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.
Algorithms for Payoff Trajectories in C 1 Parametric Games
In 2009 ([4]) a new procedure to determine the payoff space of non-parametric differentiable normal form games has been presented. Then, the new procedure has been applied (in [1]) to numerically determine an original type of 3-dimensional representation of the family of payoff spaces in normal-form C 1 parametric games, with two players. In this work, the method in [4] has been pointed out and assumed with the aim of realizing an algorithm which (computationally) gives the real geometric representation of the payoff trajectory of normal-form C 1-parametric games, with two players. The application of our algorithm to several examples concludes the paper. Our analysis of parametric games, also, allows us to pass from the standard normal-form games to their coopetitive extension, as already illustrated in several applicative papers by D. Carfi.
X-rays and photocarcinogenesis in hairless mice
It is well known that excessive X-ray radiation can cause non-melanoma skin cancers. With the increased incidence of sun-related skin cancer there is a need to investigate the combination of sunlight and X-rays. Immunocompetent C3.Cg/TifBomTac mice (n = 298) were divided into 12 groups. Mice were irradiated with 12, 29 or 50 kV X-rays. The mice received a total dose of 45 Gy. They were irradiated with 3 SED simulated solar radiation (SSR) either before or after irradiation with X-rays. The groups irradiated with X-rays alone, 0, 3, 9 and 10 mice (0, 12, 29 and 50 kV, respectively) developed squamous cell carcinoma. In the groups irradiated with SSR after X-rays the development of tumours was significantly faster in the 50 kV group than in the corresponding control group (175 vs. 194 days, p < 0.001). In the groups irradiated with SSR prior to the X-ray radiation the development of tumours was significantly faster in the 29 and the 50 kV groups than in the corresponding control group (175 vs. 202 days, p < 0.001 and 158 vs. 202 days, p < 0.001, respectively). In conclusion, X-ray radiation alone is a weak carcinogen in hairless mice. There is an added carcinogenic effect if X-ray radiation is given on prior sun-exposed skin or if the skin is sun-exposed after X-rays. We still believe that X-ray radiation is a safe and effective therapy for various dermatological diseases but caution should be observed if a patient has severely sun-damaged skin or has a high-risk sun behaviour.