The payoff trajectories in C1 parametric games

The great achievement of the game theory arises form the wide variety of fields in which it has been applied, in order to model and analyze a large collection of human and animal behavior, but economic, political, sociological and psychological ones as well. Our study pertain to normal-form C1-games in n−dimensions that is n−players normal form games whose payoff functions are at least of class C1 in a compact interval of the real line. This study includes also games depending on a parameter in a one dimensional set. In [5, 6, 8, 9, 10, 11], the authors analyze parametric games, where the parameter set is the coopetitive strategy one. It allows us to pass from the standard normal-form games to their coopetitive extension as illustrated in [3, 2, 17, 18]. In particular, in [17], a new procedure to determine the payoff space of such kind of games has been presented and it has been applied in [1] to numericaly determine the payoff space for normal-form C1 parametric games in two dimensions. In this work, the method in [17] has been pointed out and assumed with the aim of realizing an algorithm for the computational representation of the payoff trajectory in the case of normal-form C1-parametric games. To ease the reader, in the first section of the paper we bring to mind terminology and some definitions, while in the second part, the method proposed in [17] and applied in the development of our algorithm, is presented. Moreover, the particular case of two parametric games is shown in the third section. The application of our algorithm to several examples concludes the paper.