marketing - link.springer.com … · marketing is a functional area within a business firm. it...
TRANSCRIPT
Marketing
Steffen Jørgensen
Contents
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.1 A Brief Assessment of the Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2 Notes for the Reader . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2 Advertising Games . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2 Lanchester Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.3 Vidale-Wolfe (V-W) Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.4 New-Product Diffusion Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.5 Advertising Goodwill Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
3 Pricing Games . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.2 Demand Learning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.3 Government Subsidies of New Technologies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.4 Entry of Competitors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.5 Other Models of Pricing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
4 Marketing Channels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174.2 Channel Coordination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184.3 Channel Leadership . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214.4 Incentive Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214.5 Franchise Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234.6 National Brands, Store Brands, and Shelf Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244.7 The Marketing-Manufacturing Interface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
5 Stochastic Games . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276 Doing the Calculations: An Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
S. Jørgensen (�)Department of Business and Economics, University of Southern Denmark, Odense, Denmarke-mail: [email protected]
© Springer International Publishing Switzerland 2016T. Basar, G. Zaccour (eds.), Handbook of Dynamic Game Theory,DOI 10.1007/978-3-319-27335-8_22-1
1
2 S. Jørgensen
Abstract
Marketing is a functional area within a business firm. It includes all the activitiesthat the firm has at its disposal to sell products or services to other firms(wholesalers, retailers) or directly to the final consumers. A marketing managerneeds to decide strategies for pricing (toward consumers and middlemen),consumer promotions (discounts, coupons, in-store displays), retail promotions(trade deals), support of retailer activities (advertising allowances), advertising(television, internet, cinemas, newspapers), personal selling efforts, productstrategy (quality, brand name), and distribution channels. Our objective is todemonstrate that differential games have proved to be useful for the study ofa variety of problems in marketing, recognizing that most marketing decisionproblems are dynamic and involve strategic considerations. Marketing activitieshave impacts not only now but also in the future; they affect the sales and profitsof competitors and are carried out in environments that change.
KeywordsMarketing • Differential games • Advertising • Goodwill • Pricing • Demandlearning • Cost learning • Marketing channels • Channel coordination • Lead-ership
1 Introduction
Marketing is a functional area like, e.g., production and finance, in a businessfirm. It includes the activities that the firm has at its disposal to sell products orservices to other firms (wholesalers, retailers) or directly to the final consumers.The main purpose of devoting resources to marketing is to increase sales in order togenerate more revenues. The toolbox of a marketing manager includes a variety ofactivities: pricing (consumers and middlemen), consumer promotions (discounts,coupons, in-store displays), retailer promotions (trade deals), support of retaileractivities (support of retailers’ advertising expenditures), advertising (television,internet, cinemas, newspapers, outdoor displays), personal selling efforts, productstrategy (quality, brand name), and distribution channels.
The objective of the chapter is to demonstrate that differential games haveproved to be a useful methodology with which one can study decision problemsin marketing in a formal way. Such problems are dynamic and involve strategicconsiderations. Marketing activities have impacts now and in the future; they affectsales, revenues, costs, and profits of competitors and are most often carried outin environments that change over time. It seems, therefore, to be less advisable toderive recommendations for a firm’s marketing activities by using a static or a simpleperiod-by-period framework or assuming a monopolistic market and disregardexisting and/or potential competition.
Marketing 3
1.1 A Brief Assessment of the Literature
The use of differential game theory to study marketing decision problems in adynamic, competitive environment started in the 1970s. We will refer to seminalwork in the areas that are covered and give the reader an account of how researchhas progressed. There is a literature using optimal control theory that started earlierthan the dynamic game research stream. The control literature deals with marketingdecision-making in dynamic, monopolistic markets. Not surprisingly, many of theseworks have formed the basis from which game theoretic studies emanated.
Dynamic game literature in marketing has predominantly been normative,focusing on the characterization of equilibrium marketing effort strategies. Notmuch emphasis has been put on empirical work. The main part of the literatureaims at deriving predictions of what equilibrium behavior would be and to reportthese predictions as recommendations and decision support for marketing managers.Technically speaking, two options have been explored in order to derive gametheoretic equilibria:
• Analytical methods can be used to obtain closed-form characterizations ofequilibrium actions. Analytical methods may provide results of some generalitywhen they leave parameter values and functional forms – at least partly –unspecified. A major problem here is that these methods normally work onlyin models of low complexity and/or one assumes a “mathematically convenient”structure. This puts a limit to the applicability of results.
• Numerical methods have their strength when models are complex. Their disad-vantages are that results cannot be generalized beyond the scenarios consideredin the numerical calculations, functional forms in a model must be specified,and one may be unable to obtain real-life data for parameter estimations. Itis, however, worthwhile noticing that since the 1990s, a number of empiricalstudies in dynamic marketing competition have been published, particularly inthe area of advertising. This research has typically used estimated differentialgame models to (a) test the empirical validity of the prescriptions of normativemodels or (b) to assess the advantages of using information about the evolutionof the state (e.g., market shares or sales volumes) of a dynamic system.
There are not many textbooks dealing with marketing applications of differentialgames. Case (1979) was probably the first who considered many-firm advertisingproblems cast as differential games. The book by Jørgensen and Zaccour (2004)is devoted to differential games in marketing. Dockner et al. (2000) offered someexamples from marketing. See also Haurie et al. (2012). The focus of these books isnoncooperative dynamic games. The reader should be aware that in the last 10–15 years, there has been a growing interest in applying the theory of dynamiccooperative games to problems in marketing. Recent surveys of this area are Austand Buscher (2014) and Jørgensen and Zaccour (2014).
4 S. Jørgensen
At the beginning of the 1980s, most applications of differential games in mar-keting were in advertising. Jørgensen (1982a) summarized the early developmentsin this area. A few papers covered pricing, e.g., Jørgensen (1986a). Taking stock,Jørgensen and Zaccour (2004) noted that the study of competitive advertisingand pricing strategies had been continued. Moreover, new areas of marketinghave been approached using differential games. As we shall see, research activityhas increased considerably in marketing channels/supply chains where researchershave approached problems of, e.g., coordinated (cooperative) decision-making andleadership.1
The following works, covering the period from 1982 to 2014, give an account ofthe current state of the art in dynamic games in marketing:
• General: Moorthy (1993), Eliashberg and Chatterjee (1985), Rao (1990), andJørgensen and Zaccour (2004)
• Pricing: Jørgensen (1986a), Rao (1988), Kalish (1988), and Chatterjee (2009)• New-product diffusion models: Dolan et al. (1986), Mahajan et al. (1990, 1993),
Mahajan et al. (2000), and Chatterjee et al. (2000)• The production-marketing interface: Eliashberg and Steinberg (1993), and Gai-
mon (1998)• Advertising: Jørgensen (1982a), Erickson (1991, 1995a), Moorthy (1993),
Feichtinger et al. (1994), Huang et al. (2012), Jørgensen and Zaccour (2014),and Aust and Buscher (2014)
• Marketing channels: He et al. (2007) and Sudhir and Datta (2009).
1.2 Notes for the Reader
1. An overwhelming majority of the literature in the area “dynamic games inmarketing” has used deterministic differential games, and this chapter will focuson such games. A section deals with the (rather sparse) literature that has usedstochastic games to study marketing problems.
2. There is a literature in economic applications of game theory/microeconomics/in-dustrial organization that employs repeated games to study oligopolistic markets.Much work has been devoted to problems of intertemporal competition/collusionon prices, but also issues concerning, for instance, industrial structure (entry, exit,and number of firms) are treated.2 The focus and the approach in this literatureare in many respects different from that in literature on differential games inmarketing. In the latter a main emphasis is on providing prescriptions for thedecisions that individual firms should take in a market.
1Aust and Buscher (2014) present a figure showing the number of publications on cooperativeadvertising per year, from 1970 to 2013.2Examples of textbooks that deal with economic applications of repeated games are Tirole (1988),Friedman (1991), Vives (1999), and Vega-Redondo (2003).
Marketing 5
3. To benefit fully from the chapter, the reader should be familiar with the basics ofnonzero-sum, noncooperative differential game theory.
4. Many of the models we present can be/have been stated for a market with anarbitrary number of firms. To simplify the presentation, we focus on marketshaving two firms only (i.e., duopolistic markets).
5. Our coverage of the literature is not intended to be exhaustive.6. To save space and avoid redundancies, we only present – for a particular
contribution – the dynamics (the differential equation(s)) being employed. Thedynamics are a key element of any differential game model. Another element isthe objective functionals. For a specific firm, say, firm i; the objective functionaltypically is of the form
Ji D
Z T
t0
exp f��i tg ŒRi .t/ � Ci.t/� dt C exp f��iT gˆt.T /
where Œt0; T � is the planning period which can be finite or infinite. The termRi.t/ is a revenue and Ci.t/ is a cost. Function ˆt represents a salvage value tobe earned at the horizon date T: If the planning horizon is infinite, a salvage valuemakes no sense and we have ˆt D 0. In games with a finite and short horizon,discounting may be omitted, i.e., �i D 0:3
2 Advertising Games
2.1 Introduction
Advertising is an important element of a firm’s communication mix. This mix alsoincludes direct marketing (phone or e-mail), public relations, and personal selling.The planning of a firm’s advertising activities involves many issues, and differentialgame studies have studied a subset of these only. What has gained most interestis the amount and timing of advertising expenditures, the sharing of the costs ofadvertising between a manufacturer and its retailers, and the interaction betweenadvertising and other marketing activities (typically, pricing).
The literature has employed four main categories of models that are characterizedby their dynamics (state equations in differential game terminology): Lanchester,Vidale-Wolfe, new-product diffusion, and Nerlove-Arrow models. The state variablesare market shares or sales rates in the Lanchester and Vidale-Wolfe models,cumulative sales in new-product diffusion models, and advertising goodwill levelsin Nerlove-Arrow models.
3Examples are short-term price promotions or advertising campaigns.
6 S. Jørgensen
2.2 Lanchester Models
A century ago, Lanchester (1916) suggested a series of models to describe andanalyze military combat. One of these models can readily be reformulated tobecome a simple and intuitive representation of the “battles for market share” thatwe observe in real-life markets (Kimball 1957).
To present the Lanchester model of advertising competition, let Xi.t/ 2 Œ0; 1�be market share at time t for a specific product, manufactured and sold by firm i:
Denote by ai .t/ � 0 the advertising effort (or expenditure) rate of the firm: Thedynamics are, in general,
PXi.t/ D fi .ai .t//Xj .t/ � fj .aj .t//Xi .t/I i; j 2 f1; 2g ; i ¤ j
where function f .a/; called the attraction rate, is increasing and takes positivevalues. The right-hand side of the dynamics shows that a firm’s advertising has oneaim only, i.e., to steal market share from the competitor: Because of this purpose,advertising effort ai is referred to as offensive advertising. In the basic Lanchesteradvertising model, and in many differential games using this model, function fi .ai /is linear. Some authors assume diminishing returns to advertising efforts by usingfor fi .ai / the power function ˇia
˛ii ; ˛i 2 .0; 1� (e.g., Chintagunta and Jain 1995;
Jarrar et al. 2004).A seminal contribution is Case (1979) who studied a differential game with the
basic Lanchester dynamics and found a feedback Nash equilibrium (FNE). Manyextensions of the model have been suggested:
1. Multiple types of advertising (e.g., Erickson 1993)2. Industry sales expansion (e.g., Bass et al. 2005a,b)3. Multiple advertising media (e.g., Fruchter and Kalish 1998)4. Multiproduct firms (e.g., Fruchter 2001)
We confine our interest to items 1 and 2. Erickson (1993) suggested that firms usetwo kinds of effort, offensive and defensive advertising, denoted di .t/. Defensiveadvertising aims at defending a firm’s market share against attacks from the rivalfirm. Market share dynamics for firm i are
PXi.t/ D ˇiai .t/
dj .t/Xj .t/ � ˇj
aj .t/
di .t/Xi .t/
which shows that offensive efforts of firm i can be mitigated by defensive efforts offirm j . See also Martín-Herrán et al. (2012).
Industry sales are not necessarily constant (as they are in the basic Lanchestermodel). They might expand due to generic advertising, paid for by the firms in theindustry. (Industry sales could also grow from other reasons, for instance, increasedconsumer incomes or changed consumer preferences.) Generic advertising pro-motes the product category (e.g., agricultural products), not the individual brands.
Marketing 7
Technically, when industry sales expand, we replace market shares with sales ratesin the Lanchester dynamics. Therefore, let Si .t/ denote the sales rate of firm i: Thefollowing example presents two models that incorporate generic advertising.
Example 1. The dynamics used in Bass et al. (2005a,b) are
PSi .t/ D ˇiai .t/qSj .t/ � ˇj aj .t/
pSi .t/C �i Œk1g1.t/C k2g2.t/�
where gi .t/ is the rate of generic advertising, paid for by firm i .4 The third termon the right-hand side is the share of increased industry sales that accrues to firm i .The authors identified FNE with finite as well as infinite time horizons. Jørgensenand Sigué (2015) included offensive, defensive, and generic advertising efforts inthe dynamics:
PSi .t/ D�ˇai .t/ � �dj .t/
�qSj .t/ �
�ˇaj .t/ � �di .t/
�pSi .t/
Ck�gi .t/C gj .t/
�2
and identified FNE with a finite time horizon. Since players in this game aresymmetric (same parameter values) it seems plausible to allocate one half to eachplayer of the increase in industry sales.
Industry sales may decline as well. This happens if consumers stop usingthe product category as such, for example, because of technical obsolescence orchanged preferences. (Examples of the latter are carbonated soft drinks and beer inWestern countries.) A simple way to model decreasing industry sales is to include adecay term on the right-hand side of the dynamics.
Empirical studies of advertising competition with Lanchester dynamics appearedin Erickson (1985, 1992, 1996, 1997), Chintagunta and Vilcassim (1992, 1994),Mesak and Darrat (1993), Chintagunta and Jain (1995), Mesak and Calloway(1995), Fruchter and Kalish (1997, 1998), and Fruchter et al. (2001). Dynamicsare estimated from empirical data and open-loop and feedback advertising tra-jectories determined. A comparison of the costs incurred by following each ofthese trajectories, using observed advertising expenditures, gives a clue of whichtype of strategy provides the better fit (Chintagunta and Vilcassim 1992). Analternative approach assumes that observed advertising expenditures and marketshares result from decisions made by rational players in a game. The question then iswhether prescriptions derived from the game are consistent with actual advertisingbehavior. Some researchers found that feedback strategies provide the better fit to
4Letting sales appear aspS instead of linearly turns out to be mathematically expedient, cf. Sorger
(1989).
8 S. Jørgensen
observed advertising expenditures. This suggests that firms – not unlikely – act uponinformation conveyed by observed market shares when deciding their advertisingexpenditures.
Remark 2. The following dynamics are a pricing variation on the Lanchester model:
PXi.t/ D gj .pj .t//Xj .t/ � gi .pi .t//Xi .t/
where pi .t/ is the consumer price charged by firm i: Function gi .pi / takes positivevalues and is convex increasing for pi > Npi : If pi � Npi (i.e., price pi is “low”),then gi .pi / D 0 and firm i does not lose customers to firm j: If pi exceeds Npi ; firmi starts losing customers to firm j: Note that it is a high price of firm i which drivesaway its customers. It is not a low price that attracts customers from the competitor.This hypothesis is the “opposite” of that in the Lanchester advertising model whereadvertising efforts of a firm attract customers from the rival firms: They do not affectthe firm’s own customers. See Feichtinger and Dockner (1985).
2.3 Vidale-Wolfe (V-W) Models
The model of Vidale and Wolfe (1957) describes the evolution of sales of amonopolistic firm. It was extended by Deal (1979) to a duopoly:
PSi .t/ D ˇiai .t/Œm � S1.t/ � S2.t/� � ıiSi .t/
in which m is a fixed market potential (upper limit of industry sales). The firstterm on the right-hand side reflects that advertising of firm i increases its salesby inducing some potential buyers, represented by the term m � S1.t/ � S2.t/; topurchase its brand: The decay term ıiSi .t/ models a loss of sales which occurswhen some of the firm’s customers stop buying its product. These customers leavethe market for good, i.e., they do not switch to firm j: Deal characterized an open-loop Nash equilibrium (OLNE).
In the V-W model, sales of firm i are not directly affected by advertising effortsof firm j (in contrast to the Lanchester model). This suggests that a V-W modelwould be most appropriate in markets where firms – through advertising effort –can increase their sales by capturing a part of untapped market potential. It mayhappen that the market potential grows over time; this can be modeled by assuminga time-dependent market potential, m.t/; cf. Erickson (1995b).
Wang and Wu (2001) suggested a straightforward combination of the modelof Deal (1979) and the basic Lanchester model.5 Then the dynamics incorporateexplicitly the rival firm’s advertising effort. The authors made an empirical study
5An early contribution that combined elements of Lanchester and V-W models is Leitmann andSchmitendorf (1978); see also Jørgensen et al. (2010).
Marketing 9
in which sales dynamics are estimated and compared to state trajectories generatedby the Lanchester model. Data from the US cigarette and beer industries suggestthat the Lanchester model may apply to a market which has reached a steady state.(Recall that the Lanchester model assumes a fixed market potential.) The combinedmodel of Wang and Wu (2001) seems to be more appropriate during the transientmarket stages.
2.4 New-Product Diffusion Models
Models of this type describe the adoption process of a new durable product ina group of potential buyers. A good example is electronic products. The marketpotential may be fixed or could be influenced by marketing efforts and/or exogenousfactors (e.g., growing consumer incomes). When the process starts out, the majorityof potential consumers are unaware of the new product, but gradually awarenessis created and some consumers purchase the product. These “early adopters” (or“innovators”) are supposed to communicate their consumption experiences andrecommendations to non-adopters, a phenomenon known as “word of mouth.” Withthe rapidly increasing popularity of social media, word-of-mouth communication islikely to become an important driver of the adoption process of new products andservices.
The seminal new-product diffusion model is Bass (1969) who was concernedprimarily with modeling the adoption process. Suppose that the seller of a productis a monopolist and the number of potential buyers, m; is fixed. Each adopter buysone and only one unit of the product. Let Y .t/ 2 Œ0;m� represent the cumulativesales volume by time t: Bass suggested the following dynamics for a new durableproduct, introduced at time t D t0:
PY .t/ D �Œm � Y .t/�C �Y .t/
mŒm � Y .t/�; Y .t0/ D 0:
The first term on the right-hand side represents adoptions made by “innovators.” Thesecond term reflects adoptions by “imitators” who communicate with innovators.The model features important dynamic demand effects such as innovation, imita-tion, and saturation. The latter means that the fixed market potential is graduallyexhausted.
In the Bass model, the firm cannot influence the diffusion process as the modelis purely descriptive. Later on, new-product diffusion models have assumed thatthe adoption process is influenced by one or more marketing instruments, typicallyadvertising and price. For a monopolist firm, Horsky and Simon (1983) suggestedthat innovators are influenced by advertising:
PY .t/ D Œ� C ˇ ln a.t/� Œm � Y .t/�C �Y .t/
mŒm � Y .t/�
10 S. Jørgensen
where the effect of advertising is represented by the term ˇ ln a.t/: Using thelogarithm implies that advertising is subject to decreasing returns to scale.
Later on, the Bass model has been extended to multi-firm competition and studiedin a series of papers using differential games. Early examples of this literature areTeng and Thompson (1983, 1985). Dockner and Jørgensen (1992) characterized anOLNE in a game with alternative specifications of the dynamics. One example is
PY .t/ D Œ� C ˇ ln ai .t/C �Z.t/� Œm �Z.t/�
where Z.t/ , Y1.t/C Y2.t/ denotes industry cumulative sales. Equilibrium adver-tising rates decrease over time, a result which carries over from the monopoly case(cf. Horsky and Simon 1983). The reader should note that decreasing advertisingrates often are driven by (i) the saturation effect – due to the fixed and finite marketpotential – and/or (ii) the absence of salvage values in the objective functionals.
New-product diffusion models have been studied with prices instead of advertis-ing efforts as decision variables. An example is Breton et al. (1997) who studied adiscrete-time dynamic game.
2.5 Advertising Goodwill Models
Most brands enjoy some goodwill among consumers – although some brands maysuffer from negative goodwill (badwill). Goodwill can be created, for example,through good product or service quality, a fair price or brand advertising. Our focushere is on advertising.
The seminal work in the area is Nerlove and Arrow (1962) who studied a singlefirm’s dynamic optimization problem. Goodwill is modeled in a simple way, bydefining a state variable termed “advertising goodwill.” This variable represents astock and summarizes the effects of current and past advertising efforts. Goodwill,denoted G.t/; evolves over time according to the simple dynamics
PG.t/ D a.t/ � ıG.t/
that have a straightforward interpretation. The first term on the right-hand side isthe firm’s gross investment in goodwill; the second term reflects decay of goodwill.Hence the left-hand side of the dynamics represents net investment in goodwill.
Note that omitting the decay term ıG.t/ has an interesting strategic implication.When the firm has accumulated goodwill to a certain level, it is locked in and hastwo choices only: it can continue to build up goodwill or keep the goodwill levelconstant. The latter happens if the advertising rate is chosen as a.t/ D ıG.t/=;
i.e., the advertising rate at any instant of time is proportional to the current goodwilllevel.
It is a simple matter to extend the Nerlove-Arrow (N-A) dynamics to anoligopolistic market. The goodwill stock of firm i; Gi .t/, then evolves according to
Marketing 11
the dynamics PGi.t/ D ai .t/� ıGi .t/: Fershtman (1984) identified an OLNE where– not surprisingly – equilibrium advertising strategies mimic those in Nerlove andArrow (1962).
The N-A model has been employed in various contexts. We present twoillustrations.
Example 3. Chintagunta (1993) posed the following question: How sensitive area firm’s profits to deviations from equilibrium advertising strategies? After havingcharacterized an OLNE, goodwill dynamics were estimated using data for a pre-scription drug manufactured by two firms in the US pharmaceutical industry. Thenestimates of equilibrium advertising paths can be made. Numerical simulationssuggested, for a quite wide range of parameter values, that profits are not particularlysensitive to deviations from equilibrium advertising levels.6
Example 4. Buratto and Zaccour (2009) analyzed a Stackelberg game playedbetween a licensor (the leader) and a licensee (the follower), operating in the fashionindustry. The dynamics are variations of the N-A model and a time-consistent open-loop Stackelberg equilibrium (OLSE) was characterized. Time-consistency meansthat the leader has no incentive to deviate from its announced policy: This makesthe announcement credible.7 Coordination of advertising efforts can be achieved ifthe licensor uses an incentive strategy which is designed in such a way that if thelicensee sticks to its part of the cooperative agreement, the licensor will do the same(and vice versa).
As we shall see in Sect. 4, the N-A dynamics have been popular in differentialgames played by the firms forming a marketing channel (supply chain).
3 Pricing Games
3.1 Introduction
In the area of pricing, a variety of topics have been studied, for example, pricingof new products and the effects of cost experience and demand learning (word ofmouth) on pricing strategies. In Sect. 4 we look at the determination of wholesaleand retail prices in a marketing channel.
Cost experience (or cost learning) is a feature of a firm’s production processthat should be taken into consideration when a firm decides its pricing strategy. In
6This phenomenon, that a profit function is “flat,” has been observed elsewhere. One example isWilson’s formula in inventory planning.7Most often, an OLSE is not time-consistent and the leader’s announcement is incredible. Theproblem can be fixed if the leader precommits to its announced strategy.
12 S. Jørgensen
particular, cost experience has been incorporated in models of new-product adoptionprocesses. The meaning of cost experience is that the unit production cost of aproduct decreases with cumulative output, i.e., there is a learning-by-doing effect.This applies both to the production of a product and the provision of a service.Formally, let Yi .t/ represent cumulative output by time t and ci .Yi / be a functionthe values of which determine the unit production cost: Cost experience meansthat c0i .Yi / < 0; i.e., the unit cost decreases as cumulative output increases. Thisphenomenon has a long history in manufacturing and has been observed in manyindustries.
3.2 Demand Learning
The demand learning phenomenon (diffusion effect, word of mouth) was defined inSect. 2. Let pi .t/ be the consumer price charged by firm i: Dockner and Jørgensen(1988a) studied various specifications of sales rate dynamics, for example thefollowing:
PYi .t/ D�˛i � ˇipi .t/C
�pj .t/ � pi .t/
��f .Yi .t/C Yj .t// (1)
where Yi .t/ denotes cumulative sales and f is a function taking positive values.The “price differential” term pj .t/ � pi .t/ is somewhat extreme: if the price pj .t/is one cent higher than pi .t/; sales of firm i increase.8 Dockner and Jørgensenidentified an OLNE. It was shown that if f 0 < 0; and under a certain assumption,equilibrium prices are decreasing for all t . A similar result was obtained by Kalish(1983) in the case of a monopoly which illustrates the lack of strategic interactionin an OLNE. Dockner and Gaunersdorfer (1996) used the dynamics in (1) withf .Yi .t/C Yj .t// D m � ŒY1 .t/C Y2 .t/�:
A strategy of decreasing prices is known as “skimming.” The idea is to exploit thesegment of customers who are first in the market because these customers often havethe highest willingness to pay. Later on, as firms benefit from cost learning, pricescan be lowered to reach new customer segments. A good example is the marketingof electronic goods and household appliances where we see that new products oftenare introduced at a high price.
Models that incorporate demand learning have been used in various contexts.Here are two examples:
Example 5. Minowa and Choi (1996) analyzed a model in which each firm sellstwo products, one primary and one secondary (or contingent) product. The latter isuseful only if the consumer has the primary product. This is a multiproduct problemwith a diffusion process (including demand learning) of a contingent product thatdepends on the diffusion process of the primary product.
8See also Eliashberg and Jeuland (1986).
Marketing 13
Example 6. Breton et al. (1997) studied a Stackelberg differential game. Let theleader and follower be represented by subscripts l and f; respectively, such thati 2 fl; f g. The dynamics are (cf. Dockner and Jørgensen 1988a)
PYi .t/ D�˛i � ˇipi .t/C .pj .t/ � pi .t//
�ŒAi C BiYi .t/�
�m � Yl.t/ � Yf .t/
�:
Cost experience effects are included and the end result is that the model is quitecomplex. An FSE is not analytically tractable, but numerical simulations suggestthat prices decline over time and the firm with the lowest cost charges the lowestprice. The shape of price trajectories seems to be driven mainly by cost experience.
3.3 Government Subsidies of New Technologies
As is well known, governments spend money on a multitude of activities. In somecases, government funds are granted with the purpose of influencing the behaviorof citizens in one way or another. A specific situation is where a government offersa price subsidy to consumers if they are willing to adopt a new technology. A goodexample is “green energy” such as natural gas, wind power, and solar energy. InDenmark, for instance, all three types of green energy are subsidized. There aretwo main reasons why a government wishes to accelerate the adoption of suchtechnologies:
1. Increased usage of the technology is believed to benefit of the environment.2. With increased usage, manufacturers of the technology should benefit from lower
unit production costs, due to cost experience effects. This should, in turn inducemanufacturers to lower their consumer prices.
Note that consumers stand to benefit twofold: They pay a lower price to themanufacturer and receive a government subsidy.
A seminal paper in the area is Kalish and Lilien (1983) who posed the govern-ment’s problem as one of optimal control (manufacturers of the new technology arenot decision-makers in a game). Building on this work, Zaccour (1996) consideredan initial stage of the life cycle of a new technology where there still is onlyone firm in the market. Firm and government play a differential game in whichgovernment chooses a subsidy rate s.t/, while the firm decides a consumer pricep.t/. Government commits to a subsidy scheme, specified by the time path s.�/;and has allocated a fixed budget to the program. Its objective is to maximize thetotal number of units, denoted Y .T /; that have been sold by time T: The sales ratedynamics are
PY .t/ D exp f�˛.1 � s.t//p.t/gh.Y .t//
in which h is a function taking positive values. If h0 < 0 (i.e., there are saturationeffects), the subsidy increases, while the consumer price decreases over time. Both
14 S. Jørgensen
are beneficial for the diffusion of the new technology. If h0 > 0 (i.e., there aredemand learning effects), the subsidy is decreasing. This makes sense because lesssupport is needed to stimulate sales and hence production.
With a view to what happens in real life, it may be more plausible to assumethat government is a Stackelberg leader who acts first and announces its subsidyprogram. Then the firm makes its decision. This was done in Dockner et al. (1996)using the sales dynamics PY .t/ D f .p.t// Œm � Y .t/� in which f 0.p/ < 0: Inan FSE, the consumer price is nonincreasing over time if players are sufficientlyfarsighted, a result which is driven by the saturation effect.
The reader may have noticed some shortcomings of the above models. Forexample, they disregard that existing technologies are available to consumers, gov-ernment has only one instrument (the subsidy) at its disposal, firm and governmenthave same time horizon, and the government’s budget is fixed. Attempts have beenmade to remedy these shortcomings. We give two illustrations:
Example 7. In Jørgensen and Zaccour (1999b), government is a Stackelberg leaderwho offers a price subsidy to consumers and commits to buying from the man-ufacturer a number of units of the new technology for its own use. The optionof consumers buying an existing technology is also accounted for. It turns out, inequilibrium, that the subsidy and the consumer price decrease at the same rate. Thisimplies that the net consumer price remains the same over time. The program thus isfair, in the sense that a customer pays the same net price no matter when the productis bought. Janssens and Zaccour (2014) assumed that the firm and government havedifferent time horizons and the government’s budget is seen as a flexible instrument(rather than a fixed constraint) with which government can attain a desired targetlevel of the consumer price.
3.4 Entry of Competitors
In problems where an incumbent firm faces potential entry of competitors, the lattershould be viewed as rational opponents and not as exogenous decision-makers whoimplement predetermined actions. A first attempt to do this in a dynamic setup wasLieber and Barnea (1977) in which an incumbent firm sets the price of its productwhile competitors invest in productive capacity. The authors viewed, however, theproblem as an optimal control problem of the incumbent, supposing that potentialentrants believe that the incumbent’s price remains constant. Jørgensen (1982b)and Dockner (1985) solved the problem as a differential game, and Dockner andJørgensen (1984) extended the analysis to Stackelberg and cooperative differentialgames. These analyses concluded that – in most cases – the incumbent shoulddiscourage entry by decreasing its price over time.
A more “intuitive” representation of the entry problem was suggested in Eliash-berg and Jeuland (1986). A monopolist (firm 1) introduces a new durable product,anticipating the entry of a competitor (firm 2). The planning period of firm 1 is
Marketing 15
Œt0; T2� and that of firm 2 is ŒT1; T2�where T1 is the known date of entry: (An obviousmodification here would be to suppose that T1 is a random variable.) Sales dynamicsin the two time periods are, indexing firms by i D 1; 2 W
PY1.t/ D Œ˛1 � ˇ1p1.t/� Œm � Y1.t/� ; Y1.t0/ D Y2.t0/ D 0; t 2 Œt0; T1�
PYi .t/ D�˛i � ˇipi .t/C .pj .t/ � pi .t//
�Œm �Z.t/� ; Y2.T1/ D 0; t 2 ŒT1; T2�
where Z.t/ , Y1.t/C Y2.t/: Three types of incumbents are considered:
• The non-myopic monopolist correctly predicts entry at T1 and has planninghorizon T2 > T1.
• The myopic monopolist disregards the duopoly period and has planning horizonT1.
• The surprised monopolist has planning horizon T2 but does not foresee entry atT1 < T2.
It turns out that in an OLNE, the myopic monopolist sets a higher price than anon-myopic monopolist. This is the price paid for being myopic because a higherprice decreases sales and hence leaves a higher untapped market open to the entrant.The surprised monopolist charges a price that lies between these two prices.
The next two papers consider entry problems where firms control both advertis-ing efforts and consumer prices. Fershtman et al. (1990) used the N-A dynamics andassumed that one firm has an advantage from being the first in a market. Later on, acompetitor enters but starts out with a higher production cost. However, productioncosts of the two firms eventually will be the same. Steady-state market shares wereshown to be affected by the order of entry, the size of the initial cost advantage, andthe duration of the monopoly period. Chintagunta et al. (1993) demonstrated thatin an OLNE, the relationship between advertising and price can be expressed as ageneralization of the “Dorfman-Steiner formula” (see, e.g., Jørgensen and Zaccour2004, pp. 137–138). Prices do not appear in N-A dynamics which make their roleless significant.
Remark 8. Xie and Sirbu (1995) studied entry in a type of market not consideredelsewhere. The market is one in which there are positive network externalities whichmeans that a consumer’s utility of using a product increases with the number ofother users. Such externalities typically occur in telecommunication and internetservices. To model sales dynamics, a diffusion model was used, assuming thatmarket potential is a function of consumer prices as well as the size of the installedbases of the duopolists’ products. The authors asked questions like: Will networkexternalities give market power to the incumbent firm? Should an incumbent permitor prevent compatibility when a competitor enters? How should firms set theirprices?
16 S. Jørgensen
3.5 Other Models of Pricing
Some models assume that both price and advertising are control variables of afirm. A seminal paper, Teng and Thompson (1985), considered a game with new-product diffusion dynamics. For technical reasons, the authors focused on priceleadership which means that there is only one price in the market, determined bythe price leader (typically, the largest firm). OLNE price and advertising strategieswere characterized by numerical methods. See also Fershtman et al. (1990) andChintagunta et al. (1993).
Krishnamoorthy et al. (2010) studied a durable-product oligopoly. Dynamicswere of the V-W type, without a decay term:
PYi .t/ D iai .t/Di .pi .t//pm � .Y1.t/C Y2.t// (2)
where the demand function Di is linear or isoelastic. FNE advertising and pricingstrategies were determined. The latter turn out to be constant over time. In Helmesand Schlosser (2015), firms operating in a durable-product market decide on pricesand advertising efforts. The dynamics are closely related to those in Krishnamoorthyet al. (2010). Helmes and Schlosser specified the demand function as Di.pi / D
p�"ii where "i is the price elasticity of product i . The authors looked for an FNE,
replacing the square root term in the dynamics in (2) by general functional forms.One of the rare applications of stochastic differential games is Chintagunta
and Rao (1996) who supposed that consumers derive value (or utility) from theconsumption of a particular brand. Value is determined by (a function of) price; theaggregate level of consumer preference for the brand; and a random component: Theauthors looked for an OLNE and focused on steady-state equilibrium prices. Givenidentical production costs, a brand with high steady-state consumer preferencesshould set a high price which is quite intuitive. The authors used data from twobrands of yogurt in the USA to estimate model parameters. Then steady-state pricescan be calculated and compared to actual (average) prices, to see how much firms“deviated” from steady-state equilibrium prices prescribed by the model.
Gallego and Hu (2014) studied a two-firm game in which each firm has a fixedinitial stock of perishable goods. Firms compete on prices. Letting Ii .t/ denote theinventory of firm i , the dynamics are
PIi .t/ D �di .t; p.t//; Ii .t0/ D C
where di is a demand function, p.t/ the vector of prices, and C > 0 a fixedinitial inventory. OLNE and FNE strategies were identified. Although this game isdeterministic, it may shed light on situations where demand is random. For example,given that demand and supply are sufficiently large, equilibria of the deterministicgame can be used to construct heuristic policies that would be asymptotic equilibriain the stochastic counterpart of the model.
Marketing 17
4 Marketing Channels
4.1 Introduction
A marketing channel (or supply chain) is a system formed by firms that most oftenare independent businesses: suppliers, a manufacturer, wholesalers, and retailers.The last decades have seen a considerable interest in supply chain managementthat advocates an integrated view of the system. Major issues in a supply chainare how to increase efficiency by coordinating decision-making (e.g., on prices,advertising, and inventories) and information sharing (e.g., on consumer demandand inventories). These ideas are not new: The study of supply chains has a longtradition in marketing literature.
Most differential game studies of marketing channels assume a simple structurehaving one manufacturer and one retailer, although extensions have been madeto account for vertical and horizontal competition or cooperation. Examples area channel consisting of a single manufacturer and multiple (typically competing)retailers or competition between two channels, viewed as separate entities.
Decision-making in a channel may be classified as uncoordinated (decentralized,noncooperative) or coordinated (centralized, cooperative). Coordination means thatchannel members make decisions (e.g., on prices, advertising, inventories) that willbe in the best interest for the channel as an entity.9 There are various degreesof coordination, ranging from coordinating on a single decision variable (e.g.,advertising) to full coordination of all decisions. The reader should be aware thatformal – or informal – agreements to cooperate by making joint decisions (in fact,creating a cartel) are illegal in many countries. More recent literature in the area hasstudied if coordination can be achieved without making such agreements.
A seminal paper by Spengler (1950) identified “the double marginalizationproblem” which illustrates lack of coordination. Suppose that a manufacturer setsthe transfer (wholesale) price by adding a profit margin to the unit production cost.Subsequently, a retailer sets the consumer price by adding a margin to the transferprice. It is readily shown that the consumer price will be higher and hence consumerdemand lower, than if channel members had coordinated their price decisions. Thereason is that in the latter case, one margin only would be added.
In the sequel, unless explicitly stated, state equations are the N-A advertisinggoodwill dynamics (or straightforward variations on that model). We consider themost popular setup, a one-manufacturer, one-retailer channel, unless otherwiseindicated.
9It may happen that, say, a manufacturer has a strong bargaining position and can induce theother channel members to make decisions such that the payoff of the manufacturer (not that ofthe channel) is maximized.
18 S. Jørgensen
4.2 Channel Coordination
4.2.1 Full CooperationFull cooperation is understood to mean that channel members wish to maximizechannel profits. Firms remain independent, but act as one firm.10 The literatureon this issue mainly deals with situations in which channel members decide onadvertising efforts, the wholesale, and the consumer prices. The problem is to inducechannel members to make decisions that lead to maximization of the joint profits.
Models with Advertising and PricingChintagunta and Jain (1992) considered a channel in which both manufacturerand retailer advertise. Firms play OLNE strategies if they do not coordinate theiradvertising efforts. It is shown that firms use more advertising effort if theycooperate and this generates higher channel profits. These results are intuitive andhave often been noted in the literature. In Jørgensen and Zaccour (1999a) firmsdecide their respective advertising efforts. The consumer price is set by the retailer,the transfer price by the manufacturer. In an FNE, the consumer price is higherthan the one which is charged under full cooperation. This is an illustration of thedouble marginalization phenomenon. It turns out that for all t; instantaneous channelprofits under cooperation are higher than those without cooperation which gives anincentive to cooperate at any time during the play of the game.
In the pricing models that we have presented above, the consumer price has aninstantaneous influence on the dynamics and the profit functionals. To allow forlonger-term (“carry-over”) effects of retail pricing, Zhang et al. (2012) included aconsumer reference price, denotedR.t/:11 The hypothesis is that the reference pricecan be (positively) affected by channel member advertising. The consumer price pis assumed constant. Reference price dynamics are given by
PR.t/ D ˇ.p �R.t//C �mA.t/C �ra.t/ (3)
where A.t/ and a.t/ are advertising efforts of manufacturer and retailer, respec-tively. (N-A goodwill dynamics are also a part of the model.) The manufactureris a Stackelberg leader who determines the rate of advertising support to be givento the retailer. The paper proposes a “nonstandard” coordination mechanism: Eachchannel member subsidizes advertising efforts of the other member.
In Martín-Herrán and Taboubi (2015) the consumer reference price is anexponentially weighted average of past values of the actual price p.t/; as in (3),but without the advertising terms on the right-hand side. Two scenarios were
10An extreme case is vertical integration where channel members merge and become one firm.11The meaning of a reference price is the following. Suppose that the probability that a consumerchooses brand X instead of brand Y depends not only on the current prices of the brands but alsoon their relative values when compared to historic prices of the brands. These relative values, asperceived by the consumer, are called reference prices. See, e.g., Rao (2009).
Marketing 19
treated: vertical integration (i.e., joint maximization) and a Stackelberg game withmanufacturer as the leader. It turns out that – at least under certain circumstances –there exists an initial time period during which cooperation is not beneficial.
Models with Pricing OnlyChiang (2012) focused on a durable product. Retail sales dynamics are of the “new-product diffusion” variety, with a price-dependent market potential and saturationeffects only:
PY .t/ D ˛ Œm � p.t/ � Y .t/�
where Y .t/ represents cumulative sales and p.t/ the consumer price, set by theretailer. The termm�p.t/ is the market potential being linearly decreasing in price.The manufacturer sets the wholesale price: OLNE, FNE, and myopic equilibria arecompared to the full cooperation solution.12 A main result is that both channelmembers are better off, at least in the long run, if they ignore the impact of currentprices on future demand and focus on intermediate-term profits.
Zaccour (2008) asked the question whether a two-part tariff can coordinate achannel. A two-part tariff determines the wholesale price w as w D c C k=Q.t/
where c is the manufacturer’s unit production cost andQ.t/ the quantity ordered bythe retailer. The idea of the scheme clearly is to induce the retailer to order largerquantities since this will diminish the effect of the fixed ordering (processing) costk. In a Stackelberg setup with manufacturer as leader, the answer to the question isnegative: A two-part tariff will not enable the leader to impose the full cooperationsolution even if she precommits to her part of that solution.
4.2.2 Partial CooperationIn situations where full cooperation (joint profit maximization) is not an option, anobvious alternative is to cooperate partially. For such cooperation to be acceptable,it should at least be profit improving for both channel members. A pertinentresearch question then is whether there exist coordination schemes that fulfill thisrequirement. The answer is yes.
An example is cooperative advertising which is an arrangement where amanufacturer pays some (or all) of the costs incurred by the retailer who advertiseslocally to promote the manufacturer’s product. In addition, the manufacturer mayadvertise nationally (at her own expense). The terms “local” and “national” referto the type of media that are used for advertising. Local advertising most oftenhas short-term, promotional purposes, while national advertising has a longer-termobjective: to maintain or enhance brand loyalty.
12Myopic behavior means that a decision-maker disregards the dynamics when solving herdynamic optimization problem.
20 S. Jørgensen
We give below some examples of problems that have been addressed in the areaof cooperative advertising. In the first example, the manufacturer has the option ofsupporting the retailer by paying some (or all) of her advertising costs.
Example 9. In Jørgensen et al. (2000) each channel member can use long-term(typically, national) and/or short-term (typically, local) advertising efforts. Themanufacturer is a Stackelberg leader who chooses the shares that she will pay ofthe retailer’s costs of the two types of advertising. Four scenarios are studied: (i)no support at all, (ii) support of both types of retailer advertising, support of (iii)long-term advertising only, and (iv) support of short-term advertising only. It turnedout that both firms prefer full support to partial support which, in turn, is preferredto no support.
Most of the literature assumes that any kind of advertising has a favorableimpact on goodwill. The next example deals with a situation in which excessiveretailer promotions may harm the brand goodwill. The question then is: Should amanufacturer support retailer advertising?
Example 10. Jørgensen et al. (2003) imagined that frequent retailer promotionsmight damage goodwill, for instance, if consumers believe that promotions are acover-up for poor product quality. Given this, does it make sense that the manu-facturer supports retailer promotion costs? An FNE is identified if no advertisingsupport is provided, an FSE otherwise. An advertising support program is favorablein terms of profits if (a) the initial goodwill level is “low” or (b) if this level is at an“intermediate” level and promotions are not “too damaging” to goodwill.
The last example extends our two-firm channel setup to a setting with twoindependent and competing retailers.
Example 11. Chutani and Sethi (2012a) assumed that a manufacturer is a Stack-elberg leader who sets the transfer price and supports retailers’ local advertising.Retailers determine their respective consumer prices and advertising efforts as aNash equilibrium outcome in a two-player game. It turns out that a cooperativeadvertising program, with certain exceptions, does not benefit the channel as such.Chutani and Sethi (2012b) studied the same problem as the (2012a) paper. Dynamicsnow are a nonlinear variant of those in Deal (1979), and the manufacturer has theoption of offering different support rates to retailers.13
13In practice it may be illegal to discriminate retailers, for example, by offering different advertisingallowances or quoting different wholesale prices.
Marketing 21
4.3 Channel Leadership
Most of the older literature assumed that the manufacturer is the channel leader butwe note that the last decades have seen the emergence of huge retail chains havingsubstantial bargaining power. Such retail businesses may very well serve as channelleaders.
Leadership of a marketing channel could act as a coordinating device when aleader takes actions that induce the follower to decide in the best interest of thechannel.14 A leader can emerge from exogenous reasons, typically because it isthe largest firm in the channel. It may also happen that a firm can gain leadershipendogenously, for the simple reason that such an outcome is desired by all channelmembers. In situations like this, some questions arise: Given that leadership isprofitable at all, who will emerge as the leader? Who stands to benefit the mostfrom leadership? The following example addresses these questions.
Example 12. Jørgensen et al. (2001) studied a differential game with pricing andadvertising. The benchmark outcome, which obtains if no leader can be found, isan FNE. Two leadership games were studied: one having the manufacturer as leaderand another with the retailer as leader. These games are played a la Stackelbergand in each of them an FSE was identified. The manufacturer controls his profitmargin and national advertising efforts, while the retailer controls her margin andlocal advertising efforts. In the retailer leadership game, the retailer’s margin turnsout be constant (and hence its announcement is credible). The manufacturer hasthe lowest margin which might support a conjecture that a leader secures itselfthe higher margin. In the manufacturer leadership game, the manufacturer does nothave the higher margin. Taking channel profits, consumer welfare (measured by thelevel of the retail price), and individual profits into consideration, the answers to thequestions above are – in this particular model – that (i) the channel should have aleader and (ii) the leader should be the manufacturer.
4.4 Incentive Strategies
Incentive strategies can be implemented in games with a leader as well as in gameswithout a leader. The idea probably originated in Stackelberg games where, if aleader provides the right incentive to the follower, the latter can be induced to behavein accordance with the leader’s objective. This illustrates the old adage: ‘If you wishother people to behave in your own interest, then make them see things your way’(Basar and Olsder 1995, p. 396). First we look at the original setup, i.e., games witha leader.
14In less altruistic environments, a leader may wish to induce the follower to make decisions thatare in the best interest of the leader.
22 S. Jørgensen
4.4.1 Games with LeadershipSuppose that the manufacturer is the channel leader who wishes to induce the retailerto act, not in the best interest of the leader, but in the best interest of the channel.Then the retailer should be induced to choose the decisions prescribed for her by acooperative solution. We present two examples that illustrate how this can be done.
Example 13. Jørgensen and Zaccour (2003a) considered a game where aggregateretailer promotions have negative effects on goodwill, cf. Jørgensen et al. (2003),but in contrast to the latter reference, retailer promotions now have carry-overeffects. Retailers – who are symmetric – determine local promotional efforts, whilethe manufacturer determines national advertising, denoted A.t/: The manufacturerwishes to induce retailers to make promotional decisions according to the fullycooperative solution and announces the incentive strategy
a.P /.t/ D Ac.t/C �.t/ ŒP .t/ � Pc.t/� (4)
where P .t/ is aggregate retailer promotional effort and the subscript c refersto cooperation. The strategy in (4) works as follows: if retailers do as they aresupposed to do, then we shall have P .t/ D Pc.t/ and the manufacturer will chooseA.t/ D Ac.t/. Technically, to find an equilibrium of the incentive game, retailersdetermine their optimal promotion rates under the belief that the manufacturer willact in accordance with (4). This provides P .�.t// and the manufacturer selects theincentive coefficient �.t/ such that the optimal retailer promotion rate equals thedesired one, i.e., Pc.t/:
Example 14. In Jørgensen et al. (2006) a manufacturer is a Stackelberg leaderwho wishes to induce a retailer to increase her local advertising. This can beaccomplished by implementing a promotion allowance scheme such that themanufacturer pays an amount, denotedD.P /; per unit of retailer promotional effortP .t/: The incentive strategy is
D.P /.t/ D �P .t/
where � is a constant to be determined by the manufacturer before the start ofthe game. Two scenarios were considered. In the first, the manufacturer wishes theretailer to behave in accordance with the full cooperation outcome. In the second, themanufacturer is selfish and selects an incentive that will make the retailer maximizethe manufacturer’s profits.
4.4.2 Games Without LeadershipEven if there is no leader of the channel, incentive strategies may still be imple-mented. The idea is as follows. Suppose that channel members agree upon adesired outcome (e.g., the full cooperation solution) for the channel. Side payments(e.g., advertising allowances paid to the retailer) are infeasible. Nevertheless, the
Marketing 23
desired outcome could be realized if channel members play according to equilibriumincentive strategies. An incentive equilibrium has the following best-responseproperty: If firm j sticks to its incentive strategy, the best choice of firm i ¤ j
is to use its own incentive strategy. Unilateral deviations are not payoff improvingand the desired outcome is in equilibrium.
The following example shows how the fully cooperative solution can be imple-mented as an incentive equilibrium. Other solutions than full cooperation can beimplemented as equilibrium outcomes provided that channel members can agreethat such a solution is the desired one.
Example 15. In Jørgensen and Zaccour (2003b) a manufacturer controls the transferprice pM.t/ and national advertising effort aM .t/, while the retailer controls theconsumer price pR.t/ and local advertising effort aR.t/. As we have seen, anincentive strategy makes the manufacturer’s decision dependent upon the retailer’sdecision and vice versa. Considering the advertising part, this can be done by usingthe strategies
M .aR/.t/ D adM .t/C �M.t/
�aR.t/ � a
dR.t/
�R.aM /.t/ D a
dR.t/C �R.t/
�aM .t/ � a
dM .t/
�
where �M.t/; �R.t/ are incentive parameters and the superscript d refers to thedesired solution. To find the incentive parameters, one needs to solve two optimalcontrol problems. (A similar procedure is used to determine the price incentivestrategies.) A main message of the paper is that channel coordination can beachieved without a leader, using the above-incentive strategies. A related paper isDe Giovanni et al. (2015) who considered a supply chain where manufacturer andretailer invest in a product recovery program.
4.5 Franchise Systems
A franchise system is a business model that may be thought of as a mixture of acentralized and a decentralized channel. Typically, the franchisor offers a trademarkand supplies a standardized business format to be used by the franchisees. The latterare obliged to follow the business procedures outlined in their franchise contract.For instance, all franchisees must charge the same consumer price for a specificproduct. The consumer prices essentially are set by the franchisor.
We provide two examples of franchising, dealing with advertising and serviceprovision in cooperative and noncooperative situations.
Example 16. Sigué (2002) studied a system with a franchisor and two franchisees.The former takes care of national advertising efforts, while franchisees are in chargeof local advertising and service provision. Franchisees benefit from a high level
24 S. Jørgensen
of system goodwill, built up through national advertising and local service.15 Twoscenarios were examined. The first is a benchmark where a noncooperative gameis played by all three players and an FNE is identified. In the second, franchiseescoordinate their advertising and service decisions and act as one player (a coalition)in a game with the franchisor. Also here an FNE is identified. It turns out thatcooperation among franchisees leads to higher local advertising efforts (which isexpected). If franchisees do not cooperate, free riding will lead to an outcome inwhich less service than desirable is provided.
Example 17. Sigué and Chintagunta (2009) asked the following question: Whoshould do promotional and brand-image advertising, respectively, if franchisorand franchisees wish to maximize their individual profits? The problem wascast as a two-stage game. First, the franchisor selects one out of three differentadvertising arrangements, each characterized by its degree of centralization ofadvertising decisions. Second, given the franchisor’s choice, franchisees decideif they should cooperate or not. Martín-Herrán et al. (2011) considered a similarsetup and investigated, among other things, the effects of price competition amongfranchisees.
Jørgensen (2011) applied an important element of a franchise system, viz.,the contract, to the standard two-member marketing channel (not a franchisesystem). A franchise contract includes as special cases the classic wholesale-price contract, a two-part tariff (see above), and a revenue-sharing contract.16 Theretailer determines an ordering rate and the retail price, while the manufacturerdecides a production rate and the parameters of the contract that is offered to theretailer. Contract parameters are time dependent and the contract period is finite.Jørgensen characterized the fully cooperative solution and the selfish one in whichthe manufacturer wishes to maximize her own profits.
4.6 National Brands, Store Brands, and Shelf Space
Competition among national and store brands, the latter also known as private labels,is becoming more widespread. A store brand carries a name chosen by a retail chainwith the aim of creating loyalty to the chain. Store brands have existed for many
15Note that there may be a problem of free-riding if franchisees can decide independently onservice. The reason is that a franchisee who spends little effort on service gets the full benefitof her cost savings but shares the damage to goodwill with the other franchisee.16A revenue-sharing contract is one in which a retailer pays a certain percentage of her salesrevenue to the supplier who reciprocates by lowering the wholesale price. This enables the retailerto order more from the supplier and be better prepared to meet demand. A classic example hereis Blockbuster, operating in the video rental industry, that signed revenue-sharing contracts withmovie distributors. These arrangements were very successful (although not among Blockbuster’scompetitors).
Marketing 25
years and recent years, as a consequence of the increasing power of retailer chains,have seen the emergence of many new store brands.
Example 18. Amrouche et al. (2008a,b) considered a channel where a retailer sellstwo brands: one produced by a national manufacturer and a private label. Firms settheir respective prices and advertise to build up goodwill. The authors identifiedan FSE in the first paper and an FNE in the second. In the latter it was shownthat creating goodwill for both brands mitigates price competition between the twotypes of brands. Karray and Martín-Herrán (2009) used a similar setup and studiedcomplementary and competitive effects of advertising.
A line of research has addressed the problem of how to allocate shelf space tobrands in a retail outlet. Shelf space is a scarce resource of a retailer who mustallocate space between brands. With the increasing use of private labels, the retaileralso faces a trade-off between space for national and for store brands. Martín-Herránet al. (2005) studied a game between two competing manufacturers (who act asStackelberg leaders) and a single retailer (the follower). The authors identified atime-consistent OLSE. See also Martín-Herrán and Taboubi (2005), Amrouche andZaccour (2007).
4.7 The Marketing-Manufacturing Interface
Problems that involve multiple functional areas within a business firm are worth-while addressing from a research point of view and from that of a manager. Somedifferential game studies have studied problems lying in the intersection betweenthe two functional areas “marketing” and “manufacturing.” First we look at somestudies in which pricing and/or advertising games (as those encountered above) areextended with the determination of production levels, expansion and composition ofmanufacturing capacity, and inventories.
4.7.1 Production CapacityA seminal paper by Thépot (1983) studied pricing, advertising goodwill creation,and productive capacity expansion. Capacity dynamics follow the standard capitalaccumulation equations
PKi.t/ D ui .t / � ıKi .t/
where Ki.t/ is capacity, ui .t / the gross investment rate, and ı a depreciationfactor. The author identified a series of advertising-investment-pricing regimes thatmay occur as OLNE. Gaimon (1989) considered two firms that determine prices:production rates and the levels and composition of productive capacity. Three gameswere analyzed: one with feedback strategies, one with open-loop strategies, andone in which firm i uses an open loop and firm j a feedback strategy. The type
26 S. Jørgensen
of strategy critically impacts the outcome of the game, and numerical simulationssuggest that feedback strategies are superior in terms of profit.
Next we consider a branch of research devoted to the two-firm marketing channel(supply chain). The manufacturer chooses a production rate and the transfer price,while the retailer decides the consumer price and her ordering rate. Manufacturerand retailer inventories play a key role.
4.7.2 Production, Pricing, and OrderingThe following example focuses on decisions on transfer and consumer prices,production, procurement (ordering), and inventories.
Example 19. Jørgensen (1986b) studied a supply chain where a retailer determinesthe consumer price and the purchase rate from a manufacturer. The latter decidesa production rate and the transfer price: Both firms carry inventories, and themanufacturer may choose to backlog retailer’s orders. OLNE strategies werecharacterized. Eliashberg and Steinberg (1987) considered a similar setup andlooked for an OLSE with the manufacturer as leader.
Desai (1996) studied a supply chain in which a manufacturer controls theproduction rate and the transfer price, while a retailer decides its processing (orordering) rate and the consumer price. It was shown that centralized decision-making leads to a lower consumer price and higher production and processing rates,results that are expected. The author was right in stressing that channel coordinationand contracting have implications, not only for marketing decisions but also forproduction and ordering decisions.
Jørgensen and Kort (2002) studied a system with two serial inventories, onelocated at a central warehouse and another at a retail outlet. A special feature is thatthe retail inventory is on display in the outlet and the hypothesis is that having a largedisplayed inventory will stimulate demand. The retail store manager determinesthe consumer price and the ordering rate from the central warehouse. The centralwarehouse manager orders from an outside supplier. The authors first analyzed anoncooperative game played by the two inventory managers, i.e., a situation inwhich inventory decisions are decentralized. Next, a centralized system was studied,cast as the standard problem of joint profit maximization.
Eliashberg and Steinberg (1991) investigated the strategic implications of man-ufacturers having different types of production and holding cost functions. Theproduction cost of firm 1, called a “production smoother,” is convex, inventoryholding costs are linear, and the firm determines production, inventory, and pricingstrategies. Production smoothing means that the firm aims at keeping the productionrate close to the ordering rate of the retailer, trying to escape the unfortunateconsequences of the convex production cost function. The production cost of firm 2,called an “order taker,” is linear. Since this firm produces to order, it does not holdinventory. The firm determines production and pricing strategies.
Marketing 27
4.7.3 QualityGood design quality means that a product performs well in terms of durability andease of use and “delivers what it promises.” In Mukhopadhyay and Kouvelis (1997),firm i controls the rate of change of the design quality, denoted ui .t /; of its productas well as the product price pi .t/: Sales dynamics are given by a modified V-Wmodel:
PSi .t/ D ˛i Œm.t/ � Si .t/� � ıiSi .t/C i Pm.t/
in which m.t/ is the market potential and the term i Pm.t/ is the fraction of newcustomers who will purchase from firm i: A firm incurs (i) a cost of changing thedesign quality as well as (ii) a cost of providing a certain quality level: In an OLNE,two things happen during an initial stage of the game: firms use substantial effortsto increase their quality levels as fast as possible and sales grow fast as prices aredecreased. Later on, the firms can reap the benefits of having products of higherquality.
Conformance quality measures the extent to which a product conforms to designspecifications. A simple measure is the proportion of non-defective units producedduring some interval of time. In El Ouardighi et al. (2013), a manufacturer decidesthe production rate, the rate of conformance quality improvement efforts, andadvertising. Two retailers compete on prices, and channel transactions follow awholesale price or a revenue-sharing contract.
Teng and Thompson (1998) considered price and quality decisions for a newproduct, taking cost experience effects into account. Nair and Narasimhan (2006)suggested that product quality, in addition to advertising, affects the creation ofgoodwill in a duopoly.
5 Stochastic Games
As in other areas, the literature on stochastic games in marketing was precededby optimal control studies of monopolistic firms. Early work on such problemsincludes Tapiero (1975), Sethi (1983), and Rishel (1985). See also the book byTapiero (1988).
The interest in games of marketing problems played under uncertainty has beenmoderate. Among the rather few contributions are the following three:
Horsky (1988) studied a stochastic model of advertising competition with new-product diffusion dynamics represented by a Markov decision process (discretestate, continuous time). Firms can influence the transition probabilities throughtheir advertising efforts. The author identified an FNE with stationary advertisingstrategies ai .Y1; Y2/, and numerical simulations suggested that advertising rates aredecreasing in Y1; Y2:
Chintagunta and Rao (1996) considered a stochastic game in which the dynamics(representing consumer preferences) are deterministic, but the value of the firm’s
28 S. Jørgensen
brand depends on the aggregate consumer preference level, product price, and arandom term. They identified steady-state equilibrium prices.
Prasad and Sethi (2004) studied a stochastic differential game using a modifiedversion of the Lanchester advertising dynamics. Market share dynamics are givenby the stochastic differential equation
dXi D�ˇiai
pXj � ˇj aj
pXi � ı.2Xi � 1/
�dt C
�Xi ;Xj
�d!i
in which the term ı.2Xi�1/ is supposed to model decay of individual market shares.The fourth term on the right-hand side is white noise.
6 Doing the Calculations: An Example
The purpose of this section is, using a simple problem of advertising goodwillaccumulation, to show the reader how a differential game model can be formulatedand, in particular, analyzed: The example appears in Jørgensen and Gromova (2016).Key elements are the dynamics (state equations) and the objective functionals, aswell as the identification of the relevant constraints. Major tasks of the analysis arethe characterization of equilibrium strategies and the associated state trajectories aswell as the optimal profits to be earned by the players.
Consider an oligopolistic market with three firms, each selling its own particularbrand. For simplicity of exposition, we restrict our analysis to the case of symmetricfirms. The firms play a noncooperative game with an infinite horizon. Let ai .t/ � 0be the rate of advertising effort of firm i D 1; 2; 3 and suppose that the stock ofadvertising goodwill of firm i; denoted Gi.t/; evolves according to the followingNerlove-Arrow-type dynamics
PGi.t/ D �ai .t/I Gi.0/ , g0 > 0 (5)
Due to symmetry, advertising efforts of all firms are equally efficient (same value of�/, and we normalize � to one. Note, in contrast to the N-A model in Sect. 2.5,that the dynamics in (5) imply that goodwill cannot decrease: Once a firm hasaccumulated goodwill up to a certain level, it is locked in. A firm’s only optionsare to stay at this level (by refraining from advertising) or to continue to advertise(which will increase its stock of goodwill).
The dynamics in (5) could approximate a situation in which goodwill stocksdecays rather slowly. As in the standard N-A model, the evolution of a firm’sadvertising goodwill depends on its own effort only. Thus the model relies on ahypothesis that competitors’ advertising has no – or only negligible – impact on thegoodwill level of a firm.17
17A modification would be to include competitors’ advertising efforts on the right-hand side of (5),affecting negatively the goodwill stock of firm i: See Nair and Narasimhan (2006).
Marketing 29
The following assumption is made for technical reasons. Under the assumptionwe can restrict attention to strategies for which value functions are continuouslydifferentiable, and moreover, it will be unnecessary to explore the full state space ofthe system.
Assumption 1. Initial values of the goodwill stocks satisfy g0 < ˇ=6 where ˇ is aconstant to be defined below.
Let si .t/ denote the sales rate; required to be nonnegative, of firm i . The salesrate is supposed to depend on all three stocks of advertising goodwill:
si .t/ D fi .G.t// (6)
where G.t/ , .G1.t/; G2.t/; G3.t// : As we have seen, advertising of a firm doesnot affect the evolution of goodwill of its rivals, but advertising of a firm affectsits own goodwill and therefore, through (6), the sales rates of all the firms in theindustry. As in, e.g., Reynolds (1991) we specify function fi as
fi .G/ D
"ˇ �
3XhD1
Gh
#Gi (7)
where ˇ > 0 is the parameter referred to in Assumption 1. All firms must, for anyt > 0; satisfy the path constraints
3XhD1
Gh.t/ � ˇ (8)
ai .t/ � 0:
Partial derivatives of function fi are as follows:
@fi
@GiD ˇ � 2Gi �Gj �Gk
@fi
@GkD �Gi ; k ¤ i:
Sales of a firm should increase when its goodwill increases, that is, we must require
ˇ � 2Gi �Gj �Gk > 0 for any t > 0: (9)
Negativity of @fi=@Gk means that the sales rate of firm i decreases if goodwill offirm k increases.
30 S. Jørgensen
Having described the evolution of goodwill levels and sales rates, it remains tointroduce the economic components of the model. Let � > 0 be the constant profitmargin of firm i: The cost of advertising effort ai is
C.ai / Dc
2a2i
in which c > 0 is a parameter determining the curvature of the cost function.18 Let� > 0 be a discounting rate, employed by all firms. The profit of firm i then is
Ji .ai / D
Z 10
e��t
(�
"ˇ �
3XhD1
Gh.t/
#Gi.t/ �
c
2a2i .t/
)dt: (10)
Control and state constraints, to be satisfied by firm i for all t 2 Œ0;1/; are asfollows:
ai .t/ � 0I ˇ > 2Gi.t/CGj .t/CGk.t/
where we note that the second inequality impliesGi.t/CGj .t/CGk.t/ < ˇ. Thus,if sales fi increase as goodwill Gi increases, sales cannot be negative.
The model parameters are time independent and the planning horizon is infinite.In such a problem, a standard approach in the literature has been to look for sta-tionary equilibria. Stationarity means that advertising strategies and value functionswill not depend explicitly on time.
We suppose that firms cannot (or will not) cooperate when determining theiradvertising efforts and hence consider a noncooperative game. In this game weshall identify an FNE and let Vi .G/ be a continuously differentiable value functionof firm i . This function must solve the following Hamilton-Jacobi-Bellman (HJB)equation:
�Vi .G/ D maxai�0
8<:�
24ˇ �
3XjD1
Gj
35Gi � c
2a2i C
@Vi
@Giai
9=; : (11)
Performing the maximization on the right-hand side of (11) generates candidates forequilibrium advertising strategies:
ai .G/ D
(1c@Vi@Gi
> 0 if @Vi@Gi
> 0
0 if @Vi@Gi� 0
18A model with a linear advertising cost, say, cai and where ai on the right-hand side of thedynamics is replace by, say
pai ; will provide the same results, qualitatively speaking, as the
present model. To demonstrate this is left as an exercise for the reader.
Marketing 31
which shows that identical firms use the same type of strategy. Inserting thecandidate strategies into (11) provides
Vi .G/ D1
2�c
�@Vi
@Gi
2C�
�
24ˇ �
3XjD1
Gj
35Gi if ai > 0 (12)
Vi .G/ D�
�
24ˇ �
3XjD1
Gj
35Gi if ai D 0
from which it follows that
Vi .G/ jai>0 �Vi .G/ jaiD0D1
2�c
�@Vi
@Gi
2� 0:
If the inequality is strict, strategy ai > 0 payoff dominates strategy ai D 0, and allfirms will have a positive advertising rate throughout the game. If the equality signholds, any firm is indifferent between advertising and no advertising. This occurs iff@Vi=@Gi D 0; a highly unusual situation in which a firm essentially has nothing todecide.
We conjecture that the equation in the first line in (12) has the solution
Vi .G/ D ˛CAGiC A
2G2i C.�AGi C B/ .Gj CGk/C
B
2.G2
j CG2k/C�BGjGk
(13)in which ˛; A; B; A; B; �A; �B are constants to be determined. From (13) we getthe partial derivatives
@Vi
@GiD A C AGi C �A.Gj CGk/: (14)
Consider the HJB equation in the first line of (12) and calculate the left-hand sideof this equation by inserting the value function given by (13). This provides
��˛ C AGi C B.Gj CGk/C
A
2G2i C
B
2.G2
j CG2k/
C �A.GiGj CGiGk/C �BGjGk�
(15)
D1
2� AG
2i C ��AGiGj C ��AGiGk C �AGi C
1
2� BG
2j
C ��BGjGk C �BGj C1
2� BG
2k C �BGk C ˛�
32 S. Jørgensen
Calculate the two terms on the right-hand side of the HJB equation:
1
2�c
�@Vi
@Gi
2C�
�
24ˇ �
3XjD1
Gj
35Gi (16)
D1
2c
�A C AGi C �A.Gj CGk/
�2C �
�ˇ �
�Gi CGj CGk
��Gi
D1
2c2AC
1
2c 2AG
2i C
1
2c�2AG
2jC
1
2c�2AG
2kC
1
cA AGi C
1
cA�AGj C
1
cA�AGk
C1
c�2AGjGk C
1
c A�AGiGj C
1
c A�AGiGk
C �ˇGi � �G2i � �GiGj � �GiGk:
Using (15) and (16) the HJB equation becomes
1
2� AG
2i C ��AGiGj C ��AGiGk C �AGi C
1
2� BG
2j
C ��BGjGk C �BGj C1
2� BG
2k C �BGk C ˛�
D1
2c2A C
1
2c 2AG
2i C
1
2c�2AG
2j C
1
2c�2AG
2k C
1
cA AGi C
1
cA�AGj
C1
cA�AGk C
1
c�2AGjGk C
1
c A�AGiGj C
1
c A�AGiGk
C �ˇGi � �G2i � �GiGj � �GiGk
To satisfy this equation for any triple�Gi ;Gj ;Gk
�, we must have
Constant term W ˛c� D1
22A
Gi � terms W �cA D A A C �cˇ
GjGk � terms W �cB D A�A
G2i � terms W c� A D
2A � 2c�
G2j and G2
k � terms W �c B D �2A
GiGj and GiGk � terms W �c�A D A�A � c�
GjGk � terms W �c�B D �2A:
Disregarding ˛ (which follows from A/; the remaining equations admit a uniquesolution
Marketing 33
A Dˇ�p
c2�2 C 8c� � c��
4> 0I B D �
ˇ�c� �
pc2�2 C 8c�
�216c�
< 0
A Dc� �
pc2�2 C 8c�
2< 0I B D �B D
�c� �
pc2�2 C 8c�
�216c�
> 0
�A Dc� �
pc2�2 C 8c�
4< 0: (17)
It is easy to see that the solution passes the following test of feasibility (cf. Bass et al.2005a). If the profit margin � is zero, the value function should be zero because thefirm has no revenue and does not advertise.
Knowing that B D �B; the value function can be rewritten as
Vi .G/ D ˛ C AGi C A
2G2i C .�AGi C B/ .Gj CGk/C
B
2.Gj CGk/
2:
Using (17) yields A D �ˇ�A and A D 2�A, and the value function has the partialderivative
@Vi
@GiD A C AGi C �A.Gj CGk/ D �A
�2Gi CGj CGk � ˇ
�
which is positive because �A is negative and 2Gi C Gj C Gk < ˇ is imposed as astate constraint.
It remains to determine the time paths generated by the equilibrium advertisingstrategy and the associated goodwill levels. The advertising strategy is given bya�i .G/ D c
�1@Vi=@Gi and therefore
a�i .G/ D�A
c
�2Gi CGj CGk � ˇ
�: (18)
Recalling the state dynamics PGi.t/ D ai .t/, we substitute a�i .G/ into the dynamicsand obtain a system of linear inhomogeneous differential equations:
PGi.t/ D2�A
cGi .t/C
�A
c
�Gj .t/CGk.t/
��ˇ�A
c: (19)
Solving the equation in (19) yields
Gi.t/ D exp
4 �A
ct
� �g0 �
ˇ
4
Cˇ
4(20)
34 S. Jørgensen
which shows that the goodwill stock of firm i converges to ˇ=4 for t ! 1:
Finally, inserting Gi.t/ from (20) into (18) yields the time path of the equilibriumadvertising strategy:
a�i .t / D exp
4 �A
ct
��A.4g0 � ˇ/
c> 0:
Remark 20. Note that our calculations have assumed that the conditions g0 < ˇ=6
and, for all t; 2Gi .t/CGj .t/CGk.t/ < ˇ are satisfied. Using (20) it is easy to seethat the second condition is satisfied.
7 Conclusions
This section offers some comments on (i) the modeling approach in differentialgames of marketing problems as well as on (ii) the use of numerical meth-ods/algorithms to identify equilibria. Finally, we provide some avenues for futurework.
Re (i): Differential game models are stylized representations of real-life, multi-player decision problems, and they rely – necessarily – on a number of simplifyingassumptions. Preferably, assumptions should be justified theoretically as well asempirically. A problem here is that most models have not been validated empirically.Moreover, it is not always clear if the problem being modeled is likely to be adecision problem that real-life decision-makers might encounter. We give someexamples of model assumptions that may be critical.
1. Almost all of the models that have been surveyed assume deterministic consumerdemand. It would be interesting to see the implications of assuming stochasticdemand, at least for the most important classes of dynamics. Hint: In marketingchannel research, it could be fruitful to turn to the literature in operations andsupply chain management that addresses supply chain incentives, contracting,and coordination under random consumer demand.
2. In marketing channel research, almost all contributions employed Nerlove-Arrowdynamics (or straightforward modifications of these). The reason for this choicemost likely is the simplicity (and hence tractability) of the dynamics. It remains,however, to be seen what would be the conclusions if other dynamics wereused. Moreover, most models consider a simple two-firm setup. A setup withmore resemblance to real life would be one in which a manufacturer sellsto multiple retailers, but this has been considered in a minority of models.Extending the setup raises pertinent questions, for example, how a channel leadercan align the actions of multiple, asymmetric firms and if coordination requirestempering retailer competition. Another avenue for new research would be tostudy situations in which groups of retailers form coalitions, to stand united vis-a-vis a manufacturer. The literature on cooperative differential games should beuseful here.
Marketing 35
3. Many decisions in marketing affect, and are affected by, actions taken inother functional areas: procurement, production, quality management, capacityinvestment and utilization, finance, and logistics. Only a small number of studieshave dealt with these intersections
4. Quite many works study games in which firms have an infinite planning horizon.If, in addition, model parameters are assumed not to vary over time, analyticaltractability is considerably enhanced. Such assumptions may be appropriate intheoretical work in mathematical economics but seem to be less useful if we insistthat our recommendations should be relevant to real-life managers – who mustface finite planning horizons and need to know how to operate in nonstationaryenvironments.
Re (ii): In dynamic games it is most often the case that when models becomemore complex, the likelihood of obtaining complete analytical solutions (i.e., a fullcharacterization of equilibrium strategies, the associated state trajectories, and theoptimal profits of players) becomes smaller. In such situations one may resort to theuse of numerical methods. This gives rise to two problems: Is there an appropriatemethod/algorithm and are (meaningful) data available.19
There exist numerical methods/algorithms to compute, given the data, equilib-rium strategies and their time paths, the associated state trajectories, and profit/valuefunctions. For stochastic control problems in continuous time, various numericalmethods are available, see e.g., Kushner and Dupuis (2001), Miranda and Fackler(2002), and Falcone (2006). Numerical methods designed for specific problemsare treated in Cardaliaguet et al. (2001, 2002) (pursuit games, state constraints).Numerical methods for noncooperative as well as cooperative differential gamesare dealt with in Engwerda (2005).
The reader should note that new results on numerical methods in dynamicgames regularly appear in Annals of the International Society of Dynamic Games,published by Birkhäuser.
Example 21. Some studies resorted to numerical methods/algorithms to analyzedifferential games with Lanchester dynamics. For instance, Breton et al. (2006)identified an FSE (using data from the cola industry), while Jarrar et al. (2004)characterized an FNE. Both papers employed the Lanchester advertising model. Seealso Sadigh et al. (2012) who studied a problem of coordination of a two-membermarketing channel. See also Jørgensen and Zaccour (2007).
The differential game literature has left many problems of marketing strategyuntouched. In addition to what has already been said, we point to three:
19We shall disregard “numerical examples” where solutions are characterized using more or lessrandomly chosen data (typically, the values of model parameters).
36 S. Jørgensen
1. The service sector is undoubtedly gaining increasing importance. However, theprovision of services has received little attention in the literature on differentialgames in marketing. An introduction to service marketing and management canbe found in the handbook edited by Swartz and Iacobucci (2000).
2. E-Business shows substantial growth rates, but the area still has attracted littleattention from researchers working with differential games in marketing. Anintroduction to marketing channels in the E-business era is provided by thehandbook edited by Simchi-Levi et al. (2004).
3. A broad range of instruments are available to the marketing manager who tries toaffect the decisions made by wholesalers, retailers, and consumers. The literaturehas focused on a relatively small number of such instruments. For example, thearea of pricing is much broader than what our survey has shown. In real life wesee various types of price promotions toward consumers and retailers, and thereare interesting issues in, e.g., product line pricing, nonlinear pricing, advanceselling, and price differentiation. An introduction to pricing research in marketingcan be found in the handbook edited by Rao (2009).
The good news, therefore, are that there seems to be no shortage of marketingproblems awaiting to be investigated by dynamic game methods.
References
Amrouche N, Zaccour G (2007) Shelf-space allocation of national and private brands. Eur J OperRes 180:648–663
Amrouche N, Martín-Herrán G, Zaccour G (2008a) Feedback Stackelberg equilibrium strategieswhen the private label competes with the national brand. Ann Operat Res 164:79–95
Amrouche N, Martín-Herrán G, Zaccour G (2008b) Pricing and advertising of private and nationalbrands in a dynamic marketing channel. J Optim Theory Appl 137:465–483
Aust G, Buscher U (2014) Cooperative advertising models in supply chain management. Eur JOper Res 234:1–14
Basar T, Olsder GJ (1995) Dynamic noncooperative game theory, 2nd edn. Academic Press,London
Bass FM (1969) New product growth model for consumer durables. Manag Sci 15:215–227Bass FM, Krishnamoorthy A, Prasad A, Sethi SP (2005a) Generic and brand advertising strategies
in a dynamic duopoly. Mark Sci 24(4):556–568Bass FM, Krishnamoorthy A, Prasad A, Sethi SP (2005b) Advertising competition with market
expansion for finite horizon firms. J Ind Manag Optim 1(1):1–19Breton M, Chauny F, Zaccour G (1997) Leader-follower dynamic game of new product diffusion.
J Optim Theory Appl 92(1):77–98Breton M, Jarrar R, Zaccour G (2006) A note on feedback sequential equilibria in a Lanchester
Model with empirical application. Manag Sci 52(5):804–811Buratto A, Zaccour G (2009) Coordination of advertising strategies in a fashion licensing contract.
J Optim Theory Appl 142:31–53Cardaliaguet P, Quincampoix M, Saint-Pierre P (2001) Pursuit differential games with state
constraints. SIAM J Control Optim 39:1615–1632Cardaliaguet P, Quincampoix M, Saint-Pierre P (2002) Differential games with state constraints.
In: Proceedings of the tenth international symposium on dynamic games and applications,St. Petersburg
Marketing 37
Case J (1979) Economics and the competitive process. New York University Press, New YorkChatterjee R (2009) Strategic pricing of new products and services. In: Rao VR (ed) Handbook of
pricing research in marketing. E. Elgar, Cheltenham, pp 169–215Chatterjee R, Eliashberg J, Rao VR (2000) Dynamic models incorporating competition. In:
Mahajan V, Muller E, Wind Y (eds) New product diffusion models. Kluwer AcademicPublishers, Boston
Chiang WK (2012) Supply chain dynamics and channel efficiency in durable product pricing anddistribution. Manuf Serv Oper Manag 14(2):327–343
Chintagunta PK (1993) Investigating the sensitivity of equilibrium profits to advertising dynamicsand competitive effects. Manag Sci 39:1146–1162
Chintagunta PK, Jain D (1992) A dynamic model of channel member strategies for marketingexpenditures. Mark Sci 11:168–188
Chintagunta PK, Jain D (1995) Empirical analysis of a dynamic duopoly model of competition.J Econ Manag Strategy 4:109–131
Chintagunta PK, Rao VR (1996) Pricing strategies in a dynamic duopoly: a differential gamemodel. Manag Sci 42:1501–1514
Chintagunta PK, Vilcassim NJ (1992) An empirical investigation of advertising strategies in adynamic duopoly. Manag Sci 38:1230–1244
Chintagunta PK, Vilcassim NJ (1994) Marketing investment decisions in a dynamic duopoly: amodel and empirical analysis. Int J Res Mark 11:287–306
Chintagunta PK, Rao VR, Vilcassim NJ (1993) Equilibrium pricing and advertising strategies fornondurable experience products in a dynamic duopoly. Manag Decis Econ 14:221–234
Chutani A, Sethi SP (2012a) Optimal advertising and pricing in a dynamic durable goods supplychain. J Optim Theory Appl 154(2):615–643
Chutani A, Sethi SP (2012b) Cooperative advertising in a dynamic retail market oligopoly. DynGames Appl 2:347–375
De Giovanni P, Reddy PV, Zaccour G (2015) Incentive strategies for an optimal recovery programin a closed-loop supply chain. Eur J Oper Res 249(2):1–13
Deal K (1979) Optimizing advertising expenditure in a dynamic duopoly. Oper Res 27:682–692Desai VS (1996) Interactions between members of a marketing-production channel under seasonal
demand. Eur J Oper Res 90:115–141Dockner EJ (1985) Optimal pricing in a dynamic duopoly game model. Zeitschrift für Oper Res
29:1–16Dockner EJ, Gaunersdorfer A (1996) Strategic new product princing when demand obeys
saturation effects. Eur J Oper Res 90:589–598Dockner EJ, Jørgensen S (1984) Cooperative and non-cooperative differential game solutions to
an investment and pricing problem. J Oper Res Soc 35:731–739Dockner EJ, Jørgensen S (1988a) Optimal pricing strategies for new products in dynamic
oligopolies. Mark Sci 7:315–334Dockner EJ, Jørgensen S (1992) New product advertising in dynamic oligopolies. Zeitschrift für
Oper Res 36:459–473Dockner EJ, Gaunersdorfer A, Jørgensen S (1996) Government price subsidies to promote fast
diffusion of a new consumer durable. In: Jørgensen S, Zaccour G (eds) Dynamic competitiveanalysis in marketing. Springer, Berlin, pp 101–110
Dockner EJ, Jørgensen S, Van Long N, Sorger G (2000) Differential games in economics andmanagement science. Cambridge University Press, Cambridge
Dolan R, Jeuland AP, Muller E (1986) Models of new product diffusion: extension to competitionagainst existing and potential firms over time. In: Mahajan V, Wind Y (eds) Innovation diffusionmodels of new product acceptance. Ballinger, Cambridge, pp 117–150
El Ouardighi F, Jørgensen S, Pasin F (2013) A dynamic game with monopolist manufacturer andprice-competing duopolist retailers. OR Spectr 35:1059–1084
Eliashberg J, Chatterjee R (1985) Analytical models of competition with implications formarketing: issues, findings, and outlook. J Mark Res 22:237–261
38 S. Jørgensen
Eliashberg J, Jeuland AP (1986) The impact of competitive entry in a developing market upondynamic pricing strategies. Mark Sci 5:20–36
Eliashberg J, Steinberg R (1987) Marketing-production decisions in an industrial channel ofdistribution. Manag Sci 33:981–1000
Eliashberg J, Steinberg R (1991) Competitive strategies for two firms with asymmetric productioncost structures. Manag Sci 37:1452–1473
Eliashberg J, Steinberg R (1993) Marketing-production joint decision-making. In: Eliashberg J,Lilien GL (eds) Handbooks of operations research and management science: marketing. North-Holland, Amsterdam, pp 827–880
Engwerda J (2005) Algorithms for computing nash equilibria in deterministic LQ games. ComputManag Sci 4(2):113–140
Erickson GM (1985) A model of advertising competition. J Mark Res 22:297–304Erickson GM (1991) Dynamic models of advertising competition. Kluwer, BostonErickson GM (1992) Empirical analysis of closed-loop duopoly advertising strategies. Manag Sci
38:1732–1749Erickson GM (1993) Offensive and defensive marketing: closed-loop duopoly strategies. Mark
Lett 4:285–295Erickson GM (1995a) Differential game models of advertising competition. Eur J Oper Res
83:431–438Erickson GM (1995b) Advertising strategies in a dynamic oligopoly. J Mark Res 32:233–237Erickson GM (1996) An empirical comparison of oligopolistic advertising strategies. In: Jørgensen
S, Zaccour G (eds) Dynamic competitive analysis in marketing. Springer, Berlin, pp 26–36Falcone M (2006) Numerical methods for differential games based on partial differential equations.
Int Game Theory Rev 8:231–272Erickson GM (1997) Dynamic conjectural variations in a lanchester duopoly. Manag Sci 43:1603–
1608Feichtinger G, Dockner EJ (1985) Optimal pricing in a duopoly: a noncooperative differential
games solution. J Optim Theory Appl 45:199–218Feichtinger G, Hartl RF, Sethi SP (1994) Dynamic optimal control models in advertising: recent
developments. Manag Sci 40:195–226Fershtman C (1984) Goodwill and market shares in oligopoly. Economica 51:271–281Fershtman C, Mahajan V, Muller E (1990) Market share pioneering advantage: a theoretical
approach. Manag Sci 36:900–918Friedman JW (1991) Game theory with applications to economics, 2nd edn. Oxford University
Press, OxfordFruchter GE (2001) Advertising in a competitive product line. Int Game Theory Rev 3(4):301–314Fruchter GE, Kalish S (1997) Closed-loop advertising strategies in a duopoly. Manag Sci 43:54–63Fruchter GE, Kalish S (1998) Dynamic promotional budgeting and media allocation. Eur J Oper
Res 111:15–27Fruchter GE, Erickson GM, Kalish S (2001) Feedback competitive advertising strategies with a
general objective function. J Optim Theory Appl 109:601–613Gaimon C (1989) Dynamic game results of the acquisition of new technology. Oper Res 37:410–
425Gaimon C (1998) The price-production problem: an operations and marketing interface. In:
Aronson JE, Zionts S (eds) Operations research: methods, models, and applications. QuorumBooks, Westpoint, pp 247–266
Gallego G, Hu M (2014) Dynamic pricing of perishable assets under competition. Manag Sci60(5):1241–1259
Haurie A, Krawczyk JB, Zaccour G (2012) Games and dynamic games. Now/World Scientific,Singapore
He X, Prasad A, Sethi SP, Gutierrez GJ (2007) A survey of stackelberg differential models insupply and marketing channels. J Syst Sci Eng 16(4):385–413
Helmes K, Schlosser R (2015) Oligopoly pricing and advertising in isoelastic adoption models.Dyn Games Appl 5:334–360
Marketing 39
Horsky D, Mate K (1988) Dynamic advertising strategies of competing durable goods producers.Mark Sci 7:356–367
Horsky D, Simon LS (1983) Advertising and the diffusion of new products. Mark Sci 2:1–18Huang J, Leng M, Liang L (2012) Recent developments in dynamic advertising research. Eur J
Oper Res 220:591–609Janssens G, Zaccour G (2014) Strategic price subsidies for new technologies. Automatica 50:1998–
2006Jarrar R, Martín-Herrán G, Zaccour G (2004) Markov perfect equilibrium advertising strategies in
a lanchester duopoly model: a technical note. Manag Sci 50(7):995–1000Jørgensen S (1982a) A survey of some differential games in advertising. J Econ Dyn Control
4:341–369Jørgensen S (1982b) A differential game solution to an investment and pricing problem. J Operat
Res Soc 35:731–739Jørgensen S (1986a) Optimal dynamic pricing in an oligopolistic market: a survey. In: Basar T (ed)
Dynamic games and applications in economics. Springer, Berlin, pp 179–237Jørgensen S (1986b) Optimal production, purchasing, and pricing: a differential game approach.
Eur J Oper Res 24:64–76Jørgensen S (2011) Intertemporal contracting in a supply chain. Dyn Games Appl 1:280–300Jørgensen S, Gromova E (2016) Sustaining cooperation in a differential game of advertising
goodwill. Eur J Oper Res 254:294–303Jørgensen S, Kort PM (2002) Optimal pricing and inventory policies: centralized and decentralized
decision making. Eur J Oper Res 138:578–600Jørgensen S, Sigué SP (2015) Defensive, offensive, and generic advertising in a lanchester model
with market growth. Dyn Games Appl 5(4):523–539Jørgensen S, Zaccour G (1999a) Equilibrium pricing and advertising strategies in a marketing
channel. J Optim Theory Appl 102:111–125Jørgensen S, Zaccour G (1999b) Price subsidies and guaranteed buys of a new technology. Eur J
Oper Res 114:338–345Jørgensen S, Zaccour G (2003a) A differential game of retailer promotions. Automatica 39:1145–
1155Jørgensen S, Zaccour G (2003b) Channel coordination over time: incentive equilibria and
credibility. J Econ Dyn Control 27:801–822Jørgensen S, Zaccour G (2004) Differential games in marketing. Kluwer Academic Publishers,
BostonJørgensen S, Zaccour G (2007) Developments in differential game theory and numerical methods:
economic and management applications. Comput Manag Sci 4(2):159–181Jørgensen S, Zaccour G (2014) A survey of game-theoretic models of cooperative advertising. Eur
J Oper Res 237:1–14Jørgensen S, Sigue SP, Zaccour G (2000) Dynamic cooperative advertising in a channel. J Retail
76:71–92Jørgensen S, Sigue SP, Zaccour G (2001) Stackelberg leadership in a marketing channel. Int Game
Theory Rev 3(1):13–26Jørgensen S, Taboubi S, Zaccour G (2003) Retail promotions with negative brand image effects: is
cooperation possible. Eur J Oper Res 150:395–405Jørgensen S, Taboubi S, Zaccour G (2006) Incentives for retailer promotion in a marketing channel.
In: Haurie A, Muto S, Petrosjan LA, Raghavan TES (eds) Advances in dynamic games. Annalsof ISDG, vol 8, Birkhäuser, Boston, Ma., USA
Jørgensen S, Martín-Herrán G, Zaccour G (2010) The Leitmann-Schmitendorf advertising differ-ential game. Appl Math Comput 217:1110–1118
Kalish S (1983) Monopolist pricing with dynamic demand and production costs. Mark Sci 2:135–160
Kalish S (1988) Pricing new products from birth to decline: an expository review. In: DevinneyTM (ed) Issues in pricing. Lexington Books, Lexington, pp 119–144
40 S. Jørgensen
Kalish S, Lilien GL (1983) Optimal price subsidy for accelerating the diffusion of innovations.Mark Sci 2:407–420
Karray S, Martín-Herrán G (2009) A dynamic model for advertising and pricing competitionbetween national and store brands. Eur J Oper Res 193:451–467
Kimball GE (1957) Some industrial applications of military operations research methods. OperRes 5:201–204
Krishnamoorthy A, Prasad A, Sethi SP (2010) Optimal pricing and advertising in a durable-goodduopoly. Eur J Oper Res 200(2):486–497
Kushner HJ, Dupuis PG (2001) Numerical methods for stochastic optimal control problems incontinuous time. Springer, Berlin
Lanchester FW (1916) Aircraft in warfare: the dawn of the fourth arm. Constable, LondonLeitmann G, Schmitendorf WE (1978) Profit maximization through advertising: nonzero-sum
differential game approach. IEEE Trans Autom Control 23:645–650Lieber Z, Barnea A (1977) Dynamic optimal pricing to deter entry under constrained supply. Oper
Res 25:696–705Mahajan V, Muller E, Bass FM (1990) New-product diffusion models: a review and directions for
research. J Mark 54:1–26Mahajan V, Muller E, Bass FM (1993) New-product diffusion models. In: Eliashberg J, Lilien GL
(eds) Handbooks in operations research and management science: marketing. North-Holland,Amsterdam
Mahajan V, Muller E, Wind Y (eds) (2000) New-product diffusion models. Springer, New YorkMartín-Herrán G, Taboubi S (2005) Shelf-space allocation and advertising decisions in the
marketing channel: a differential game approach. Int Game Theory Rev 7(3):313–330Martín-Herrán G, Taboubi S (2015) Price coordination in distribution channels: a dynamic
perspective. Eur J Oper Res 240:401–414Martín-Herrán G, Taboubi S, Zaccour G (2005) A time-consistent open-loop stackelberg equilib-
rium of shelf-space allocation. Automatica 41:971–982Martín-Herrán G, Sigué SP, Zaccour G (2011) Strategic interactions in traditional franchise
systems: are franchisors always better off? Eur J Oper Res 213:526–537Martín-Herrán G, McQuitty S, Sigué SP (2012) Offensive versus defensive marketing: what is the
optimal spending allocation? Int J Res Mark 29:210–219Mesak HI, Calloway JA (1995) A pulsing model of advertising competition: a game theoretic
approach, part B – empirical application and findings. Eur J Oper Res 86:422–433Mesak HI, Darrat AF (1993) A competitive advertising model: some theoretical and empirical
results. J Oper Res Soc 44:491–502Minowa Y, Choi SC (1996) Optimal pricing strategy and contingent products under a duopoly
environment. In: Jørgensen S, Zaccour G (eds) Dynamic competitive analysis in marketing.Springer, Berlin, pp 111–124
Miranda MJ, Fackler PL (2002) Applied computational economics and finance. The MIT Press,Cambridge
Moorthy KS (1993) Competitive marketing strategies: game-theoretic models. In: Eliashberg J,Lilien GL (eds) Handbooks in operations research and management science: marketing. North-Holland, Amsterdam, pp 143–190
Mukhopadhyay SK, Kouvelis P (1997) A differential game theoretic model for duopolisticcompetition on design quality. Oper Res 45:886–893
Nair A, Narasimhan R (2006) Dynamics of competing with quality- and advertising-basedGoodwill. Eur J Oper Res 175:462–476
Nerlove M, Arrow KJ (1962) Optimal advertising policy under dynamic conditions. Economica39:129–142
Prasad A, Sethi SP (2004) Competitive advertising under uncertainty: a stochastic differential gameapproach. J Optim Theory Appl 123(1):163–185
Rao RC (1988) Strategic pricing of durables under competition. In: Devinney TM (ed) Issues inpricing. Lexington Books, Lexington, pp 197–215
Marketing 41
Rao RC (1990) The impact of competition on strategic marketing decisions. In: Day G, Weitz B,Wensley R (eds) The interface of marketing and strategy. JAI Press, Greenwich, pp 101–152
Rao VR (2009) Handbook of pricing research in marketing. Edward Elgar, CheltenhamReynolds SS (1991) Dynamic oligopoly with capacity adjustment costs. Journal of Economic
Dynamics & Control 15:491–514Rishel R (1985) A partially observed advertising model. In: Feichtinger G (ed) Optimal control
theory and economic analysis. North-Holland, Amsterdam, pp 253–262Sadigh AN, Mozafari M, Karimi B (2012) Manufacturer-retailer supply chain coordination: a bi-
level programming approach. Adv Eng Softw 45(1):144–152Sethi SP (1983) Deterministic and stochastic optimization of a dynamic advertising model. Optim
Control Methods Appl 4:179–184Sigué SP (2002) Horizontal strategic interactions in franchising. In: Zaccour G (ed) Decision and
control in management science: essays in honor of Alain Haurie. Kluwer, BostonSigué SP, Chintagunta P (2009) Advertising strategies in a franchise system. Eur J Oper Res
198:655–665Simchi-Levi D, Wu SD, Shen Z-J (2004) Handbook of quantitative supply chain analysis: modeling
in the e-business era. Springer, BerlinSorger G (1989) Competitive dynamic advertising: a modification of the case game. J Econ Dyn
Control 13:55–80Spengler JJ (1950) Vertical integration and antitrust policy. J Polit Econ 58:347–352Sudhir K, Datta S (2009) Pricing in marketing channels. In: Rao VR (ed) Handbook of pricing
research in marketing. Elgar E, Cheltenham, pp 169–215Swartz TA, Iacobucci D (2000) Handbook of services marketing & management. Sage Publica-
tions, LondonTapiero CS (1975) Random walk models of advertising, their diffusion approximation and
hypothesis testing. Ann Econ Soc Meas 4:293–309Tapiero CS (1988) Applied stochastic models and control in management. North-Holland,
AmsterdamTeng JT, Thompson GL (1983) Oligopoly models for optimal advertising when production costs
obey a learning curve. Manag Sci 29:1087–1101Teng JT, Thompson GL (1985) Optimal strategies for general price-advertising models. In:
Feichtinger G (ed) Optimal control theory and economic analysis, vol 2. Elsevier, Amsterdam,pp 183–195
Teng JT, Thompson GL (1998) Optimal strategies for general price-quality decision models of newproducts with learning production costs. In: Aronson JE, Zionts S (eds) Operations research:methods, models and applications. Greenwood Publishing Co., Westport
Thépot J (1983) Marketing and investment policies of duopolists in a growing industry. J EconDyn Control 5:387–404
Tirole J (1988) The theory of industrial organization. The MIT Press, CambridgeVega-Redondo F (2003) Economics and the theory of games. Cambridge University Press,
CambridgeVidale ML, Wolfe HB (1957) An operations research study of sales response to advertising. Oper
Res 5:370–381Vives X (1999) Oligopoly pricing: old ideas and new tools. MIT Press, CambridgeWang Q, Wu Z (2001) A duopolistic model of dynamic competitive advertising. Eur J Oper Res
128:213–226Xie J, Sirbu M (1995) Price competition and compatibility in the presence of positive demand
externalities. Manag Sci 41(5):909–926Zaccour G (1996) A differential game model for optimal price subsidy of new technologies. Game
Theory Appl 2:103–114Zaccour G (2008) On the coordination of dynamic marketing channels and two-part tariffs.
Automatica 44:1233–1239Zhang J, Gou Q, Liang L, Huang Z (2012) Supply chain coordination through cooperative
advertising with reference price effect. Omega 41(2):345–353