An empirical study on the Lanchester model of combat for competitive advertising decisions

This paper presents an empirical study on the Lanchester model of combat for competitive advertising decisions. Three issues are evaluated: (i) the specification of the market share response model; (ii) the effect of inflation on the estimation of the response model; and (iii) the performance of competitive strategies. It is shown that (a) the square root function that is used in previous studies is often inappropriate to characterize the market share response model; (b) market share variations are more responsive to current advertising expenditures; (c) closed-loop Nash equilibrium strategies are better competitive advertising strategies for firms to maximize profits than open-loop Nash equilibrium strategies; and (d), finally, general perfect equilibria Nash equilibrium strategies developed by Case are usually not good competitive advertising strategies for firms to maximize profits.     

Competitive Advertising Under Uncertainty: A Stochastic Differential Game Approach

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.     

Feedback Competitive Advertising Strategies with a General Objective Function

We introduce a general objective function, which incorporates competitive situations, such as conservative, punitive, and predatory advertising. Linking together the particular situations into a two-parameter family of max–min problems, and using the Lanchester model to describe the dynamics of the market, a bilinear-quadratic differential game is obtained. For this game, we find saddle-point feedback time-invariant advertising strategies and show when these strategies are Nash equilibrium strategies. In an empirical application involving duopolistic competition in the cola market, we find evidence of a punitive motivation for the advertising strategies.     

Advertising in a Differential Oligopoly Game

We illustrate a differential oligopoly game where firms compete à la Cournot in homogeneous goods in the market phase and invest in advertising activities aimed at increasing the consumers reservation price. Such investments produce external effects, characterizing the advertising activity as a public good. We derive the open-loop and closed-loop Nash equilibria, and show that the properties of the equilibria depend on the curvature of the market demand function. The comparative assessment of these equilibria shows that the firms advertising efforts are larger in the open-loop equilibrium than in the closed-loop equilibrium. We also show that a cartel involving all the firms, setting both output levels and advertising efforts, may produce a steady state where the social welfare level is higher than the social welfare levels associated with both open-loop and closed-loop noncooperative settings.     

On the generalizability of advertising pulsation monopoly results to an oligopoly

This paper introduces, for the first time, an analytical framework for modeling and analyzing sales response to advertising pulsation for competitive markets of mature products in continuous time for discrete, piecewise policies that allow unequal cycle lengths. Employing a Nash equilibrium solution concept, this study supported with numerical analysis demonstrates that there is an oligopolistic, as opposed to a monopolistic, justification for advertising at a constant rate in the presence of concave response functions or advertising according to a pulsing policy in the presence of S-shaped response functions. The article employs a modeling framework whereby each of a set of competitors' responses to advertising spending is governed mainly by Lanchester type response is used to analyze the game resulting from cyclical advertising policies that are strictly adhered to by all players. A non-discounted averaged performance is used to assess the purported optimality of different variants of piecewise-constant policies.     

Linear–quadratic stochastic two-person nonzero-sum differential games: Open-loop and closed-loop Nash equilibria

In this paper, we consider a linear–quadratic stochastic two-person nonzero-sum differential game. Open-loop and closed-loop Nash equilibria are introduced. The existence of the former is characterized by the solvability of a system of forward–backward stochastic differential equations, and that of the latter is characterized by the solvability of a system of coupled symmetric Riccati differential equations. Sometimes, open-loop Nash equilibria admit a closed-loop representation, via the solution to a system of non-symmetric Riccati equations, which could be different from the outcome of the closed-loop Nash equilibria in general. However, it is found that for the case of zero-sum differential games, the Riccati equation system for the closed-loop representation of an open-loop saddle point coincides with that for the closed-loop saddle point, which leads to the conclusion that the closed-loop representation of an open-loop saddle point is the outcome of the corresponding closed-loop saddle point as long as both exist. In particular, for linear–quadratic optimal control problem, the closed-loop representation of an open-loop optimal control coincides with the outcome of the corresponding closed-loop optimal strategy, provided both exist.     

A Competitive Advertising Model: Some Theoretical and Empirical Results

Using a modified Lanchester model, this article demonstrates that the policy of constant advertising spending is superior to a cyclic counterpart, provided that the advertising response functions of competing firms are concave. The competitive model is estimated using filter and non-filter cigarette data. The empirical results suggest the presence of over-advertising and that the relative effectiveness and elasticity of filter cigarette advertising are both higher than those of non-filter cigarette advertising.     

A class of differential games for which the closed-loop and open-loop Nash equilibria coincide

It is well known that, in general, Nash equilibria in open-loop strategies do not coincide with those in closed-loop strategies. This note identifies a class of differential games in which the Nash equilibrium in closed-loop strategies is degenerate, in the sense that it depends on time only. Consequently, the closed-loop equilibrium is also an equilibrium in open-loop strategies.The helpful comments of Professors Y. C. Ho, G. Leitmann, H. Y. Wan, Jr., and an anonymous referee are gratefully acknowledged.     

Changing market conditions and advertising strategies: theory and evidence

Using a modified continuous Lanchester model of advertising competition and following a game theoretic approach, this paper investigates analytically the sensitivity of optimal advertising to changes in model parameters in the cases of symmetric and semi-symmetric competition. The theoretical results show that non-dominant firms in both market structures behave similarly whereas the dominant firm in the latter structure behaves considerably differently.We estimate a modified Lanchester model using ready-to-eat cereal data. Based on the obtained estimated parameters, we conduct an empirical analysis to assess the robustness of the theoretical estimated results. In the reported application, we demonstrate that when the model parameters change, non-dominant competitors as well as the dominant rival behave in a manner consistent with the theoretical results related to the semi-symmetric competitive structure.     

Dynamics of competing with quality- and advertising-based goodwill

Goodwill formation is a complex process and many factors influence the formation of goodwill of a firm. The implications of advertising enabled goodwill formation are reported in several articles in the research literature. In this paper, we extend this stream of research by including quality in the goodwill formation process. We adopt a dynamic model of competition utilizing a differential game approach and derive expressions for open-loop Markovian Nash equilibrium investments in advertising and quality. The insights gained from the analysis of our model and from the equilibrium solutions are presented in the form of research propositions.     

N-person differential games governed by semilinear stochastic evolution systems

In this paper the problem ofN-person infinite-dimensional stochastic differential games governed by semilinear stochastic evolution control systems is discussed. First the minimax principle which is the necessary condition for the existence of open-loop Nash equilibrium is proved. Then the necessary and sufficient conditions of open-loop and closed-loop Nash equilibrium for linear quadratic infinite-dimensional stochastic differential games are derived.     

New product advertising in dynamic oligopolies

The success of the introduction of a new product in a market is very sensitive to the marketing decision variables adopted by the firm. In the present paper we are concerned with the question of new product advertising in a heterogeneous oligopoly market consisting of N firms. A dynamic game is formulated to model strategic as well as sales interactions in such a market. Optimal advertising strategies are identified as open-loop Nash solutions.The comments of two anonymous referees are appreciated. The first author wishes to acknowledge support from NSERC (Grant No. OGP0037342).     

On a class of nonzero-sum,linear-quadratic differential games

A class of two-player, nonzero-sum, linear-quadratic differential games is investigated for Nash equilibrium solutions when both players use closed-loop control and when one or both of the players are required to use open-loop control. For three formulations of the game, necessary and sufficient conditions are obtained for a particular strategy set to be a Nash equilibrium strategy set. For a fourth formulation of the game, where both players use open-loop control, necessary and sufficient conditions for the existence of a Nash equilibrium strategy set are developed. Several examples are presented in order to illustrate the differences between this class of differential games and its zero-sum analog.This research was supported by the National Science Foundation under Grant No. GK-3341.     

Differential game approach to pricing and advertising decisions

This study proposes a model to make concurrent decisions on dynamic pricing and advertising to maximise firms' profitability over an infinite time horizon in a duopoly market. To this end, the Nerlove-Arrow pricing and advertising model is designed in the presence of shifting costs in a dynamic duopolistic competition as a differential game. The Nash equilibrium solution is defined based upon a set of Hamilton–Jacobi–Bellman. Four scenarios are applied for economic interpretations and the efficacy of the model.     

Systemic Risk and Stochastic Games with Delay

We propose a model of inter-bank lending and borrowing which takes into account clearing debt obligations. The evolution of log-monetary reserves of banks is described by coupled diffusions driven by controls with delay in their drifts. Banks are minimizing their finite-horizon objective functions which take into account a quadratic cost for lending or borrowing and a linear incentive to borrow if the reserve is low or lend if the reserve is high relative to the average capitalization of the system. As such, our problem is a finite-player linear–quadratic stochastic differential game with delay. An open-loop Nash equilibrium is obtained using a system of fully coupled forward and advanced-backward stochastic differential equations. We then describe how the delay affects liquidity and systemic risk characterized by a large number of defaults. We also derive a closed-loop Nash equilibrium using a Hamilton–Jacobi–Bellman partial differential equation approach.     

Hybrid subgames and copycat games in a pulsing model of advertising competition

Two recent papers,^6,7 introduced the game of pulsing competition (PC) in advertising together with its related subgames of alternating pulsing competition (APC) and matching pulsing competition (MPC) for a duopoly. Following a game theoretic approach in conjunction with a continuous Lanchester model, the above authors basically concluded that when at least one of the response functions is convex, generalising monopolistic advertising pulsation results to a competitive setting might not be adequate. This paper expands the scope of the PC game by incorporating in its structure for the first time in the literature, two versions of a hybrid pulsing competition (HPC) subgame. The article compares the payoffs of the four alternative subgames and provides an analytical solution of a special case of the PC game. In addition, the article also introduces for the first time a variant of the PC game designated by ‘the copycat advertising game’ and shows analytically that for such a game the policy of constant advertising spending over time is optimal for both firms irrespective of the shape of their advertising response functions. The paper illustrates at its end how to solve numerically the expanded PC game in its general form using linear programming and how to derive a solution for a copycat advertising game.     

A Differential Games Solution to a Logarithmic Advertising Model

A logarithmic excess-advertising model of a duopoly is presented, and Nash optimal open-loop advertising strategies are determined. It turns out that if the two firms use different discount rates, then the optimal strategies will be exponentially decreasing. However, in this case the state equation has no nice solution and must be solved by numerical methods. When both firms use the same discount rate, then the state equation has a simple solution. This solution is also valid for the case where no discounting is performed. Furthermore, when no discounting is performed, the optimal strategies will be simple time-linear decreasing strategies. Finally, it is studied how the optimal strategies and trajectories depend on the parameters of the model.     

A dynamic model for advertising and pricing competition between national and store brands

We study the relationship between the pricing and advertising decisions in a channel where a national brand is competing with a private label. We consider a differential game that incorporates the carryover effects of brand advertising over time for both the manufacturer and the retailer and we account for the complementary and competitive roles of advertising. Analysis of the obtained equilibrium Markov strategies shows that the relationship between advertising and pricing decisions in the channel depends mainly on the nature of the advertising effects. In particular, the manufacturer reacts to higher competitive retailer’s advertising levels by offering price concessions and limiting his advertising expenditures. The retailer’s optimal reaction to competitive advertising effects in the channel depends on two factors: (1) the price competition level between the store and the national brands and (2) the strength of the competitive advertising effects. For example, in case of intense price competition between the two brands combined with a strong manufacturer’s competitive advertising effect, the retailer should lower both the store and the national brands’ prices as a reaction to higher manufacturer’s advertising levels. For the retailer, the main advantage from boosting his competitive advertising investments seems to be driven by increased revenues from the private label. The retailer should however limit his investments in advertising if the latter generates considerable competitive effects on the national brand’s sales.     

Recent developments in dynamic advertising research

A variety of continuous-time differential functions have been developed to investigate dynamic advertising problems in business and economics fields. Since major dynamic models appearing before 1995 have been reviewed by a few survey papers, we provide a comprehensive review of the dynamic advertising models published after 1995, which are classified into six categories: (i) Nerlove–Arrow model and its extensions, (ii) Vidale–Wolfe model and its extensions, (iii) Lanchester model and its extensions, (iv) the diffusion models, (v) dynamic advertising-competition models with other attributes, and (vi) empirical studies for dynamic advertising problems. For each category, we first briefly summarize major relevant before-1995 models, and then discuss major after-1995 models in details. We find that the dynamic models reviewed in this paper have been extensively used to analyze various advertising problems in the monopoly, duopoly, oligopoly, and supply chain systems. Our review reveals that the diffusion models have not been used to analyze advertising problems in supply chain operations, which may be a research direction in the future. Moreover, we learn from our review that very few publications regarding dynamic advertising problems have considered the supply chain competition. We also find that very few researchers have used the diffusion model to investigate the dynamic advertising problems with product quality as a decision variable; and, the pricing decision has not been incorporated into any extant Lanchester model. The paper ends with a summary of our review and suggestions on possible research directions in the future.     

The Leitmann-Schmitendorf advertising game with n players and time discounting

The extension of the Leitmann-Schmitendorf advertising game to n players and positive time discounting is investigated. We show that the strong time consistency of the open-loop Nash equilibrium is preserved. As to optimal controls, while the boundary solution is unaffected by the number of firms as well as discounting, the inner solution depends on industry structure. The fully symmetric version of the game allows us to identify the parameter regions wherein both solutions are sustainable.