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 \(\sigma \)-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 \(\sigma \)-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 \(\sigma \)-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 \(\sigma \)-algebras.
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Notes
For a pair of partitions, the strictly finer partition distinguishes more elements, implying that a DM can say more accurately about the true state (the state in which she lies). For a pair of \(\sigma \)-algebras, the larger one contains more sets. For larger number of sets, a DM is able to say whether it contains the true state or not, thus having more information.
By the smallest \(\sigma \)-algebra generated by the partition, we mean that the \(\sigma \)-algebra contains all the complements and the countable unions of partition cells.
A countably generated \(\sigma \)-algebra is the smallest \(\sigma \)-algebra generated by a collection of countably many subsets of the state space.
A null set is a set to which a DM ascribes zero probability. HM refers to it as a negligible set of states.
The existence of non-measurable sets can be addressed by Theorem 4 and the following Remark 3 in HM. However, the notion of an informed set, as it is defined in HM, fails to accommodate this: the collection of informed sets in Billingsley’s example, according to HM, is the power set even when the underlying \(\sigma \)-algebra is strongly Blackwell (See Example 4 in HM). In fact, we propose the notion of an informed event to accommodate Theorem 4 and Remark 3 in HM.
The posterior completion of a \(\sigma \)-algebra is to create the smallest \(\sigma \)-algebra by adding events that are either measure zero or one with a proper regular conditional probability measure, into a given \(\sigma \)-algebra. The posterior completion of a partition is to add in those events to the partition.
A \(\sigma \)-algebra is a strongly Blackwell \(\sigma \)-algebra if it is separable and every two countably generated sub-\(\sigma \)-algebras with the same atom coincide.
Interested readers may see, for example, Bacharach (1985).
This term originates in Aumann (1999a). Whenever a self-evident event occurs, it informs the DM of its occurrence. The self-evident event, therefore, is the knowledge about itself.
The collection of informed sets suggested by HM consists of \(\phi \)-invariant subsets of \(\varOmega \). The collection includes non-measurable subsets, and the collection of informed events excludes those non-measurable subsets as it is obvious from Lemma 3.
This reveals why we need to restrict \(\mathcal {F}\) to be a Borel \(\sigma \)-algebra, instead of being a strongly Blackwell \(\sigma \)-algebra in this subsection. If \(\mathcal {F}\) is not a Borel \(\sigma \)-algebra, a proper regular conditional probability may not exist. See Shortt (1984).
The equivalence relation \(\sim \) between any two partitions \(\varPi \) and \(\varPi '\) is defined so that the smallest \(\sigma \)-algebras generated by these partitions, denoted by \(\sigma (\varPi )\) and \(\sigma (\varPi ')\), have the same posterior-completed \(\sigma \)-algebra, i.e., \(\sigma (\varPi )\sim \sigma (\varPi ')\).
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I am deeply grateful to M. Ali Khan for his guidance and support. I am also thankful to Adam Brandenburger and Juan Dubra for discussion and encouragement during their stimulating visits to Johns Hopkins. This version of the paper has benefited from the careful reading and constructive comments of the three anonymous referees of the journal.
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Lee, J.J. Formalization of information: knowledge and belief. Econ Theory 66, 1007–1022 (2018). https://doi.org/10.1007/s00199-017-1078-4
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DOI: https://doi.org/10.1007/s00199-017-1078-4