Decision-form games

In this paper we formalize a new form of two-player game, that we call decision- form. A two-player decision-form game consists in a pair of decision rules, representing the rationality of each player. We develop the basic facts of this type of games, showing that this form of game generalizes the normal-form . Indeed, we show that with a normal form game it is possible, in a natural way, to associate a decision form game. In the paper we give examples of decision-form games not-deriving, a priori, by normal-form games. We observe that it is possible to associate with a normal-form game several decision-form games, each representing a possible decisional behavior of the pair of players. The classic best-response behavior is only one of these possible behaviors.2-person games; Applications of game theory

Fibrations of financial events

In this paper we shall prove that the plane of financial events, introduced and applied to financial problems by the author himself (see [2], [3] and [4]) can be considered as a fiber bundle in two different ways. The first one, the natural one, reveals itself to be isomorphic to the tangent bundle of the real line, when the last one is considered as a differentiable manifold in the natural way; the second one is a fibration induced by the status of compound interest capitalization at a given rate i in the interval ] − 1, → [. Moreover, in the paper we define on the first vector bundle an affine connection, also in this case induced by the compound interest capitalization at a given rate i. The final goal of this paper is the awareness that all the effects determined by compound interest capitalization are nothing but the consequences of the fact that the space of financial events is a fibration endowed with a particular affine connection, so they are consequences of purely geometric properties, at last, depending upon the curvature determined by the connection upon the fibration. A natural preorder upon the set of fibers of the second fibration is considered. Some remarks about the applicability to economics and finance of the theories presented in the paper and about the possible developments are given in the directions followed in papers [1], [5], [6], [7], [8] of the author himself.Financial event; compound interest; capital evolution; accumulation function; force of interest; fiber bundle; vector bundle; affine connection; Christoffel bilinear form

Reactivity in decision-form games

In this paper we introduce the reactivity in decision-form games. The concept of reactivity allows us to give a natural concept of rationalizable solution for decision-form games: the solubility by elimination of sub- reactive strategies. This concept of solubility is less demanding than the concept of solubility by elimination of non-reactive strategies (introduced by the author and already studied and applied to economic games). In the work we define the concept of super-reactivity, the preorder of re- activity and, after a characterization of super-reactivity, we are induced to give the concepts of maximal-reactivity and sub-reactivity; the latter definition permits to introduce the iterated elimination of sub-reactive strategies and the solubility of a decision-form game by iterated elimina- tion of sub-reactive strategies. In the paper several examples are devel- oped. Moreover, in the case of normal-form games, the relation between reactivity and dominance is completely revealed.Decision form games; reactivity; dominance

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.Tourism fir; differentiable game; strategic interaction; non-cooperative behaviour; cooperative behavior; Pareto efficiency.

Optimal boundaries for decisions

In this paper we state and prove some new results about the optimal boundaries. These boundaries (also called Pareto boundaries or efficiency boundaries or maximal/minimal boundaries) are of increasing importance in the applications to Decision Theory and Economics. First of all the Pareto boundaries are the first and most important generalization of the optima of decision constraints. On the other hand, if f is a real functional (utility function) defined on a non empty set X (of choices or economic strategies) and K is a part of X, the determination of the optimal boundaries of the part K, with respect to some preference relation ≤ of X for which the function f is strictly increasing, allows to reduce the optimization problem of finding the minimum of the functional f upon the part K to the problem of finding the minimum of f upon the minimal boundary of K. We note that the minimal boundary of K is, in general, greatly smaller than the initial decision constraint K. An economic application to the Cournot duopoly is presented.Optimal strategy, Pareto efficiency, cofinality, decision problem, utility function, Cournòt duopoly

Asymmetric Bertrand duopoly: game complete analysis by algebra system Maxima

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.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

A coopetitive model for the green economy

The paper proposes a coopetitive model for the Green Economy. It addresses the issue of the climate change policy and the creation and diffusion of low-carbon technologies. In the present paper the complex construct of coopetiton is applied at macroeconomic level. The model, based on Game Theory, enables us to offer a set of possible solutions in a coopetitive context, allowing to find a Pareto solution in a win-win scenario. The model, which is based on the assumption that each country produces a level of output which is determined in a non-cooperative game of Cournot-type and that considers at the same time a coopetitive strategy regarding the low carbon technologies, will suggest a solution that shows the convenience for each country to participate actively to a program of low carbon technologies within a coopetitive framework to address a policy of climate change, thus aiming at balancing the environmental imbalances.coopetition; game theory; green economy; energy-saving technologies; policy of climate change

Crisis in the Euro area: coopetitive game solutions as new policy tools

The crisis within the Euro area have become frequent during 2010. First was the Greek economy to face a default problem of its sovreign debt, in November it was Ireland who has been in a serious financial situation at the verge of collapse causing difficulties to the euro. In this contribution we focus on the Greek crisis and we suggest, through a model of coopetition based on game theory and conceived at a macro level, feasible solutions in a cooperative perspective for the divergent interests which drive the economic policies in Germany and Greece, with the aim of improving the position of Greece, Germany and the whole Euro area and also giving a contribution to expand the set of macroeconomic policy tools. By means of our general analytical framework of coopetition, we show the strategies that could bring to feasible solutions in a cooperative perspective for Germany and Greece, where these feasible solutions aim at offering a win-win outcome for both countries, letting them to share the pie fairly within a growth path represented by a non-zero sum game. A remarkable analytical result of our work consists in the determination of the win-win solution by a new selection method on the transferable utility Pareto boundary of the coopetitive game.European monetary Union, Coopetitive Games, Macroeconomic Policy

Game complete analysis for financial markets stabilization

The aim of this paper is to propose a methodology to stabilize the financial markets using Game Theory and in particular the Complete Study of a Differentiable Game, introduced in the literature by David Carfì. Specifically, we will focus on two economic operators: a real economic subject and a financial institute (a bank, for example) with a big economic availability. For this purpose we will discuss about an interaction between the two above economic subjects: the Enterprise, our first player, and the Financial Institute, our second player. The only solution which allows both players to win something, and therefore the only one desirable, is represented by an agreement between the two subjects: the Enterprise artificially causes an inconsistency between spot and future markets, and the Financial Institute, who was unable to make arbitrages alone, because of the introduction by the normative authority of a tax on economic transactions (that we propose to stabilize the financial market, in order to protect it from speculations), takes the opportunity to win the maximum possible collective (social) sum, which later will be divided with the Enterprise by contract.Financial Markets; Game Theory; Stabilization of Financial Markets; arbitrages