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An algorithm for dynamical games with fractal-like trajectories. (English) Zbl 1321.91009
Carfì, David (ed.) et al., Fractal geometry and dynamical systems in pure and applied mathematics II: Fractals in applied mathematics. Selected papers based on three conferences following the passing of Benoît Mandelbrot in October 2010. 1st PISRS 2011 international conference on analysis, fractal geometry, dynamical systems and economics, Messina, Italy, November 8–12, 2011, AMS special session on fractal geometry in pure and applied mathematics, in memory of Benoît Mandelbrot, Boston, MA, USA, January 2012, AMS special session on geometry and analysis on fractal spaces, Honolulu, HI, USA, March 2012. Providence, RI: American Mathematical Society (AMS) (ISBN 978-0-8218-9148-3/pbk; 978-1-4704-1083-4/ebook). Contemporary Mathematics 601, 95-112 (2013).
Summary: We propose an algorithm to represent the payoff trajectory 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 $$C^1$$. 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 $$\dots$$ – 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 nonlinear 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 the first author [Appl. Sci. 11, 35–47 (2009; Zbl 1185.91018)] in several applicative papers by Carfì, Musolino and Perrone 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: on the contrary, our algorithm provides the direct and one shot graphical representation of the entire evolution of those games (by movies) and consequently 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.
For the entire collection see [Zbl 1276.00023].

##### MSC:
 91A25 Dynamic games 28A80 Fractals 91A05 2-person games 91A10 Noncooperative games 91A50 Discrete-time games 91-04 Software, source code, etc. for problems pertaining to game theory, economics, and finance 68W30 Symbolic computation and algebraic computation 91A80 Applications of game theory