Generalizations of the Optimal Control Problem for the Vidale-Wolfe Advertising Model.

The purpose of this dissertation is to study two different generalizations of the optimal control problem based on the Vidale-Wolfe Advertising Model. The first problem is an infinite horizon free endpoint one-dimensional version of the Vidale-Wolfe optimal control problem of advertising in time varying markets. A solution is obtained for this problem. Then a thorough proof using the method of dynamic programming is presented to verify that this solution is optimal under reasonable market conditions. The second problem is a finite time fixed endpoint two-dimensional version of the Vidale-Wolfe optimal control problem. Normal optimal trajectories are obtained for this problem. A thorough proof using the Pontryagin Maximum Principle is then presented under reasonable market conditions

Find, Process, and Share: An Optimal Control in the Vidale-Wolfe Marketing Model

The Vidale-Wolfe marketing model is a first-order, linear, non-homogeneous ordinary differential equation (ODE) where the forcing term is proportional to advertising expenditure. With an initial response in sales as the initial condition, the solution of the initial value problem is straightforward for a first undergraduate ODE course. The model serves as an excellent example of many relevant topics for those students whose interests lie in economics, finance, or marketing. Its inclusion in the curriculum is particularly rewarding at an institution without a physics program. The model is not new, but it was novel to us when a group of students chose it for an exploratory project that we designed in order to help students acquire the ability to interpret and communicate mathematical results. In addition to describing the project in this work, we discuss the Vidale-Wolfe model and show how it can lead one to use Green\u27s theorem in a real situation

Optimal control of the vidale-wolfe-deal and three populations models of market share dynamics

The purpose of this dissertation is to study the optimal response of salesadvertising models for duopolies using modern optimization software (JModelica.org and JuMP). Specifically, the models adopted were the Vidale-Wolfe-Deal and the Three Populations, a Lotka-Volterra type model. Duopoly analysis is split in two parts: one that solves an optimal control problem with the objective of maximizing the net profit of both firms in one-shot, referred to as simultaneous co-operation, and the other that places the two firms as opponents in a sequential game, each with the individual goal of maximizing net profit on its turn, referred to as sequential competition. The contributions of this dissertation are: providing a stability analysis for the Vidale-Wolfe-Deal model, which shows that any control attaining final constant positive values leads to a stable equilibrium of market shares, proposing a duopolistic version of the Three Populations model, and, lastly, proposing and solving a sequential game based on Leader-Follower iteration for the Vidale-Wolfe-Deal model.O propósito desta dissertação é estudar a resposta ótima dos modelos de venda-publicidade em duopólios utilizando modernos software de otimização (JModelica.org e JuMP). Especificamente, os modelos adotados foram Vidale-Wolfe-Deal e Três Populações (um modelo do tipo Lotka-Volterra). As análises de duopólio são divididas em duas partes: uma, que soluciona o problema de controle ótimo com o objetivo de maximizar o lucro líquido de ambas as empresas de uma só vez, referido como cooperação simultânea, e outra, que coloca as duas empresas como oponentes em um jogo sequencial, cada uma com o objetivo individual de maximizar o lucro líquido durante seu turno, referido como competição sequencial. As contribuições da dissertação são: prover análise de estabilidade do modelo Vidale-Wolfe-Deal mostrando que qualquer controle que atinja valores constantes positivos leva a um equilíbrio estável das fatias de mercado, propor uma versão do modelo de Três Populações para o duopólio, e, por fim, propor e solucionar um jogo sequencial baseado em iterações de Líder-Seguidor para o modelo Vidale-Wolfe-Deal

Certainty equivalence principle in stochastic differential games: An inverse problem approach

Producción CientíficaThis paper aims to characterize a class of stochastic differential games, which satisfy the certainty equivalence principle beyond the cases with quadratic, linear, or logarithmic value functions. We focus on scalar games with linear dynamics in the players' strategies and with separable payoff functionals. Our results are based on the resolution of an inverse problem that determines strictly concave utility functions of the players so that the game satisfies the certainty equivalence principle. Besides establishing necessary and sufficient conditions, the results obtained in this paper are also a tool for discovering new closed-form solutions, as we show in two specific applications: in a generalization of a dynamic advertising model and in a game of noncooperative exploitation of a productive asset.Este trabajo se ha hecho con ayuda de los proyectos del Ministerio de Economía, Industria y Competitividad, Grant/Award Number: ECO2017-86261-P, ECO2014-56384-P, y MDM 2014-0431, de la Consejería de Educación, Juventud y Deporte de la Comunidad de Madrid, Grant/Award Number: MadEco-CM S2015/HUM-3444, y de la Consejería de Educación de la Junta de Castilla y León VA148G18

Modeling and analysis of market share dynamics in a duopoly subject to affine feedback advertising policies and delays

Presents extensions of the Vidale-Wolfe and Lanchester models for market share duopoly dynamics. The novelties in the proposed extensions are the explicit introduction of a set of undecided clients into existing models, which consider only the sets of clients of the two competing firms, as well as the use of decentralized affine feedback advertising policies. It is shown that, under the proposed class of advertising policies, the extended Vidale-Wolfe and Lanchester models, despite having different dynamics, have equilibria in identical locations, with the same stability properties. The introduction of a third set of undecided clients also motivates the introduction of a more elaborate model of market share dynamics based on the replicator-mutator model from evolutionary game theory. This is done by identifying strategies with the entries of a preference matrix consisting of the choice preferences of firms by clients and the mutation matrix representing transition probabilities from one set of clients to another. The proposed model is analysed with respect to equilibria and their stability properties, as well as parametric sensitivity, under the proposed advertising policies. All proposed models are analysed for stability, the presence of oscillations, existence of Hopf bifurcations when implementation or adoption delays are introduced. The extended models of Vidale-Wolfe and Lanchester are robust to implementation delays, while for adoption delays, bifurcations can occur. The proposed replicator-mutator model is robust for both types of delays.Apresenta extensões aos modelos de Vidale-Wolfe e Lanchester para a dinâmica de duopólios. As novidades nas extensões propostas são a introdução explícita de um conjunto de clientes indecisos nos modelos existentes, os quais consideram apenas os conjuntos de clientes das duas empresas concorrentes, e o uso de políticas de publicidade afins com realimentação. Demonstra-se que sob a classe proposta de políticas de publicidade, os modelos estendidos de Vidale-Wolfe e Lanchester, apesar de terem dinâmicas diferentes, apresentam pontos de equilíbrio idênticos com as mesmas propriedades de estabilidade. A introdução de um terceiro conjunto de clientes indecisos também motiva a introdução de um modelo mais elaborado da dinâmica de mercado baseado no modelo Replicador-Mutador da teoria dos jogos evolucionários. A proposta do modelo é realizada identificando estratégias com elementos de uma matriz de preferência consistindo nas preferências de escolha das empresas pelos clientes e a matriz de mutação representa probabilidades de transição de um conjunto de clientes para outro. O modelo proposto é analisado em relação aos pontos de equilíbrios e suas propriedades de estabilidade, bem como a sensibilidade paramétrica sob as políticas de publicidade propostas. Todos os modelos propostos são analisados quanto à estabilidade, a presença de oscilações, a existência de bifurcações de Hopf quando atrasos de implementação ou de adoção são introduzidos. Os modelos estendidos de Vidale-Wolfe e Lanchester são robustos para atrasos em implementação enquanto que para o atraso de adoção apresentam a existência de bifurcações. O modelo Replicador-Mutador proposto é robusto para ambos tipos de atrasos

Optimum advertising pulsation strategies: A dynamic programming approach

This study, using the dynamic programming approach, has addressed the problem of optimally allocating a fixed advertising budget of a monopolistic firm over a planning horizon comprised of n equal periods to maximize two popular measures of advertising performance: (1) profits related to the advertising effort (discount factor r = 0), and (2) present value of profits related to the advertising effort (discount factor r \u3e 0). Two dynamic programming models that use the modified Vidale-Wolfe model to represent sales response to advertising are formulated with respect to whether the time value of money is considered. For a planning horizon comprised of four equal time periods, computing routines are developed to solve two sample problems with respect to the dynamic programming models. Sensitivity analyses are performed to assess the impacts of a change in some key model parameters upon the behavior patterns of the optimum dynamic programming advertising policy and the associated total return. Four alternative types of traditional advertising pulsation policies are modeled for the purpose of comparing their performance with the optimum advertising policy determined by dynamic programming. For a planning horizon comprised of four equal time periods, computing routines are also developed to generate total returns under these traditional advertising pulsation policies. Computational results show that the performance under the optimal advertising policy determined by dynamic programming, as expected, is at least as good as the maximum performance among the four traditional advertising pulsation policies. The plausibility of the modified Vidale-Wolfe model is empirically examined using the well-known Lydia Pinkham vegetable compound annual data covering the period from 1907 to 1960. Model parameters have been estimated using the Gauss-Newton algorithm related to nonlinear regression. The model selected is one corrected for first-order autoregressive residuals. The empirical results indicate that the model parameters are statistically significant and of the expected signs. More important, it is found that the advertising response function is concave