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Abstract:
Firms frequently utilize multiple communications instruments as part of their marketing campaign. Interactions between these instruments suggest that firms should apply Integrated Marketing Communications (IMC) to benefit from the synergies. We review different IMC models and then present a stochastic IMC model for which explicit closed-loop solutions of the optimal advertising and market share are obtained. This enables us to understand the role of firm and market parameters, such as synergy, on the optimal advertising budget and allocation. For the proposed and existing IMC models we show that the budget and allocation decisions can be made independently, greatly simplifying the implementation of IMC. We also show that there is an optimal long-run market share that the firm should try to maintain through appropriate use of IMC. Finally, the model and results are generalized to multiple (>2) instruments and multiple competitors.

Advertising, Dynamics, Budgeting, Integrated Marketing Communications, Decisions under Uncertainty, Dynamic Optimization

Abstract:
Using a two-product and two-period model, we investigate optimal capacity acquisition and capacity cost allocation for product costing and pricing decisions for existing and new products under long-term capacity commitment. The existing product is produced in both periods, while the new product is produced only in the second period. The insights we obtain differ from those in earlier literature and complement them. We find conditions under which the capacity charge for existing products when idle capacity exists can be more or less than the capacity charge when the firm introduces new products to better utilize the capacity. A necessary requirement for the capacity charge for the existing product in the first period when there is idle capacity to be greater than the capacity charge in the second period when the capacity is better utilized is that the demand for the two products be negatively correlated. The result provides a rationale for the variety of cost allocation schemes used in practice.

product costing, fixed costs, pricing, capacity, new product introduction

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

Stackelberg differential games, supply chain management, marketing channels, open-loop equilibria, feedback policies, channel coordination, optimal control

Abstract:
Firms have the choice of developing software as either open source or closed source. The open-source approach to software development has been advocated as a new and better method for developing high quality software than the traditional closed-source approach. In open source, volunteer programmers freely contribute code to develop and improve the software. This paper describes the key nonpecuniary motivations for these programmers. They are less motivated to contribute if they observe commercial marketing of the open-source software they helped create, leading to a reduction in improvements to the software. A primary concern for software firms seeking to develop and market open-source software is, thus, how the motivation of contributors should be managed. We examine optimal pricing strategies for open-source and closed-source software keeping in mind the distinct motivations of programmers in the two cases. We compare profits and software qualities from the two approaches and provide implications for firms in the software industry.

Open source, Pricing, altruism, optimal control, software management, the maximum principle, chattering control

Abstract:
We analyze optimal advertising spending in a duopolistic market where each firm's market share depends on its own and its competitor''s advertising decisions, and is also subject to stochastic disturbances. We develop a differential game model of advertising in which the dynamic behavior is based on the Sethi stochastic advertising model and the Lanchester model of combat. Particularly important to note is the morphing of the sales decay term in the Sethi model into decay caused by competitive advertising and noncompetitive churn that acts to equalize market shares in the absence of advertising. We derive closed-loop Nash equilibria for symmetric as well as asymmetric competitors. For all cases, explicit solutions and comparative statics are presented.

Advertising, advertising budgeting, competitive strategy, stochastic differential game, stochastic calculus, duopoly, stochastic differential equations, Ito equations, dynamic programming, the Sethi model, Nash equilibrium

Abstract:
Companies spend hundreds of millions of dollars annually on advertising to build and maintain awareness for their brands in competitive markets. However, awareness formation models in the marketing literature ignore the role of competition. Consequently, we lack both the empirical knowledge and normative understanding of building brand awareness in dynamic oligopoly markets. To address this gap, we propose an N-brand awareness formation model, design an extended Kalman filter to estimate the proposed model using market data for five car brands over time, and derive the optimal closed-loop Nash equilibrium strategies for every brand. The empirical results furnish strong support for the proposed model in terms of both goodness-of-fit in the estimation sample and cross-validation in the out-of-sample data. In addition, the estimation method offers managers a systematic way to estimate ad effectiveness and brands as well as competitors' brands. Finally, the normative analysis reveals an inverse allocation principle that suggests - contrary to the proportional-to-sales or competitive parity heuristics - that large (small) brands should invest in advertising proportionally less (more) than small (large) brands.

marketing, competitive strategy, advertising, media, Kalman filter, dynamic games, differential games, Nash equilibrium, Optimal control, Sethi model, Feedback Nash, competative advertising model, car companies data from Italy

Abstract:
A model of new-product adoption is proposed that incorporates price and advertising effects. An optimal control problem that uses the model as its dynamics is explicitly solved to obtain the optimal price and advertising effort over time. The model has a great potential to be used in obtaining solutions and insights in a variety of differential game settings.

new-product adoption model, durable goods model, pricing, advertising, optimal control

Abstract:
We examine an oligopoly model of advertising competition where each firm's market share depends on its own and its competitors' advertising decisions. A differential game model is developed and used to derive the closed-loop Nash equilibrium under symmetric as well as asymmetric competition. We obtain explicit solutions under certain plausible conditions, and discuss the effects of an increase in the number of competing firms on advertising expenditure, market share and profitability.

Advertising, Oligopoly, Differential games, Optimization

Abstract:
Cooperative (co-op) advertising is an important instrument for aligning manufacturer and retailer decisions in supply chains. In this, the manufacturer announces a co-op advertising policy, i.e., a participation rate that specifies the percentage of the retailer's advertising expenditure that it will provide. In addition, it also announces the wholesale price. In response, the retailer chooses its optimal advertising and pricing policies. We model this supply chain problem as a stochastic Stackelberg differential game whose dynamics follows Sethi's stochastic sales-advertising model. We obtain the condition when offering co-op advertising is optimal. We provide in feedback form the optimal advertising and pricing policies for the manufacturer and the retailer. We contrast the results with the advertising and price decisions of the vertically integrated channel, and suggest a method for coordinating the channel.

Co-op Advertising, Cooperative advertising, Sales-Advertising Dynamics, Differential Games, Sethi Model, Distribution Channel, Stackelberg equilibrium, Feedback Stackelberg strategy, Sales-advertising model, advertising participation rate, optimal advertising, optimal pricing

Abstract:
This paper analyzes dynamic advertising and pricing policies in a durable-good duopoly. The proposed infinite-horizon model, while general enough to capture dynamic price and advertising interactions in a competitive setting, also permits closed-form solutions. We use differential game theory to analyze two different demand specifications - linear demand and isoelastic demand - for symmetric and asymmetric competitors. We find that the optimal price is constant and does not vary with cumulative sales, while the optimal advertising is decreasing with cumulative sales. Comparative statics for the results are presented.

Optimal Control, Dynamic programming, Game theory, Differential games, Pricing, Advertising, Nash equilibrium, the Sethi model, Durable goods, Innovative products

Abstract:
The open source paradigm is often defined as a collaborative effort, implying that firms and consumers come together in a non-competitive climate. We show here that open source development can arise from a competitive climate. Under competition, we find that open source is the surplus maximizing outcome and can be in equilibrium if cost asymmetries are small. However, when cost asymmetries are large, contradictions between equilibrium and welfare maximization result. Considerations typical to public good problems arise, with issues of asymmetric contributions and freeriding. These issues should guide the firm's as well as the society's decisions to implement open source in particular environments. We analyze this problem in the framework of a dynamic duopolistic competition, with firms controlling their investments in software.

Differential Games, Public Goods, Open source, Software, Nash equilibrium, Two point boundary Value Problems, Optimal control, welfare maximization

Abstract:
We examine an oligopoly model of advertising competition where each firm's market share depends on its own and its competitors' advertising decisions. A differential game model is developed and used to derive the closed-loop Nash equilibrium under symmetric as well as asymmetric competition. We obtain explicit solutions under certain plausible conditions, and discuss the effects of an increase in the number of competing firms on advertising expenditure, market share and profitability.

Sethi model, advertising, dynamic games, game theory, differential games, optimal control, close-loop Nash equilibrium, oligopoly, Feedback Nash equilibrium

Abstract:
Firms that want to increase the sales of their brands through advertising have the choice of capturing market share from their competitors through brand advertising, or increasing primary demand for the category through generic advertising. In this paper, differential game theory is used to analyze the effects of the two types of advertising decisions made by firms offering a product in a dynamic duopoly. Each firm's sales depend not only on its own and its competitor's brand advertising strategies, but also on the generic advertising expenditures of the two firms. Closed-loop Nash equilibrium solutions are obtained for symmetric and asymmetric competitors in a finite-horizon setting. The analysis for the symmetric case results in explicit solutions, and numerical techniques are employed to solve the problem for asymmetric firms.

Advertising, Generic Advertising, Brand Advertising, duopoly, dynamic model, optimal control, differential games, Nash equilibrium. Feedback Nash, marketing mix

Abstract:
The presentation of Table 2 in the original version of this article ("A Survey of Stackelberg Differential Game Models in Supply and Marketing Channels", Journal of Systems Science and Systems Engineering, Vol. 16, No. 4, pp., 385-413, 2007) contained a few typos. The corrected Table 2 is given below.

Stackelberg differential games, supply chain management, marketing channels, open-loop equilibria, feedback policies, channel coordination, optimal control

Abstract:
Retailers often face a newsvendor problem, i.e., they must order their inventory prior to a short selling period with uncertain demand. The uncertainty can be reduced by advance selling because not only are advance orders certain, but the remaining demand can be better forecasted. Consumers, however, may prefer not to purchase in advance unless given a discount because they are uncertain about their valuation for the product in advance. It is then unclear whether advance selling to pass some uncertainty risk to consumers is optimal for the retailer.

This paper examines the advance selling price and inventory decisions in a two-period setting, where the first period is the advance selling period and the second is the selling (and consumption) period. We find that the advance selling strategy is not always optimal, but is contingent on parameters of the market (e.g., market potential, uncertainty), the consumers (e.g., valuation, risk aversion and heterogeneity). For example, we find that retailers should sell in advance if the consumers' expected valuation exceeds consumers' expected surplus when not buying early by a certain threshold at least, and that this threshold increases with risk aversion but decreases with stockout risks.

Advance selling, Newsvendor, Consumer valuation, Uncertainty, Pricing

Abstract:
This paper examines open source software development in a competitive environment. The quality of open source software improves over time based upon contributions by firms and users. A firm's decision to contribute is interesting because it also augments competitors' software quality in future periods subject to compatibility considerations with their existing software. A differential game model is developed to understand why firms are increasingly involved in open source software development by determining the optimal contributions and software quality over time. We obtain a closed-loop Nash equilibrium solution. Examples are given to derive insights from this model.

Open sourcd software, dynamic games, game theory, differential games, optimal control, close-loop Nash equilibrium, software quality

Abstract:
To increase the sales of their products through advertising, firms must integrate their brand-advertising strategy for capturing market share from competitors and their generic-advertising strategy for increasing primary demand for the category. This paper examines whether, when, and how much brand advertising versus generic advertising should be done. Using differential game theory, optimal advertising decisions are obtained for a dynamic duopoly with symmetric or asymmetric competitors. We show how advertising depends on the cost and effectiveness of each type of advertising for each firm, the allocation of market expansion benefits, and the profit margins determined endogenously from price competition. We find that generic advertising is proportionally more important in the short term and that there are free-riding effects leading to suboptimal industry expenditure on generic advertising that worsen as firms become more symmetric. Due to free-riding by the weaker firm, its instantaneous profit and market share can actually be higher. The effectiveness of generic advertising and the allocation of its benefits, however, have little effect on the long-run market shares, which are determined by brand-advertising effectiveness. Extensions of the model show that market potential saturation leads to a decline in generic advertising over time.

Advertising, Generic Advertising, Brand Advertising, duopoly, dynamic model, optimal control, differential games, Nash equilibrium. Feedback Nash, marketing mix

Abstract:
To have a productive sales force, firms must provide their salespeople with sales training. But from a profit-maximizing perspective, there are also reasons to limit training: training is expensive, it has diminishing returns, and trained salespeople need to be compensated at a higher level since their value in the outside labor market has increased. Due to these reasons, the following inter-related questions are not straightforward to answer: (1) How much training should be provided and how should training be scheduled over time? (2) How should compensation vary with training? (3) Should salespeople be asked to pay for some or all of their training? An analytical model is developed and analyzed using optimal control theory to provide answers to these questions. Thereafter, an empirical investigation is undertaken that broadly corroborates the analytical findings.

Sales force, Training, Compensation, Optimal control