linear-Quadratic Stochastic Stackelberg Differential Game with one Leader and Two Followers

This paper is concerned with a linear-quadrati stochastic Stackelberg differential game with one leader and two followers,where the game system is governed by a mean-field stochastic differential equatio.By maximum principle and verification theorem,the open-loop Stackelberg solution is expressed as a feedback form of the state and its mean with the help of three systems of Riccati equations.

A STACKELBERG DIFFERENTIAL GAMES SOLUTION TO A PROBLEM OF OPTIMAL INTERTEMPORAL INVESTMENT AND TAX RATE DESIGN

Using the Stackelberg differential games(SDG) theory,we quantitatively study a problem of optimal intertemporal investment and tax rate design.Under some appropriate assumptions,the open-loop Stackelberg equilibrium solutions are obtained.Equilibrium solutions show that:1.The optimal strategies derived from differential game and unilateral optimal control approaches are different;2.It is not always the best strategy for the government to use a constant tax rate over the whole time period;3.The admissible size of tax rate adjustment may have great effect on the government's optimal strategy;4.SDG approach has no significant effect on the firm's optimal investment strategy.

A simplified differential game model for the optimal choice of tax rate

A classical problem on optimal choice of tax rate from the perspective of differential game approach is studied. Under some appropriate assumptions on the profit and utility functions, the open-loop Stackelberg equilibrium solution which is time- dependent is obtained. Result shows that 1) the optimal strategies derived from differential game and traditional unilateral optimal control approaches are different; 2) both marginal profit rate and the market rate of interest have great effect on the equilibrium solution; and 3) the government should think about the firm’s potential reaction when selecting tax rates and the timing of taxation.

Mean-field type FBSDEs in a domination-monotonicity framework and LQ multi-level Stackelberg games

Motivated by various mean-field type linear-quadratic(MF-LQ,for short)multilevel Stackelberg games,we propose a kind of multi-level self-similar randomized dominationmonotonicity structures.When the coefficients of a class of mean-field type forwardbackward stochastic differential equations(MF-FBSDEs,for short)satisfy this kind of structures,we prove the existence,the uniqueness,an estimate and the continuous dependence on the coefficients of solutions.Further,the theoretical results are applied to construct unique Stackelberg equilibria for forward and backward MF-LQ multi-level Stackelberg games,respectively.