GAME COMPLETE ANALYSIS OF BERTRAND DUOPOLY
In this paper we apply the Complete Analysis of Differentiable Games (introduced by D. Carfi in [3], [6], [8] and [9]) and al-ready employed by himself and others in [4], [5], [7]) to the classic Bertrand Duopoly (1883), 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 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 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. Carfi, 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 computers) and this total view allows us to analyze critically 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 are considered, and we show how the complete study gives a real extremely extended comprehension of the classic model.
Differentiable game complete analysis for tourism firm decisions
In this paper we apply the complete analysis of a differentiable game (recently introduced by the author) to determine possible suitable behaviors (actions) of tourism firms during strategic interactions with other tourism firms, from both non-cooperative and cooperative point of view. To associate with a real strategic interaction among tourism firms a differentiable game any player’s strategy-set must, for instance, be a part of a topological vector space, closure of an open subset of the space. The most frequent case is that in which the strategy-sets are compact intervals of the real line. On the other hand, very often, the actions at disposal of a player can form a finite set, and in this case a natural manner to construct a game representing the economic situation is the von Neumann convexification (also known as canonical extension) that leads to a differentiable game with probabilistic scenarios, and thus even more suitable for the purpose of represent real interactions. For what concerns the complete analysis of a differentiable game, its first goal is the precise knowledge of the Pareto boundaries (maximal and minimal) of the payoff space, this knowledge will allow us to evaluate the quality of the different Nash equilibria (by the distances from the Nash equilibria themselves to Pareto boundaries, with respect to appropriate metrics), in order to determine some “focal” equilibrium points collectively more satisfactory than each other. Moreover, the complete knowledge of the payoff-space will allow to develop explicitly the cooperative phase of the game and the various bargaining problems rising from the strategic interaction of the tourist firms (Nash bargaining problem, Kalai-Smorodinski bargaining problem and so on). In the paper we shall deal with some practical study cases.
S-diagonalizable operators in quantum mechanics
In this paper we study a certain class of endomorphisms on the space of tempered distributions. More precisely, the core of the paper deals with endomorphisms, dened on the whole space of tempered distributions, for which there exists an S-basis of the space (see section 5) formed by their eigenvectors. We call these operators S-diagonalizable operators. One of the goals of the paper is the realization that this class of endomorphisms represents in the innite-dimension al case what in nite- dimension is represented by the diagonalizable matrices. We concentrate our examination on two aspects: the study of the spectrum of these operators and the foundation of a functional calculus for them. Concerning the rst aspect, we do not assume nothing about the spectrum of these operators. The circumstance that the eigenvalue- spectrum of these operators will be continuous (more precisely it will be a connected subset of the complex plane, as it is proved in the present note) is a consequence of our denition. Moreover, the spectral measures will be not used in the construction of the functional calculus. In such a way, the denition of the function of an operator, presented in the paper, diers deeply from the usual one, in which the spectral measures of the operators play a fundamental role (as in the spectral decomposition of an operator). Note that, even in the case in which the eigenvalues-spectrum is a subset of the real line, we show that it is not necessarily coinciding with the whole real line.