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L. Fiaschi, M. Cococcioni. "Generalizing Pure and Impure Iterated Prisoner's Dilemmas to the Case of Infinite and Infinitesimal Quantities", Proc. of 3rd Int. Conf. on Numerical Computations: Theory and Algorithms (NUMTA'19), July 2019.

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In this work we aim at generalizing Pure and Impure PD iterated games when using finite/infinite/infinitesimal payoffs and finite/infinitesimal probabilities, i.e., dealing with non-Archimedean quantities (realm so far investigated only on the theoretical side). In our work we are able to generalize Pure and Impure iterated PD games, both from the theoretical and operational standpoints, by means of the Sergeyev’s Infinity Computer and his Grossone Methodology. Indeed, the theoretical novelties introduced have been also validated numerically in Matlab, by means of an Infinity Computer simulator. Pragmatically, we have analytically proved that in Pure and Impure Iterated PD games involving non-Archimedean quantities, if the latter have different orders of infinite than the magnitude of the players average expectations per iteration can span all such orders (i.e., can be either infinite, finite or infinitesimal) depending on the adopted strategies. Moreover, exploiting the Infinity Computer simulator we have plot such expectations for both the PD scenarios, numerically validating the theoretical results. Finally, we have shown that the new graphic thus obtained is a generalization of the finite case, proving the maintaining of the requested properties of linearity, continuity and proper slope of its edges. This new approach opens the door to a more precise modeling of non-Archimedean scenarios, as demonstrated in a previous work of the same authors. Here, the existence and the relative numerical treatment of a new class of PD Tournaments (which previous approaches were not able to discover, nor to manage) has been demonstrated by means of such finer modeling paradigm. 



Game Theory; Prisoner’s Dilemma; Non-Archimedean Payoffs and Probabilities; Infinity Computer; Grossone Methodology