## 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 language | English |
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Pages (from-to) | 2724-2732 |

Number of pages | 9 |

Journal | Mathematics and Computers in Simulation |

Volume | 79 |

Issue number | 9 |

DOIs | |

Publication status | Published - 2009 May 1 |

## Keywords

- Nash equilibrium
- Oligopoly
- Replicator dynamics

## ASJC Scopus subject areas

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