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January 2017, 22(1): 125-159. doi: 10.3934/dcdsb.2017007
A review of dynamic Stackelberg game models
Tao Li 1,, and Suresh P. Sethi 2,
Santa Clara University, 500 El Camino Real, Santa Clara, CA 95053, USA
The University of Texas at Dallas, 800 W Campbell Rd, Richardson, TX 75080, USA
* Corresponding author: Tao Li
Received January 2016 Revised June 2016 Published December 2016
Dynamic Stackelberg game models have been used to study sequential decision making in noncooperative games in various fields. In this paper we give relevant dynamic Stackelberg game models, and review their applications to operations management and marketing channels. A common feature of these applications is the specification of the game structure: a decentralized channel consists of a manufacturer and independent retailers, and a sequential decision process with a state dynamics. In operations management, Stackelberg games have been used to study inventory issues, such as wholesale and retail pricing strategies, outsourcing, and learning effects in dynamic environments. The underlying demand typically has a growing trend or seasonal variation. In marketing, dynamic Stackelberg 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 classic advertising capital models or sales-advertising response models. We begin each section by introducing the relevant dynamic Stackelberg game formulation along with the definition of the equilibrium used, and then review the models and results appearing in the literature.
Keywords: Dynamic game, Stackelberg game, Nash game, non-cooperative game, operations management, marketing, open-loop equilibrium, feedback equilibrium, maximum principle, time consistency.
Mathematics Subject Classification: Primary:91A25, 91A15, 91A23;Secondary:91A1.
Citation: Tao Li, Suresh P. Sethi. A review of dynamic Stackelberg game models. Discrete & Continuous Dynamical Systems - B, 2017, 22 (1) : 125-159. doi: 10.3934/dcdsb.2017007
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Figure 1. Optimal Policies with Promotion
Table 1. Notations
$*$ Optimal/ equilibrium levels $\hat{c}_{i}$ Unit advertising cost
$i$ Denotes the $i^{th}$ player $f$ Production function, $\dot{x}$
$A$ , $\hat{A}$ Advertising level $h_{i}$ Unit inventory/ backlog cost
$B$ , $\hat{B}$ Local advertising $h_{i}^{+}$ Unit inventory holding cost
$C_{i}$ Cost of production/advertising $h_{i}^{-}$ Unit backlog cost
$D$ Demand, Revenue rate $m_{i}$ Margin
$G$ , $G_{i}$ Goodwill $p$ , $p_{i}$ Retail price
$H_{i}$ , $\bar{H}_{i}$ Current -value Hamiltonians $q$ Manufacturer's share of revenue
$I_{i}$ Inventory level $r_{1}$ , $r_{2}$ Effectiveness of advertising
$J_{i}$ Objective functional $t_{1}^{d}$ , $t_{2}^{d}$ Time parameters
$K$ Infrastructure capital $t_{s}$ , $t_{f}$ Start & end of promotional period
$L_{i}$ Labor force $u$ Leader's control variable
$M$ Market size $v$ , Follower's control variable
$N$ Number of firms $v^0$ Optimal response of the follower
$Q_{i}$ Production rate, Processing rate $w$ Wholesale price
$\bar{Q}_{i}$ Capacity limit $x$ State variable; Sales rate
$S$ Shelf space $\alpha$ External market influence
$S_{i}$ Salvage value, Unit salvage value $\alpha_{i}$ Demand parameters
$T$ Planning horizon $\beta$ Internal market influence
$r$ Optimal response of the follower $\delta$ Decay rate
$V_{i}$ Value function $\theta$ , $\hat{\theta}$ M's share of R's advertising cost
$X$ Cumulative sales $\lambda_{i}$ Adjoint variable, Shadow price
$a$ , Market potential/Advertising effectiveness $\pi_{i}$ Instantaneous profit rate
$a_{i}$ , $b_{i}$ , $d$ , $e_{i}$ Problem parameters $\rho$ Discount rate
$a_{_{l}}$ , $a_{_{s}}$ , $b_{_{l}}$ , $b_{_{s}}$ Advertising effectiveness $\phi$ , $\psi$ Adjoint variable, Shadow price
$b$ Price sensitivity/Advertising effectiveness $\omega_{i}$ Coefficient of incentive strategy
$c_{i}$ Unit production/advertising cost $\mathcal {U}$ , $\mathcal {V}$ Feasible set of controls
$q_i$ Order quantity $\Lambda$ , $\gamma_i$ Learning efficiency
Table 2. Summary of Model Descriptions
Section Dynamics L's decisions F's decisions Solution $^{1}$
2.2.1 Seasonal Production rate, Price Price OLSE
2.2.4 General Price, Production rate Price OLSE
2.2.5 General Price Order quantity OLSE
2.2.7 General Price Price OLSE
2.2.8 General Labor, investment Labor, investment OLSE
2.2.9 Bass-type Price Price OLSE
2.2.10 Bass type Price Price OLSE
2.3.1 NA dynamics Participation rate, Ad. effort Ad. effort FSE
2.3.2 NA dynamics Ad. effort, Price Ad. effort, Price FSE
2.3.3 NA dynamics Participation rate, Ad. effort Ad. effort, Price FSE
2.3.4 NA dynamics Ad. effort, (coop), Price Ad. effort, Price FSE
2.3.5 NA dynamics Ad. effort, Incentive Shelf-space FSE
2.3.6 NA dynamics Ad. effort Ad. effort FSE
2.3.7 Lanchester type Ad. effort Ad. effort FSE
2.3.8 Sethi 1983 Participation rate, Price Ad. effort, Price FSE
2.3.9 Sethi 1983 Participation&Ad. rate Participation&Ad. rate FSE
2.5.1 Inventory Price Order Quantity, Price OLSE, FSE
2.5.2 Production Cost Price Order quantity FSE
2.5.3 Inv., Prod. Cost Price, Production quantity Price, Order quantity FSE
1 The symbol OLSE = Open-loop Stackelberg equilibrium and FSE = Feedback Stackelberg equilibrium.
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Tao Li Suresh P. Sethi