Non-Archimedean game theory: A numerical approach
Introduction
Game Theory (GT) is the study of mathematical models of strategic interaction between rational decision-makers. One of the most known and simplest models is the so called Prisoner’s Dilemma (PD), along with its variants (e.g, the Iterated Prisoner’s Dilemma, IPD). Usually, the payoffs in a PD are assumed to be Archimedean, i.e., commensurable to each other. There are situations, however, where such an assumption can be too strong, or, put another way, it could be helpful or useful to relax this constraint, allowing payoffs to span different orders of magnitude. For instance, we could be interested in separating the order of magnitude of a payoff related to physical things (e.g., gaining/loosing value on owned stock options from loosing owns life or that of a relative). The same could be said for the subjective probability attached to an event. We might want to distinguish probable events (i.e., those ones that have a positive, finite probability in (0,1] ) from improbable events (i.e., those ones that have, in comparison, an infinitely smaller probability to happen). Such events are sometimes called black swans or perfect storms.
From a theoretical point of view, a lot of work has already been done in non-Archimedean quantities modeling. For example, the concept of infinitesimal probability is now formally well established [1], and we may think to use it in an IPD, provided the capability to handle them not only from the theoretical side but also on the operational side (i.e., on a computer). Indeed, non-standard analysis (NSA, [2]) would be enough to extend GT to the case of non-Archimedean quantities. However, such an extension would be only theoretical and thus of none help for empirical and pragmatical studies. In this work, instead, we aim at doing it operationally and numerically, by devising a software library able to verify if the theoretical results are true. This goal can be achieved using Sergeyev’s Infinity Computer [3], a patented computer able to handle numerically arithmetic operations between non-Archimedean quantities. In a recent work [4] we have already extended PD tournaments using Sergeyev’s approach, showing how new PD tournaments can be conceived and solved using infinitesimal numbers. In the present work we are going to make a different generalization of PDs, namely, we are going to generalize Pure and Impure IPD games (see Section 2 for their formal introduction) to the case of non-Archimedean quantities. In particular, we will extend the well-known results in both Pure and Impure IPDs when i) the payoffs are infinite, finite or infinitesimal and ii) the probabilities of cooperation are finite or infinitesimal. The numerical results attained by an Infinity Computer simulator implemented in Matlab are perfectly in line with the literature and with our theoretical speculations, paving the way for a direct application of such new family of models. The results obtained in the present work have already been partially presented at NUMTA’19 conference [5].
Section snippets
Pure and impure PDs: Standard formulation
In the most general form of the PD, two players (℘1 and ℘2) have to independently and simultaneously decide whether to cooperate (C) or to defect (D) [6], [7]. The first player will get one of the following payoffs or depending on his and the other player’s choice. Similarly, the second player will get or . In this work we assume the game is symmetric, i.e., etc. In addition, a game is called PD if the payoffs follow the relation T > R > P
The infinity computer and the grossone methodology
In 2003 Sergeyev introduced the Arithmetic of Infinity (AoI) [28], a novel way to numerically deal with and characterize infinite and infinitesimal quantities. More recent references on the methodology are [29], [30], while [28] contains an introduction to the topic written in a popular way. It is also worth to mention since the beginning that the numerical aspect of this methodology and its formal approach put it apart from NSA, as discussed in [31].
This computational methodology has already
Non-Archimedean quantities: Their meaning and their modelling power
In this section we aim to provide an overview of what a non-Archimedean quantity may mean and what are the advantages coming from a non-Archimedean approach. Within GT, non-Archimedean quantities can be used to model probabilities and payoffs.
Extending pure and impure IPDs to the case of non-Archimedean quantities
The goal of this section and that of the next one is to extend the Pure and Impure IPDs to cases where infinite and infinitesimal quantities are involved, i.e., when we let the payoffs to be infinite, finite, infinitesimal and the probabilities to be finite and infinitesimal. By implementing such an extension, we are able to show that each player’s expectation can be either finite, infinite or infinitesimal, i.e, Ec (see Section 2) can have components with different orders of magnitude. This
Theoretical properties of the obtained diagrams
In this section we prove three properties of the two diagrams of the G-PIPD and G-IIPD games. The results have been proved only for the payoffs and probabilities configuration we used in the simulations. With a little more effort and formalism the demonstrations can be generalized to any possible configuration.
Possible economic applications
In this section we want to focus our attention on the possible existence of real competitive scenarios involving non-Archimedean quantities. Such scenarios would be better modeled and studied by means of the theory presented in this work, both from a theoretical and numerical standpoint.
Conclusions
In this work we have demonstrated the possibility to computationally and numerically deal with Pure and Impure IPDs when payoffs and probabilities are not required to be finite numbers. In particular, we have been able to provide a more general theory for both Pure and Impure IPDs which includes, as a special case, the standard IPD theory. The theoretical consistency of the proposed extended theory has been proved and validated numerically in Matlab using an Infinity Computer simulator. With
Acknowledgments
Work partially supported by the Italian Ministry of Education and Research (MIUR) in the framework of the CrossLab project (Departments of Excellence) and partially supported by the University of Pisa funded project PRA_2018_81 “Wearable sensor systems: personalized analysis and data security in healthcare”.
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