Abstract
What does it mean for an agent faced with choice under uncertainty to “know” something? While a variety of mathematical methods are available to construct formal models to answer this question, the combination of different approaches may lead to unsettling paradoxes. I propose a unified theory that eliminates such inconsistencies by relying on a sharp conceptual distinction between information the decision-maker observes and how much of that information she can cognitively process. The resulting model allows for natural decision-theoretic characterizations of comparing different amounts of information.
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Notes
Recall that a \(\sigma\)-algebra (or \(\sigma\)-field) on a non-empty set is a non-empty collection of subsets of the set that is closed under complementation and countable unions.
The findings of Dubra and Echenique (2004) make the authors so skeptical about \(\sigma\)-algebras as an epistemological model as to claim already in the title of their paper that “[i]nformation is not about measurability.”
Many random phenomena require a model involving large state spaces, making the concern about uncountability a relevant one. For example, Aumann (1999a) shows that a satisfactory canonical model of interactive knowledge featuring more than one agent requires a state space that has the cardinality of at least the continuum, even if there are only finitely many elementary occurrences of interest.
The state space is said to be a Polish space if it is endowed with a separable and completely metrizable topological structure. A \(\sigma\)-algebra is defined to be a strongly Blackwell \(\sigma\)-algebra if (i) it is countably generated; (ii) it contains the singletons; and (iii) any two countably generated \(\sigma\)-subalgebras associated with the same partition coincide.
Note that Definition 1 imposes no a priori measure-theoretic structure on partitions. In particular, each cell of a given partition is allowed to be any non-empty subset of the state space—it need not be measurable with respect to any given \(\sigma\)-algebra.
Indeed, even though the partition \({\mathscr {P}}\) is finer than \({\mathscr {Q}}\) in Example 2, the \(\sigma\)-algebra \(\sigma ({\mathscr {P}})\) does not contain (nor is it contained in) \(\sigma ({\mathscr {Q}})\).
At the extreme, if \({\mathscr {P}}\) is the discrete partition, then \({\mathscr {U}}({\mathscr {P}})\) coincides with the power set \(2^{\Omega }\). If one accepts the continuum hypothesis, then there exists no probability measure on \((\mathbb {R},2^{\mathbb {R}})\) for which the measure of each singleton vanishes (Ulam 1930; see also Remark 2 and Theorem 2 in Hervés-Beloso and Monteiro 2013).
Roughly speaking, if \(E\in {\mathscr {M}}\), then the observer is capable of comprehending in a meaningful way the notion of whether the true state of the world \(\omega\) in \(\Omega\) is contained in the set E, whereas if \(E\notin {\mathscr {M}}\), then the structure of the set E is too complicated for the observer to grasp the sheer concept of whether \(\omega \in E\) or \(\omega \in E^{\mathsf c}\).
This procedure can be regarded as the dual operation of the usual method of generating a \(\sigma\)-algebra on the domain of a function given a \(\sigma\)-algebra on its codomain—in this case, the \(\sigma\)-algebra \({\mathscr {M}}\) is given on the domain \(\Omega\) of f, and \(f({\mathscr {M}})\) is newly constructed on the codomain Y (see also Billingsley 1995, p. 186).
It is worth mentioning in this context the characterization by Fukuda (2019) of when an observer’s knowledge corresponds to a \(\sigma\)-algebra. More precisely, consider an operator \(K:{\mathscr {M}}\rightarrow {\mathscr {M}}\) satisfying the axioms of (i) truth: \(K(E)\subseteq E\) for every \(E\in {\mathscr {M}}\) (this is the same condition as non-delusion according to the terminology of Lee 2018—see Definition 7); (ii) monotonicity: \(E\subseteq F\) implies \(K(E)\subseteq K(F)\) for every \(E,F\in {\mathscr {M}}\); and (iii) positive introspection: \(K(E)\subseteq K(K(E))\) for every \(E\in {\mathscr {M}}\). Then the family \(\{E\in {\mathscr {M}}\,|\,E=K(E)\}\) of self-evident events forms a \(\sigma\)-subalgebra of \({\mathscr {M}}\) if and only if the axiom of (iv) negative introspection: \(K(E)^{\mathsf c}\subseteq K(K(E)^{\mathsf c})\) for every \(E\in {\mathscr {M}}\) is also satisfied.
As Dubra and Echenique (2004) argue, a reasonable mathematical model of knowledge should be closed under arbitrary unions, not just under countable ones, in order to faithfully preserve the decision-maker’s information. The concept of generated information satisfies a weaker version of this natural requirement: the decision-maker “knows” any event expressible as an arbitrary union of partition cells, subject to measurability with respect to \({\mathscr {M}}\).
It is natural to take the family of Borel sets as the base \(\sigma\)-algebra in this context, as it is the smallest \(\sigma\)-algebra on state space (0,1] that contains its subintervals. This corresponds to the implicit assumption that the observer is cognitively capable of performing ordinal comparisons on the unit interval.
By Proposition 4 in Hervés-Beloso and Monteiro (2013), a sufficient condition for \(\sigma ({\mathscr {P}})=\Sigma ({\mathscr {P}}|{\mathscr {M}})\) is that (i) the partition cells be measurable: \({\mathscr {P}}\subseteq {\mathscr {M}}\); and (ii) \({\mathscr {P}}\) be countable.
It should be mentioned that in their Proposition 2, Hervés-Beloso and Monteiro (2013) characterize the interaction of joins of partitions and the informed-set operation in the following way:
$$\begin{aligned} {\mathscr {U}}\left( \bigvee _{i\in I}{\mathscr {P}}_{(Y_i,f_i)}\right) =\sigma ^{\mathsf u}\left( \bigcup _{i\in I}{\mathscr {U}}\left( {\mathscr {P}}_{(Y_i,f_i)}\right) \right) , \end{aligned}$$where \(\sigma ^{\mathsf u}(\cdot )\) denotes the smallest \(\sigma\)-algebra that contains a given family of sets and is closed under arbitrary (as opposed to countable) unions.
I thank a referee for suggesting this example.
Cf. the remark by Dubra and Echenique (2004, p. 183): “different partitions generate different \(\sigma\)-algebras, but different \(\sigma\)-algebras may generate the same partition.”
See also Proposition 5 of Hervés-Beloso and Monteiro (2013).
It is interesting to mention that if such a function \(\ell :Y\rightarrow Z\) satisfying \(g=\ell \circ f\) exists, then it is automatically \(f({\mathscr {M}})/g({\mathscr {M}})\)-measurable for any base \(\sigma\)-algebra \({\mathscr {M}}\), where the \(\sigma\)-algebra \(f({\mathscr {M}})\) on Y and the \(\sigma\)-algebra \(g({\mathscr {M}})\) on Z are as defined in (3). I am grateful to a referee for pointing this out.
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Acknowledgements
I am very grateful to the Associate Editor and three referees for their excellent comments and suggestions, which have led to substantial improvement of the paper. All remaining errors and omissions are mine.
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Tóbiás, Á. A unified epistemological theory of information processing. Theory Decis 90, 63–83 (2021). https://doi.org/10.1007/s11238-020-09769-x
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DOI: https://doi.org/10.1007/s11238-020-09769-x