Non-Archimedean Game Theory: A Numerical Approach

In this paper we consider the Pure and Impure Prisoner's Dilemmas. Our purpose is to theoretically extend them when using non-Archimedean quantities and to work with them numerically, potentially on a computer. The recently introduced Sergeyev's Grossone Methodology proved to be effective in addressing our problem, because it is both a simple yet effective way to model non-Archimedean quantities and a framework which allows one to perform numerical computations between them. In addition, we could be able, in the future, to perform the same computations in hardware, resorting to the infinity computer patented by Sergeyev himself. After creating the theoretical model for Pure and Impure Prisoner's Dilemmas using Grossone Methodology, we have numerically reproduced the diagrams associated to our two new models, using a Matlab simulator of the Infinity Computer. Finally, we have proved some theoretical properties of the simulated diagrams. Our tool is thus ready to assist the modeler in all that problems for which a non-Archimedean Pure/Impure Prisoner's Dilemma model provides a good description of reality: energy market modeling, international trades modeling, political merging processes, etc.