Abstract
In this work, a generalization of both Pure and Impure iterated Prisoner’s Dilemmas is presented. More precisely, the generalization concerns the use of non-Archimedean quantities, i.e., payoffs that can be infinite, finite or infinitesimal and probabilities that can be finite or infinitesimal. This new approach allows to model situations that cannot be adequately addressed using iterated games with purely finite quantities. This novel class of models contains, as a special case, the classical known ones. This is an important feature of the proposed methodology, which assures that we are proposing a generalization of the already known games. The properties of the generalized models have also been validated numerically, by using a Matlab simulator of Sergeyev’s Infinity Computer.
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References
Benci, V., Horsten, L., Wenmackers, S.: Infinitesimal probabilities. Br. J. Philos. Sci. 69(2), 509–552 (2018)
Cococcioni, M., Cudazzo, A., Pappalardo, M., Sergeyev, Y.D.: Solving the lexicographic mixed-integer linear programming problem using Branch-and-Bound and Grossone methodology. Commun. Nonlinear Sci. Numer. Simul. 84, 105177 (2020). https://doi.org/10.1016/j.cnsns.2020.105177
Cococcioni, M., Pappalardo, M., Sergeyev, Y.D.: Towards lexicographic multi-objective linear programming using grossone methodology. In: Sergeyev, Y.D., Kvasov, D., Dell’Accio, F., Mukhametzhanov, M. (eds.) Proceedings of the 2nd International Conference “Numerical Computations: Theory and Algorithms”, vol. 1776, p. 090040. AIP Publishing, New York (2016)
Cococcioni, M., Pappalardo, M., Sergeyev, Y.D.: Lexicographic multi-objective linear programming using grossone methodology: theory and algorithm. Appl. Math. Comput. 318, 298–311 (2018)
Fiaschi, L., Cococcioni, M.: Numerical asymptotic results in game theory using sergeyev’s infinity computing. Int. J. Unconv. Comput. 14(1), 1–25 (2018)
Fiaschi, L., Cococcioni, M.: Non-archimedean game theory: a numerical approach. Appl. Math. Comput. (submitted)
Fudenberg, D., Tirole, J.: Game Theory. MIT Press, Cambridge (1991)
Gale, D., Stewart, F.M.: Infinite Games with Perfect Information, vol. 2. Princeton University Press, Princeton (1953)
Hilbe, C., Traulsen, A., Sigmund, K.: Partners or rivals? Strategies for the iterated prisoner’s dilemma. Games Econ. Behav. 92(Suppl. C), 41–52 (2015)
Lai, L., Fiaschi, L., Cococcioni, M.: Solving mixed pareto-lexicographic multi-objective optimization problems: the case of priority chains (submitted)
Lambertini, L.: Prisoners’ dilemma in duopoly (super) games. J. Econ. Theory 77, 181–191 (1997)
Lambertini, L., Sasaki, D.: Optimal punishments in linear duopoly supergames with product differentiation. J. Econ. 69(2), 173–188 (1999)
Rizza, D.: A study of mathematical determination through bertrand’s paradox. Philos. Math. 26(3), 375–395 (2017)
Scott, D.: Continuous lattices. In: Lawvere, F.W. (ed.) Toposes, Algebraic Geometry and Logic. LNM, vol. 274, pp. 97–136. Springer, Heidelberg (1972). https://doi.org/10.1007/BFb0073967
Sergeyev, Y.D.: Numerical infinities and infinitesimals: methodology, applications, and repercussions on two Hilbert problems. EMS Surv. Math. Sci. 4, 219–320 (2017)
Sergeyev, Y.D.: Independence of the grossone-based infinity methodology from non-standard analysis and comments upon logical fallacies in some texts asserting the opposite. Found. Sci. 24, 153–170 (2019)
Sergeyev, Y.D., Mukhametzhanov, M., Mazzia, F., Iavernaro, F., Amodio, P.: Numerical methods for solving initial value problems on the infinity computer. Int. J. Unconv. Comput. 12(1), 3–23 (2016)
Steven, K.: Prisoner’s Dilemma. The Stanford Encyclopedia of Philosophy, Spring 2017. https://plato.stanford.edu/archives/spr2017/entries/prisoner-dilemma/
Vickers, S.: Topology via Logic. Cambridge Tracts in Theoretical Computer Science, vol. 5. Cambridge University Press, Cambridge (1989)
Zeng, W., Li, M., Chen, F., Nan, G.: Risk consideration and cooperation in the iterated prisoner’s dilemma. Soft. Comput. 20(2), 567–587 (2016)
Acknowledgment
Work 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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Fiaschi, L., Cococcioni, M. (2020). Generalizing Pure and Impure Iterated Prisoner’s Dilemmas to the Case of Infinite and Infinitesimal Quantities. In: Sergeyev, Y., Kvasov, D. (eds) Numerical Computations: Theory and Algorithms. NUMTA 2019. Lecture Notes in Computer Science(), vol 11974. Springer, Cham. https://doi.org/10.1007/978-3-030-40616-5_32
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