### 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 |
---|---|

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 |

### Fingerprint

### 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

*Mathematics and Computers in Simulation*,

*79*(9), 2724-2732. https://doi.org/10.1016/j.matcom.2008.10.009

**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.** / Aiyoshi, Eitaro; Maki, Atsushi.

Research output: Contribution to journal › Article

*Mathematics and Computers in Simulation*, vol. 79, no. 9, pp. 2724-2732. https://doi.org/10.1016/j.matcom.2008.10.009

}

TY - JOUR

T1 - A Nash equilibrium solution in an oligopoly market

T2 - The search for Nash equilibrium solutions with replicator equations derived from the gradient dynamics of a simplex algorithm

AU - Aiyoshi, Eitaro

AU - Maki, Atsushi

PY - 2009/5

Y1 - 2009/5

N2 - 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.

AB - 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.

KW - Nash equilibrium

KW - Oligopoly

KW - Replicator dynamics

UR - http://www.scopus.com/inward/record.url?scp=67349244701&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=67349244701&partnerID=8YFLogxK

U2 - 10.1016/j.matcom.2008.10.009

DO - 10.1016/j.matcom.2008.10.009

M3 - Article

AN - SCOPUS:67349244701

VL - 79

SP - 2724

EP - 2732

JO - Mathematics and Computers in Simulation

JF - Mathematics and Computers in Simulation

SN - 0378-4754

IS - 9

ER -