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Formalization of information: knowledge and belief

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  • Jong Jae Lee

    (Wuhan University)

Abstract

Billingsley (Probability and measure, Wiley, New Jersey, 1995) and Dubra and Echenique (Math Soc Sci 47(2):177–185, 2004) provide an example to show that the formalization of information by $$\sigma $$ σ -algebras and by partitions need not be equivalent. Although Hervés-Beloso and Monteiro (Econ Theory 54(2):405–418, 2013) provide a method to generate a $$\sigma $$ σ -algebra from a partition and another method for going in the opposite direction, we show that their two methods are in fact based on two different notions of information: (i) information as belief, (ii) information as knowledge. If information is conceived to allow for falsehoods, case (i) above, the equivalence between $$\sigma $$ σ -algebras and partitions holds after applying the notion of posterior completion suggested by Brandenburger and Dekel (J Math Econ 16(3):237–245, 1987). If information is conceived not to allow for falsehoods, case (ii) above, the equivalence holds only for measurable partitions and countably generated $$\sigma $$ σ -algebras.

Suggested Citation

  • Jong Jae Lee, 2018. "Formalization of information: knowledge and belief," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 66(4), pages 1007-1022, December.
    • Handle: RePEc:spr:joecth:v:66:y:2018:i:4:d:10.1007_s00199-017-1078-4
    DOI: 10.1007/s00199-017-1078-4
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    References listed on IDEAS

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    1. Robert J. Aumann, 1999. "Interactive epistemology II: Probability," International Journal of Game Theory, Springer;Game Theory Society, vol. 28(3), pages 301-314.
    2. Robert J. Aumann, 1999. "Interactive epistemology I: Knowledge," International Journal of Game Theory, Springer;Game Theory Society, vol. 28(3), pages 263-300.
    3. Bacharach, Michael, 1985. "Some extensions of a claim of Aumann in an axiomatic model of knowledge," Journal of Economic Theory, Elsevier, vol. 37(1), pages 167-190, October.
    4. Brandenburger, Adam & Dekel, Eddie, 1987. "Common knowledge with probability 1," Journal of Mathematical Economics, Elsevier, vol. 16(3), pages 237-245, June.
    5. Joseph Y. Halpern, 1999. "Hypothetical knowledge and counterfactual reasoning," International Journal of Game Theory, Springer;Game Theory Society, vol. 28(3), pages 315-330.
    6. Nabil I. Al-Najjar, 2009. "Decision Makers as Statisticians: Diversity, Ambiguity, and Learning," Econometrica, Econometric Society, vol. 77(5), pages 1371-1401, September.
    7. Dubra, Juan & Echenique, Federico, 2004. "Information is not about measurability," Mathematical Social Sciences, Elsevier, vol. 47(2), pages 177-185, March.
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    Cited by:

    1. Fukuda, Satoshi, 2019. "Epistemic foundations for set-algebraic representations of knowledge," Journal of Mathematical Economics, Elsevier, vol. 84(C), pages 73-82.
    2. Áron Tóbiás, 2021. "A unified epistemological theory of information processing," Theory and Decision, Springer, vol. 90(1), pages 63-83, February.
    3. Áron Tóbiás, 2023. "Cognitive limits and preferences for information," Decisions in Economics and Finance, Springer;Associazione per la Matematica, vol. 46(1), pages 221-253, June.
    4. Áron Tóbiás, 2021. "Meet meets join: the interaction between pooled and common knowledge," International Journal of Game Theory, Springer;Game Theory Society, vol. 50(4), pages 989-1019, December.

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    More about this item

    Keywords

    Information; $$sigma $$ σ -Algebras; S5 knowledge; KD45 belief; Formalization of information;
    All these keywords.

    JEL classification:

    • C60 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - General
    • C70 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - General

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