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A review of dynamic Stackelberg game models

  • * Corresponding author: Tao Li

    * Corresponding author: Tao Li 
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  • 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.

    Mathematics Subject Classification: Primary:91A25, 91A15, 91A23;Secondary:91A10.


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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
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    Table 2.  Summary of Model Descriptions

    SectionDynamicsL's decisionsF's decisionsSolution $^{1}$
    2.2.1SeasonalProduction rate, PricePriceOLSE
    2.2.2SeasonalProduction rate, PricePriceOLSE
    2.2.3SeasonalProduction rate, PricePriceOLSE
    2.2.4GeneralPrice, Production ratePriceOLSE
    2.2.5GeneralPriceOrder quantityOLSE
    2.2.6GeneralPriceOrder quantityOLSE
    2.2.8GeneralLabor, investmentLabor, investmentOLSE
    2.2.10Bass typePricePriceOLSE
    2.3.1NA dynamicsParticipation rate, Ad. effortAd. effortFSE
    2.3.2NA dynamicsAd. effort, PriceAd. effort, PriceFSE
    2.3.3NA dynamicsParticipation rate, Ad. effortAd. effort, PriceFSE
    2.3.4NA dynamicsAd. effort, (coop), PriceAd. effort, PriceFSE
    2.3.5NA dynamicsAd. effort, IncentiveShelf-spaceFSE
    2.3.6NA dynamicsAd. effortAd. effortFSE
    2.3.7Lanchester typeAd. effortAd. effortFSE
    2.3.8Sethi 1983Participation rate, PriceAd. effort, PriceFSE
    2.3.9Sethi 1983Participation&Ad. rateParticipation&Ad. rateFSE
    2.5.1InventoryPriceOrder Quantity, PriceOLSE, FSE
    2.5.2Production CostPriceOrder quantityFSE
    2.5.3Inv., Prod. CostPrice, Production quantityPrice, Order quantityFSE
    1 The symbol OLSE = Open-loop Stackelberg equilibrium and FSE = Feedback Stackelberg equilibrium.
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