Approaches to Studying Collective Behavior Models in Competitive Markets: Peculiarities and Insights

Abstract

The paper addresses the problem of achieving Nash equilibrium in an oligopoly market where common knowledge is absent. It explores two approaches to study the convergence conditions of the dynamics in a linear oligopoly model with any number of Cournot or Stackelberg reflexive agents.
 The first approach utilizes indicator functions to guide agents in adjusting their outputs to reach the optimum under existing competitors' outputs. The second approach uses norms of transition matrices from iteration to iteration in the computational process. The paper presents the authors' research results, including necessary mathematical lemmas and statements, and computational experiments with the models.
 The paper emphasizes several important features of the approaches, such as the conditions for individual agent choices that ensure the convergence to equilibrium of collective behavior dynamics for duopoly and any number of agents in the market, the convergence criteria of the dynamics, and the calculation of indicator functions and norms of transition matrices.