A Categorical Characterization of a ①-Iteratively Defined State of Common Knowledge

Authors

  • Fernando Tohmé Universidad Nacional Del Sur, Department of Economics Bahia Blanca, Argentina
  • Gianluca Caterina Center for Diagrammatic and Computational Philosophy, Endicott College, Beverly, MA, U.S.A. https://orcid.org/0000-0002-5677-2232
  • Rocco Gangle Center for Diagrammatic and Computational Philosophy, Endicott College, Beverly, MA, U.S.A. https://orcid.org/0000-0003-0545-9278

DOI:

https://doi.org/10.21638/11701/spbu31.2021.24

Abstract

We present here a novel approach to the analysis of common knowledge based on category theory. In particular, we model the global epistemic state for a given set of agents through a hierarchy of beliefs represented by a presheaf construction. Then, by employing the properties of a categorical monad, we prove the existence of a state, obtained in an iterative fashion, in which all agents acquire common knowledge of some underlying statement. In order to guarantee the existence of a fixed point under certain suitable conditions, we make use of the properties entailed by Sergeyev's numeral system called grossone, which allows a finer control on the relevant structure of the infinitely nested epistemic states.

Keywords:

common knowledge, category theory, grossone

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References

Aumann, R. (1976). Agreeing to Disagre. Annals of Statistics, 4, 1236–1239

Barwise, J. (1988). Three Views of Common Knowledge, in M. Vardi (ed.), Proceedings of the Second Conference on Theoretical Aspects of Reasoning About Knowledge, Morgan Kaufman, pp. 365–379

Barwise, J. and Moss, L. (1996). Vicious Circles: On the Mathematics of Non-Wellfounded Phenomena. CSLI Publications

Bicchieri, C. (1993). Rationality and Coordination. Cambridge University Press

Bonanno, G. and Battigalli, P. (1999). Recent Results on Belief, Knowledge and the Epistemic Foundations of Game Theory. Research in Economics, 53, 149–225

Brandenburger, A. and Dekel, E. (1987). Common Knowledge with Probability. Journal of Mathematical Economics, 16, 237–245

Brandom, R. (2019). A Spirit of Trust: A Reading of Hegel's Phenomenology, Belknap/Harvard University Press

Echenique, F. (2005). A Short and Constructive Proof of Tarski's Fixed-Point Theorem. International Journal of Game Theory, 33, 215–218

Fagin, R., Halpern, J., Moses, Y. and Vardi, M. (1995). Reasoning About Knowledge. MIT Press

Geanakoplos, J. (1994). Common Knowledge, in R. Aumann and S. Hart (eds.), Handbook of Game Theory (Volume 2), North-Holland, pp. 1438–1496

Heifetz, A. (1999). Iterative and Fixed Point Common Belief. Journal of Philosophical Logic, 28, 61–79

Hintikka, J. (1962). Knowledge and Belief. Cornell University Press

Johnston, P. (2002). Sketches of an Elephant: A Topos Theory Compendium. Oxford University Press

Manchak, J.B. and Roberts, B. W.: Supertasks, in E. Zalta (ed.) The Stanford Encyclopedia of Philosophy (Winter 2016 Edition), https://plato.stanford.edu/archives/win2016/entries /spacetime-supertasks/

Lewis, D. (1969). Convention: A Philosophical Study, Harvard University Press

Lolli, G. (2015). Metamathematical investigations on the theory of Grossone. Applied Mathematics and Computation, 255, 3–14

Mac Lane, S. (2013). Categories for the Working Mathematician, Springer-Verlag

Rizza, D. (2018). A Study of Mathematical Determination through Bertrand's Paradox. Philosophia Mathematica, 26, 375–395

Rizza, D. (2019). Numerical Methods for Infinite Decision-making Processes. Int. Journal of Unconventional Computing, 14, 139–158

Manes, E. and Arbib, M. (1986). Algebraic Approaches to Program Semantics. Springer-Verlag

Meyer, J.-J.Ch. and van der Hoek, W. (1995). Epistemic Logic for Computer Science and Artificial Intelligence. Cambridge University Press.

Milewski, B. (2017). Category Theory for Programmers. Creative Commons

Moss, L. and Viglizzo, I. (2004). Harsanyi Type Spaces and Final Coalgebras constructed from Satisfied Theories. Electronic Notes in Theoretical Computer Science, 106, 279–295

Osborne, M. and Rubinstein, A. (1994). A Course in Game Theory. MIT Press

Sergeyev, Y. (2008). A New Applied Approach for Executing Computations with Infinite and Infinitesimal Quantities. Informatica, 19, 567–596

Sergeyev, Y. (2013). Arithmetic of Infinity. Edizioni Orizzonti Meridionali, 2nd edition

Sergeyev, Y. (2020). Novel local tuning techniques for speeding up one-dimensional algorithms in expensive global optimization using Lipschitz derivatives. Journal of Computational and Applied Mathematics, 383(1), 113–134

Sergeyev, Y. (2017). Numerical Infinities and Infinitesimals: Methodology, Applications, and Repercussions on Two Hilbert Problems. EMS Surveys in Mathematical Sciences, 4, 219–320

Spivak, D. (2014). Category Theory for the Sciences. MIT Press

Tohmé, F., Caterina, G. and Gangle, R. (2020). Computing Truth Values in the Topos of Infinite Peirce's α-Existential Graphs. Applied Mathematics and Computation, in press

Vassilakis, S. (1992). Some Economic Applications of Scott Domains. Mathematical Social Sciences, 24, 173–208

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Published

2021-10-30

How to Cite

Tohmé, F., Caterina, G., & Gangle, R. (2021). A Categorical Characterization of a ①-Iteratively Defined State of Common Knowledge. Contributions to Game Theory and Management, 14, 329–341. https://doi.org/10.21638/11701/spbu31.2021.24

Issue

Vol. 14 (2021)

Section

Articles

License

Articles of "Contributions to Game Theory and Management" are open access distributed under the terms of the License Agreement with Saint Petersburg State University, which permits to the authors unrestricted distribution and self-archiving free of charge.