KRIPKE SEMANTICS, COMMON KNOWLEDGE, BOUNDED RATIONALITY AND AUMANN’S AGREEMENT THEOREM
Keywords:
kripke semantics, common knowldedge, kantian equilibrium, nash equilibrium, agreement theorem, bounded rationality
Abstract
The first model with k-level thinking with Gaussian noise shows that small deviations for common knowledge led to “almost” common knowledge equilibria. The second model demonstrated the semantic economy idea: as agents exchange and adapt beliefs, they create shared informational value by reaching a consensus that reflects network-wide insights rather than mere individual optimization. Higher reasoning depth does not change Nash equilibria but shifts up Kantian beliefs. The shift up in Kantian beliefs suggests a greater alignment toward strategies that maximize collective welfare, rather than purely individualistic or competitive outcomes. This doesn't alter the Nash equilibrium, where players still act independently, but it emphasizes a higher baseline of cooperative or altruistic expectations among players due to more profound belief hierarchies in the reasoning process. In the third model: networked economic context where agents interact in an economic network where competitive advantage depends on the informational value generated across the network, results differ from the second example: Nash beliefs adjust based on others' best responses (shift up), while Kantian beliefs account for mutual benefit, dampening large shifts. Nash and Kantian equilibria differ when only three agents exist versus network economy.
References
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2. Aumann, R. J. (1976). Agreeing to Disagree. The Annals of Statistics, 4(6), 1236–1239. http://www.jstor.org/stable/2958591
3. Cooper, D.,Fatas, E., Morales, A. , Qi, S. (2024). Consistent Depth of Reasoning in Level-k Models. American Economic Journal: Microeconomics. 16. 40-76. 10.1257/mic.20210237.
4. Dickson,M.(2007). Non-relativistic quantum mechanics, In Handbook of the Philosophy of Science,Philosophy of Physics,North-Holland,Pages 275-415,
5. Fourny, G. (2018). Kripke Semantics of the Perfectly Transparent Equilibrium. 10.3929/ethz-b-000277118.
6. Geanakoplos, J. (1989/2021). Game Theory Without Partitions, and Applications to Speculation and Consensus.The B.E. Journal of Theoretical Economics, vol. 21, no. 2, 2021, pp. 361-394. https://doi.org/10.1515/bejte-2019-0010
7. Geanakoplos, J. (1992). Common Knowledge. The Journal of Economic Perspectives, 6(4), 53–82. http://www.jstor.org/stable/2138269
8. Harsanyi, J. C. (1967). Games with Incomplete Information Played by “Bayesian” Players, I-III. Part I. Part I. The Basic Model. Management Science, Vol 14 No 3 November, 1967
9. Harsanyi, J. C. (1968). Games with Incomplete Information Played by “Bayesian” Players, I-III. Part II. Bayesian Equilibrium Points. Management Science, 14(5), 320–334. http://www.jstor.org/stable/2628673
10. Harsanyi, J. C. (1968). Games with Incomplete Information Played by “Bayesian” Players, I-III. Part III. The Basic Probability Distribution of the Game. Management Science, 14(7), 486–502. http://www.jstor.org/stable/2628894
11. Hintikka, J. (1962). Knowledge and Belief. Ithaca, N.Y.: Cornell University Press
12. Ilik,D. Lee,G., Herbelin,H. (2010). Kripke Models for Classical Logic, Annals of Pure and Applied Logic, Volume 161, Issue 11, Pages 1367-1378.
13. Jech, T. (1997) Set Theory, 2nd ed. New York: Springer-Verlag.
14. Kreps, D. (1977).A Note on Fulfilled Expectations Equilibria, J.Econ. Theory.
15. Kripke, S.A. (1959). A completeness theorem in modal logic" J. Symbolic Logic , 24 (1959) pp. 1–14
16. Kripke, S.A. (1962). The undecidability of monadic modal quantification theory. Z. Math. Logik Grundl. Math., 8, pp. 113–116
17. Kripke, S.A. (1963), Semantical analysis of modal logic, I, Z. Math. Logik Grundl. Math., 9 pp. 67–96
18. Kripke, S.A. (1965). Semantical analysis of modal logic, II. J.W. Addison (ed.) L. Henkin (ed.) A. Tarski (ed.), The theory of models, North-Holland pp. 206–22.
19. Levin, J. (2016). Knowledge and Equilibrium. Online lecture notes at: https://web.stanford.edu/~jdlevin/Econ%20286/Knowledge%20and%20Equilibrium.pdf
20. Luce, R. D. (1959/2005) Individual Choice Behavior: A Theoretical Analysis. New York: Wiley. Reprinted by Dover Publications.
21. Luce, R. D. (1977). The choice axiom after twenty years. Journal of Mathematical Psychology, 15(3), 215–233.
22. McKenzie, Lionel W. (1954). On Equilibrium in Graham's Model of World Trade and Other Competitive Systems. Econometrica. 22 (2): 147–161.
23. Milgrom, P. and N. Stokey (1982). Information, Trade and Common Knowledge, J. Econ. Theory.
24. Myerson, R. (1991), Game Theory: The Analysis of Conflict, Harvard University Press, Cambridge, MA
25. Nagel, R. (1995). Unraveling in Guessing Games: An Experimental Study. The American Economic Review, 85(5), 1313–1326. http://www.jstor.org/stable/2950991
26. Osborne,M.J., Rubinstein,A.(2023). Models in Microeconomic Theory, (Expanded Second Edition), Cambridge, UK: Open Book Publishers
27. Rényi, A.(1955).On a new axiomatic theory of probability, Acta Mathematica, 6,pp.285–335
28. Roemer, J. E. (2010). Kantian Equilibrium. Scandinavian Journal of Economics, 112(1), 1–24.
29. Roemer, J. E. (2019).How We Cooperate: A Theory of Kantian Optimization. Yale University Press
30. Rubinstein, A. (1989). The Electronic Mail Game: Strategic Behavior Under “Almost Common Knowledge. The American Economic Review, 79(3), 385–391. http://www.jstor.org/stable/1806851
31. Rubinstein,A.(1998).Modeling Bounded Rationality, The MIT Press
32. Rubinstein,A.(2021).Modeling Bounded Rationality in Economic Theory: Four Examplesin Routledge Handbook of Bounded Rationality, edited by Riccardo Viale, Routledge , 423-435. pdf
33. Selten, R. (1998). Features of experimentally observed bounded rationality. European Economic Review, 42(3-5), 413–436. doi:10.1016/s0014-2921(97)00148-7
34. Simon, H.A.(1957). Models of Man: Social and Rational. Wiley, New York.
35. Stalnaker R (1994) On the evaluation of solution concepts. Theory and Decision 37(1):49–73, DOI 10.1007/BF01079205, URL https://doi.org/10.1007/BF01079205
36. Stone, Jon R. (1996). Latin for the Illiterati: Exorcizing the Ghosts of a Dead Language. London: Routledge.
37. Strzalecki, T. (2014). Depth of reasoning and higher order beliefs, Journal of Economic Behavior & Organization, Volume 108, Pages 108-122,
38. Tamuz,O.(2024). Undergraduate game theory lecture notes. California Institute of Technology caltech
39. Tirole, J. (1982).On the Possibility of Speculation under Rational Expectations, Econometrica.
40. Zermelo, E.(1930). Über Grenzzahlen und Mengenbereiche. Fund. Math. 16, pp. 29-47.
Published
2024-12-21
Section
Articles
