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

Abstract

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.