Asymmetric Bertrand duopoly: game complete analysis by algebra system Maxima
AbstractIn this paper we apply the Complete Analysis of Differentiable Games (introduced by D. Carfì in , ,  and ) and already employed by himself and others in , , ) and some new algorithms employing the software wxMaxima 11.04.0 in order to reach a total knowledge of the classic Bertrand Duopoly (1883), viewed as a complex interaction between two competitive subjects, in a particularly difficult asymmetric case. The software wxMaxima is an interface for the computer algebra system Maxima. Maxima is a system for the manipulation of symbolic and numerical expressions, including differentiation, systems of linear equations, polynomials, and sets, vectors, matrices. Maxima yields high precision numeric results by using exact fractions, arbitrary precision integers, and variable precision floating point numbers. Maxima can plot functions and data in two and three dimensions. The Bertrand Duopoly is a classic oligopolistic market in which there are two enterprises producing the same commodity and selling it in the same market. In this classic model, in a competitive background, the two enterprises employ as possible strategies the unit prices of their products, contrary to the Cournot duopoly, in which the enterprises decide to use the quantities of the commodity produced as strategies. The main solutions proposed in literature for this kind of duopoly (as in the case of Cournot duopoly) are the Nash equilibrium and the Collusive Optimum, without any subsequent critical exam about these two kinds of solutions. The absence of any critical quantitative analysis is due to the relevant lack of knowledge regarding the set of all possible outcomes of this strategic interaction. On the contrary, by considering the Bertrand Duopoly as a differentiable game (games with differentiable payoff functions) and studying it by the new topological methodologies introduced by D. Carfì, we obtain an exhaustive and complete vision of the entire payoff space of the Bertrand game (this also in asymmetric cases with the help of wxMaxima 11.04.0) and this total view allows us to analyze critically the classic solutions and to find other ways of action to select Pareto strategies, in the asymmetric cases too. In order to illustrate the application of this topological methodology to the considered infinite game we show how the complete study gives a real extremely extended comprehension of the classic model.
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Bibliographic InfoPaper provided by University Library of Munich, Germany in its series MPRA Paper with number 35417.
Date of creation: 2011
Date of revision:
Asymmetric Bertrand Duopoly; Normal-form games; software algorithms in microeconomic policy; Complete Analysis of a normal-form complex interaction; Pareto optima; valuation of Nash equilibriums; bargaining solutions;
Find related papers by JEL classification:
- C88 - Mathematical and Quantitative Methods - - Data Collection and Data Estimation Methodology; Computer Programs - - - Other Computer Software
- C7 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory
- C8 - Mathematical and Quantitative Methods - - Data Collection and Data Estimation Methodology; Computer Programs
- D43 - Microeconomics - - Market Structure and Pricing - - - Oligopoly and Other Forms of Market Imperfection
- D4 - Microeconomics - - Market Structure and Pricing
- L1 - Industrial Organization - - Market Structure, Firm Strategy, and Market Performance
- C72 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Noncooperative Games
This paper has been announced in the following NEP Reports:
- NEP-ALL-2012-01-03 (All new papers)
- NEP-CMP-2012-01-03 (Computational Economics)
- NEP-COM-2012-01-03 (Industrial Competition)
Please report citation or reference errors to , or , if you are the registered author of the cited work, log in to your RePEc Author Service profile, click on "citations" and make appropriate adjustments.:
- David CARFÃŒ & Daniele SCHILIRÃ’, 2011.
"Crisis In The Euro Area. Coopetitive Game Solutions As New Policy Tools,"
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- Carfì, David & Schilirò, Daniele, 2010. "Crisis in the Euro area: coopetitive game solutions as new policy tools," MPRA Paper 27138, University Library of Munich, Germany.
- Carfí, David & Musolino, Francesco, 2014. "Speculative and hedging interaction model in oil and U.S. dollar markets with financial transaction taxes," Economic Modelling, Elsevier, vol. 37(C), pages 306-319.
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