Carfì, David and Perrone, Emanuele (2011): Asymmetric Bertrand duopoly: game complete analysis by algebra system Maxima.

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Abstract
In this paper we apply the Complete Analysis of Differentiable Games (introduced by D. Carfì in [3], [6], [8] and [9]) and already employed by himself and others in [4], [5], [7]) 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.
Item Type:  MPRA Paper 

Original Title:  Asymmetric Bertrand duopoly: game complete analysis by algebra system Maxima 
Language:  English 
Keywords:  Asymmetric Bertrand Duopoly; Normalform games; software algorithms in microeconomic policy; Complete Analysis of a normalform complex interaction; Pareto optima; valuation of Nash equilibriums; bargaining solutions 
Subjects:  C  Mathematical and Quantitative Methods > C8  Data Collection and Data Estimation Methodology ; Computer Programs > C88  Other Computer Software C  Mathematical and Quantitative Methods > C7  Game Theory and Bargaining Theory C  Mathematical and Quantitative Methods > C8  Data Collection and Data Estimation Methodology ; Computer Programs D  Microeconomics > D4  Market Structure, Pricing, and Design > D43  Oligopoly and Other Forms of Market Imperfection D  Microeconomics > D4  Market Structure, Pricing, and Design L  Industrial Organization > L1  Market Structure, Firm Strategy, and Market Performance C  Mathematical and Quantitative Methods > C7  Game Theory and Bargaining Theory > C72  Noncooperative Games 
Item ID:  35417 
Depositing User:  DAVID CARFì 
Date Deposited:  15. Dec 2011 02:35 
Last Modified:  23. Sep 2015 15:23 
References:  1. Aubin Jean Pierre, Mathematical Methods of Game and Economic Theory (Revised Edition), (NorthHolland). 2. Aubin Jean Pierre, Optima and Equilibria, (Springer Verlag, 1998). 3. Carfì David, Topics in Game Theory, (Edizioni Il gabbiano, 2010). 4. Carfì David, Schilirò Daniele, Crisis in the Euro area: coopetitive game solutions as new policy tools, TPREF  Theoretical and Practical Research in Economic Fields, Summer issue 2011 http://mpra.ub.unimuenchen.de/27138/ 5. Carfì David, Ricciardello Angela, An algorithm for payoff space in C1 games, AAPP  Physical, Mathematical, and Natural Sciences, Vol. LXXXVIII, issue 1, 2010 http://cab.unime.it/journals/index.php/AAPP/article/view/C1A1001003/ 6. Carfì David, Differentiable game complete analysis for tourism firm decisions, Proceedings of THE 2009 INTERNATIONAL CONFERENCE ON TOURISM and WORKSHOP on Sustainable tourism within High Risk areas of environmental crisis, Messina, April 22/25 (2009) http://mpra.ub.unimuenchen.de/29193/ 7. Carfì David, Globalization and Differentiable General Sum Games, Proceeding of 3rd International Symposium Globalization and convergence in economic thought, Faculty of Economics  Communication and Economic Doctrines Department  The Bucharest Academy of Economic Studies  Bucharest December 11/12 (2009) 8. Carfì David, Payoff space and Pareto boundaries of C1 Games, APPS vol. 2009 http://www.mathem.pub.ro/apps/v11/a11.htm 9. Carfì David, Complete study of linear infinite games, Proceedings of the International Geometry Center  Prooceeding of the International Conference “Geometry in Odessa 2009”, Odessa, 25  30 May 2009  vol. 2 n. 3 (2009) http://proceedings.domega.org/ 10. Carfì David, Optimal boundaries for decisions, Atti dell'Accademia Peloritana dei Pericolanti  Classe di Scienze Fisiche, Matematiche e Naturali Vol. LXXXVI, (2008) http://cab.unime.it/journals/index.php/AAPP/article/view/376 11. Carfì David, Ricciardello Angela, Nonreactive strategies in decisionform games, AAPP  Physical, Mathematical, and Natural Sciences, Vol. LXXXVII, issue 1, pp. 118 http://cab.unime.it/journals/index.php/AAPP/article/view/C1A0902002/0 12. Carfì David, Decisionform games, Proceedings of the IX SIMAI Congress, Rome, 22  26 September 2008, Communications to SIMAI congress, vol. 3, (2009) pp. 112 http://cab.unime.it/journals/index.php/congress/article/view/307 13. Carfì David, Reactivity in decisionform games, MPRA  paper 29001, University Library of Munich, Germany http://mpra.ub.unimuenchen.de/29001/ 14. Carfì David, Ricciardello Angela, Mixed Extensions of decisionform games, MPRA  paper 28971, University Library of Munich, Germany http://mpra.ub.unimuenchen.de/28971/ 
URI:  https://mpra.ub.unimuenchen.de/id/eprint/35417 