A review of dynamic Stackelberg game models

Tao Li; Suresh P. Sethi

doi:10.3934/dcdsb.2017007

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2017, Volume 22, Issue 1: 125-159. Doi: 10.3934/dcdsb.2017007

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A review of dynamic Stackelberg game models

Tao Li^1, , and
Suresh P. Sethi^2,

1.
Santa Clara University, 500 El Camino Real, Santa Clara, CA 95053, USA
2.
The University of Texas at Dallas, 800 W Campbell Rd, Richardson, TX 75080, USA

* Corresponding author: Tao Li
* Corresponding author: Tao Li

Received: January 2016

Revised: June 2016

Published: November 2017

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Abstract

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.

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

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Figure 1. Optimal Policies with Promotion

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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

Section	Dynamics	L's decisions	F's decisions	Solution $^{1}$
2.2.1	Seasonal	Production rate, Price	Price	OLSE
2.2.2	Seasonal	Production rate, Price	Price	OLSE
2.2.3	Seasonal	Production rate, Price	Price	OLSE
2.2.4	General	Price, Production rate	Price	OLSE
2.2.5	General	Price	Order quantity	OLSE
2.2.6	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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