Abstract 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.

中文翻译：

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非阿基米德博弈论：一种数值方法

摘要 在本文中，我们考虑纯囚徒困境和非纯囚徒困境。我们的目的是在使用非阿基米德量时在理论上扩展它们，并在数字上处理它们，可能是在计算机上。最近引入的 Sergeyev 的 Grossone Methodology 被证明可以有效地解决我们的问题，因为它既是一种对非阿基米德量建模的简单而有效的方法，也是一种允许在它们之间执行数值计算的框架。此外，我们可以在未来使用 Sergeyev 本人拥有专利的无穷大计算机在硬件中执行相同的计算。在使用格罗松方法论为纯囚徒困境和非纯囚徒困境创建理论模型后，我们在数值上复制了与我们的两个新模型相关的图表，使用 Infinity 计算机的 Matlab 模拟器。最后，我们证明了模拟图的一些理论性质。因此，我们的工具已准备好帮助建模者解决非阿基米德纯/不纯囚徒困境模型提供的对现实的良好描述的所有问题：能源市场建模、国际贸易建模、政治合并过程等。