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.
A model of oligopoly with an arbitrary number of rational agents that are reflexive according to Cournot or Stackelberg, under the conditions of incomplete information for the classical case of linear functions of costs and demand is considered. The problem of achieving equilibrium based on mathematical modeling agents' decision-making processes is investigated. Works in this direction are relevant due to the importance of understanding the processes in real markets and the convergence of theoretical models with them. In the framework of a dynamic model of reflexive collective behavior, each agent at each moment adjusts its output, making a step in the direction of output maximizing its profit under the expected choice of competitors. The permissible step value is set by the range. This article sets and solves the problem of finding the ranges of permissible steps of agents, which are formulated as conditions that guarantee the convergence of dynamics to equilibrium. The novelty of the study is determined by the use of the norm of the error transition matrix from the t-th to (t+1)-moment of time as a criterion of the dynamics convergence. It is shown that the dynamics converge if the norm is less than unity, starting at some point in time, and the failure to fulfill this criterion especially manifests itself in multidirectional choice, when some agents choose "big" steps towards their current goals, while others, on the contrary, choose "small" steps. Failure to meet the criterion also increases as the market grows. The general conditions for the ranges of convergence of dynamics for an arbitrary number of agents are established, and a method for constructing the maximum such ranges is proposed, which also constitutes the novelty of the study. The results of solving the above problems for particular cases of oligopoly, which are the most widespread in practice, are presented.
Рассматривается проблема достижения равнове-сия Нэша на рынке олигополии фирмами-олигопо-листами, действующими по Штакельбергу, с приме-нением рефлексивных повторяющихся игр и моделей динамики коллективного поведения. Олигополисты, основываясь на наблюдении за текущей ценой товара и собственных размышлениях о наилучшем собствен-ном действии с учетом наилучших откликов осталь-ных фирм, при повторении игры уточняют по модели коллективного поведения свои объемы выпуска, делая шаги в направлении текущего оптимального выпуска. Развитие динамики может направляться правилами игры на величины шагов. В нашем случае рефлек-сивной игры длины шагов могут меняться во вре-мени, но фирмы в каждый момент времени должны следовать политике единой длины шага. В классе ли-нейных функций спроса и издержек олигополистов получены необходимые и достаточные условия схо-димости динамик в дискретном времени к равнове-сию Нэша. Динамики сходятся при любых первона-чальных значениях выпуска и цены. Перспективными представляются исследования сходимости динамик, в которых каждая фирма может придерживаться сво-ей собственной политики выбора величины шага, от-личной от политики других фирм.Ключевые слова: олигополия, поведение по Штакель-бергу, равновесие по Нэшу, рефлексивные игры, коллек-тивное поведение, текущие цены, уточнение выпусков, сходимость динамики.The problem of achieving Nash equilibrium in the oligopoly market with oligopolistic firms that acted on Stackelberg using reflexive repetitive games and models of collective behavior dynamics is considered.Oligopolists observe the product current price and use their own reflections on what actions should produce the best response from remaining firms. They repeat the game and clarify output using collective behavior models, making steps towards the current optimal release. The development of the dynamics can be directed by rules of the game with the step size. In case of reflexive games, step sizes can change over time, but firms at each time point must follow a unified policy of the same step size. The necessary and sufficient conditions for the convergence of dynamics in discrete time to the Nash equilibrium are obtained in the class of linear demand and cost functions of oligopolists. Dynamics converge for any initial output and price. Studies of the convergence dynamics when each firm may follow its own independent policy of step size choice are seemed to be prospective.Key words: oligopoly, Stackelberg behavior, Nash equilibrium, reflexive games, collective behavior, current prices, clarifying output, dynamic convergence.На конкурентном олигопольном рынке каждая из рациональных фирм, желая увеличить собствен-ную прибыль, стремится стать лидером. Лидерские амбиции обязывают их правильно прогнозировать поведение и выбор конкурентов. Однако, наблюдая текущее состояние рынка, одна, несколько или все фирмы могут убедиться в том, что из-за неправильно-го прогноза их текущие объемы выпуска продукции не являются оптимальными. Естественно, что у них возникает желание уточнить свои будущие объ...
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