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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

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  • Aiyoshi, Eitaro
  • Maki, Atsushi

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.

Suggested Citation

  • Aiyoshi, Eitaro & Maki, Atsushi, 2009. "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," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 79(9), pages 2724-2732.
    • Handle: RePEc:eee:matcom:v:79:y:2009:i:9:p:2724-2732
    DOI: 10.1016/j.matcom.2008.10.009
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    References listed on IDEAS

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    1. Fairen, Victor, 1996. "A replicator theory model of competition through imitation in the automobile market," Journal of Economic Behavior & Organization, Elsevier, vol. 29(1), pages 141-157, January.
    2. Montobbio, Fabio, 2002. "An evolutionary model of industrial growth and structural change," Structural Change and Economic Dynamics, Elsevier, vol. 13(4), pages 387-414, December.
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    1. Aiyoshi, Eitaro & Maki, Atsushi & Okamoto, Takashi, 2011. "The robustness of a Nash equilibrium simulation model: Game-theoretic approach using variable metric projection method," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 81(7), pages 1518-1526.

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