An algorithm for dynamical games with fractal-like trajectories

Inthispaper,weproposeanalgorithmtorepresentthepayofftra- jectory of two-player discrete-time dynamical games. Specifically, we consider discrete dynamical games which can be modeled as sequences of normal-form games (the states of the dynamical game) with payoff functions of class C1. In this context, the payoff evolution of such type of dynamical games is the sequence of the payoff spaces of their game-states and the payoff trajectory of such games is the union of the members of the evolution. The formulation of the algorithm is motivated - especially in several applicative contexts such as Economics, Finance, Politics, Management Sciences, Medicine and so on ... - by the need of a complete knowledge of the payoff evolution (problem which is still open in the most part of the cases), when the real problem requires a Complete Analysis of the interactions, beyond the study of just the Nash equilibria. We consider, to prove the efficiency and strength of our method, the development (by the algorithm itself) of some non-linear dynamical games taken from applications to Microeconomics and Finance. The dynamical games that we shall examine are already deeply studied and represented, at least at their initial state - by the application of the topological method presented by Carf`ı in [7] - in several applicative papers by Carf`ı, Musolino and Perrone (see [10], [11–20] by a long, quite indirect and step by step implementations of other standard computational softwares (such as AutoCad, Derive, Grapher, Graph and Maxima) or following a pure mathematical way (see for example [8]): on the contrary, our algorithm provides the direct and one shot graphical representation of the entire evolution of those games (by movies) and conse- quently of the entire trajectory. Moreover, the applicative games we consider in the paper (inspired and suggested by Economics and Finance) have a natural dynamics having fractal-like trajectories.