An algorithm for payoff space in <em>C</em>¹-games

Abstract

In this paper we present an algorithm implemented by MATLAB, and several examples completely realized by this algorithm, based on a method developed by one of the authors to determine the payoff-space of certain normal-form C1-games. Specifically, our study is based on a method able to determine the payoff space of normal form C1-games in n dimensions, that is for n-players normal form games whose payoff functions are defined on compact intervals of the real line and of class at least C1. In this paper we will determine the payoff space of such normal form C1-games in the particular case of two dimensions. The implementation of the algorithm gives the parametric form of the critical zone of a game in the bistrategy space and in the payoff space and their graphical representations. Moreover, we obtain the parametric form of the transformation of the topological boundary of the bistrategy space and of the transformation of the critical zone. The final aim of the program is to plot the entire payoff space of the considered games. One of the main motivations of our paper is that the mixed extension of a bimatrix game - the most used in the application of Game Theory - is a game of the type considered. For this reason we realized an algorithm that produces the payoff space and the critical zone of a game in normal form supported by a finite family of compact intervals of the real line. Resuming in details, the algorithm returns: the parametric form of the critical zone; the parametric form of the transformation of the topological boundary of the bistrategy space; the parametric form of the transformation of the critical zone. All of them are graphically represented. To prove the efficiency of the algorithm, we show several examples. Our final goal is to provide a valuable tool to study simply but completely normal form C1-games in two dimensions.