Geodesic
A categorical characterization of a~~~${\scriptsize1}\!\!\!\!\bigcirc$-iteratively defined state of common knowledge
Contributions to game theory and management, Tome 14 (2021), pp. 329-341.

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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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Fernando Tohmé; Gianluca Caterina; Rocco Gangle. A categorical characterization of a~~~${\scriptsize1}\!\!\!\!\bigcirc$-iteratively defined state of common knowledge. Contributions to game theory and management, Tome 14 (2021), pp. 329-341. http://geodesic.mathdoc.fr/item/CGTM_2021_14_a24/

[1] Aumann R., “Agreeing to Disagre”, Annals of Statistics, 4 (1976), 1236–1239 | MR | Zbl

[2] Barwise J., “Three Views of Common Knowledge”, Proceedings of the Second Conference on Theoretical Aspects of Reasoning About Knowledge, ed. M. Vardi, Morgan Kaufman, 1988, 365–379 | MR

[3] Barwise J., Moss L., Vicious Circles: On the Mathematics of Non-Wellfounded Phenomena, CSLI Publications, 1996 | MR | Zbl

[4] Bicchieri C., Rationality and Coordination, Cambridge University Press, 1993 | MR | Zbl

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

[6] Brandenburger A., Dekel E., “Common Knowledge with Probability $1$”, Journal of Mathematical Economics, 16 (1987), 237–245 | MR | Zbl

[7] Brandom R., A Spirit of Trust: A Reading of Hegel's Phenomenology, Belknap/Harvard University Press, 2019 | MR

[8] Echenique F., “A Short and Constructive Proof of Tarski's Fixed-Point Theorem”, International Journal of Game Theory, 33 (2005), 215–218 | MR | Zbl

[9] Fagin R., Halpern J., Moses Y., Vardi M., Reasoning About Knowledge, MIT Press, 1995 | MR | Zbl

[10] Geanakoplos J., “Common Knowledge”, Handbook of Game Theory, v. 2, eds. R. Aumann, S. Hart, North-Holland, 1994, 1438–1496 | MR

[11] Heifetz A., “Iterative and Fixed Point Common Belief”, Journal of Philosophical Logic, 28 (1999), 61–79 | MR | Zbl

[12] Hintikka J., Knowledge and Belief, Cornell University Press, 1962

[13] Johnston P., Sketches of an Elephant: A Topos Theory Compendium, Oxford University Press, 2002 | MR

[14] Manchak J.B., Roberts B.W., “Supertasks”, The Stanford Encyclopedia of Philosophy, Winter 2016 Edition, ed. E. Zalta, 2016 https://plato.stanford.edu/archives/win2016/entries/spacetime-supertasks/ | Zbl

[15] Lewis D., Convention: A Philosophical Study, Harvard University Press, 1969

[16] Lolli G., “Metamathematical investigations on the theory of Grossone”, Applied Mathematics and Computation, 255 (2015), 3–14 | MR | Zbl

[17] Mac Lane S., Categories for the Working Mathematician, Springer-Verlag, 2013 | MR

[18] Rizza D., “A Study of Mathematical Determination through Bertrand's Paradox”, Philosophia Mathematica, 26 (2018), 375–395 | MR | Zbl

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

[20] Manes E., Arbib M., Algebraic Approaches to Program Semantics, Springer-Verlag, 1986 | MR | Zbl

[21] Meyer J.-J.Ch., van der Hoek W., Epistemic Logic for Computer Science and Artificial Intelligence, Cambridge University Press, 1995 | MR

[22] Milewski B., Category Theory for Programmers, Creative Commons, 2017

[23] Moss L., Viglizzo I., “Harsanyi Type Spaces and Final Coalgebras constructed from Satisfied Theories”, Electronic Notes in Theoretical Computer Science, 106 (2004), 279–295 | MR | Zbl

[24] Osborne M., Rubinstein A., A Course in Game Theory, MIT Press, 1994 | MR | Zbl

[25] Sergeyev Y., “A New Applied Approach for Executing Computations with Infinite and Infinitesimal Quantities”, Informatica, 19 (2008), 567–596 | MR | Zbl

[26] Sergeyev Y., Arithmetic of Infinity, 2nd edition, Edizioni Orizzonti Meridionali, 2013 | MR

[27] Sergeyev Y., “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 (2020), 113–134 | MR

[28] Sergeyev Y., “Numerical Infinities and Infinitesimals: Methodology, Applications, and Repercussions on Two Hilbert Problems”, EMS Surveys in Mathematical Sciences, 4 (2017), 219–320 | MR | Zbl

[29] Spivak D., Category Theory for the Sciences, MIT Press, 2014 | MR | Zbl

[30] Tohmé F., Caterina G., Gangle R., “Computing Truth Values in the Topos of Infinite Peirce's $\alpha$-Existential Graphs”, Applied Mathematics and Computation, 2020 (to appear) | MR

[31] Vassilakis S., “Some Economic Applications of Scott Domains”, Mathematical Social Sciences, 24 (1992), 173–208 | MR | Zbl