A Nash equilibrium solution in an oligopoly market: The search for Nash equilibrium solutions with replicator equations derived from the gradient dynamics of a simplex algorithm

Eitaro Aiyoshi, Atsushi Maki

Research output: Contribution to journalArticle

6 Citations (Scopus)

Abstract

The present analysis applies continuous time replicator dynamics to the analysis of oligopoly markets. In the present paper, we discuss continuous game problems in which decision-making variables for each player are bounded on a simplex by equalities and non-negative constraints. Several types of problems are considered under conditions of normalized constraints and non-negative constraints. These problems can be classified into two types based on their constraints. For one type, the simplex constraint applies to the variables for each player independently, such as in a product allocation problem. For the other type, the simplex constraint applies to interference among all players, creating a market share problem. In the present paper, we consider a game problem under the constraints of allocation of product and market share simultaneously. We assume that a Nash equilibrium solution can be applied and derive the gradient system dynamics that attain the Nash equilibrium solution without violating the simplex constraints. Models assume that three or more firms exist in a market. Firms behave to maximize their profits, as defined by the difference between their sales and cost functions with conjectural variations. The effectiveness of the derived dynamics is demonstrated using simple data. The present approach facilitates understanding the process of attaining equilibrium in an oligopoly market.

Original languageEnglish
Pages (from-to)2724-2732
Number of pages9
JournalMathematics and Computers in Simulation
Volume79
Issue number9
DOIs
Publication statusPublished - 2009 May

Fingerprint

Oligopoly
Simplex Algorithm
Equilibrium Solution
Nash Equilibrium
Gradient
Cost functions
Profitability
Sales
Dynamical systems
Decision making
Non-negative
Game
Replicator Dynamics
Gradient System
Market
System Dynamics
Profit
Cost Function
Continuous Time
Equality

Keywords

  • Nash equilibrium
  • Oligopoly
  • Replicator dynamics

ASJC Scopus subject areas

  • Modelling and Simulation
  • Numerical Analysis
  • Applied Mathematics
  • Theoretical Computer Science
  • Computer Science(all)

Cite this

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