Game Complete Analysis of Bertrand Duopoly

In this paper, we have applied the Complete Analysis of Differentiable Games (introduced by David Carfì in 2009 and 2010) and already employed by himself and others in Carfi, and Schilliro, 2011; Carfi, and Ricciardello, 2010; Carfi, 2009) to the classic Bertrand Duopoly‟s (1883) classic oligopolistic market in which two enterprises are 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 product, 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 the 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 David Carfì, we obtain an exhaustive and complete vision of the entire payoff space of the Bertrand game (also in asymmetric cases with the help of computers) and this total view allows us to critically analyze the classic solutions and to find other ways of action to select Pareto strategies. In order to illustrate the application of this topological methodology to the considered infinite game, several compromise pricing-decisions have been considered, and we show how the complete study gives a real and extremely extended comprehension of the classic model.