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Cognitive limits and preferences for information

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Abstract

The structure of uncertainty underlying certain decision problems may be so complex as to elude decision makers’ full understanding, curtailing their willingness to pay for payoff-relevant information—a puzzle manifesting itself in, for instance, low stock-market participation rates. I present a decision-theoretic method that enables an analyst to identify decision makers’ information-processing abilities from observing their preferences for information. A decision maker who is capable of understanding only those events that either almost always or almost never happen fails to attach instrumental value to any information source. On the other hand, non-trivial preferences for information allow perfect identification of the decision maker’s technological capacity.

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Change history

  • 06 June 2023

    typo in the text has been corrected.

Notes

  1. See, among others, Campbell (2006), Dhar and Zhu (2006), Calvet et al. (2007, 2009a, 2009b), Christiansen et al. (2008), Christelis et al. (2010), Kimball and Shumway (2010), Grinblatt et al. (2011, 2012), van Rooij et al. (2011), Klapper et al. (2013), Lusardi and Mitchell (2014), Jappelli and Padula (2015), and von Gaudecker (2015). Furthermore, The Economist (2020, p. 62) argues that private asset markets promise better returns than do public markets, but “data are hard to come by” and “assets are complex and value is hard to appraise” when it comes to private-equity funds, which is one of the reasons why they are utilized primarily by highly sophisticated institutional investors.

  2. De Oliveira et al. (2017) present an analysis that is similar in spirit, showing that the observing of a decision maker’s preferences over menus of state-contingent lotteries allows the identification of canonical costs of collecting information (modeled as noisy signals). Such information-cost functions, which the analyst usually cannot observe directly, are the methodological backbone of the rational-inattention literature originating in Sims (2003).

  3. See, among others, Aumann (1974, 1976, 1999a, 1999b), Wilson (1978), Milgrom (1981), Milgrom and Stokey (1982), Brandenburger and Dekel (1987), Gilboa and Lehrer (1991), Yannelis (1991), and Krishna and Morgan (2001).

  4. Aumann (1999a, 1999b) calls this formalism the semantic approach to modeling uncertainty.

  5. The fundamental issue with the \(\sigma \)-algebra \(\sigma ({\mathscr {P}})\) generated by the partition \({\mathscr {P}}\) is that its parsimoniousness leads to the omission of “too many” sets. As a result, \(\sigma ({\mathscr {P}})\) may fail to include important events that a satisfactory measure-theoretic model of knowledge of the partition \({\mathscr {P}}\) is supposed to include and enable the decision maker to cognitively process.

  6. See also Berti et al. (2020), who review certain well-known paradoxical features of proper regular conditional distributions (such as non-existence, to start with) and argue that such issues can be mitigated if the customary measurability requirement imposed on conditional probabilities is replaced with one that is formulated in terms of partitions rather than \(\sigma \)-algebras.

  7. As Lee (2018, p. 1013) puts it, a decision maker’s technological capacity can be conceived of as “the collection of sets of states which [she] is able to recognize.” Villegas (1964), Shafer (1986), and Heifetz et al. (2006) suggest similar interpretations of imposing limits on the family of sets that are to be considered as events from the decision maker’s point of view.

  8. Formally:

    $$\begin{aligned}{\mathscr {U}}({\mathscr {P}})\equiv \left\{ \bigcup _{P\in {\mathscr {R}}}P\, |\,{\mathscr {R}}\subseteq {\mathscr {P}}\right\} . \end{aligned}$$

    If \({\mathscr {R}}=\varnothing \), then the “empty union” is interpreted as the empty set by convention:

    $$\begin{aligned} \bigcup _{P\in \varnothing }P\equiv \varnothing . \end{aligned}$$
  9. Conversely, it is notoriously difficult to exhibit subsets of the unit interval that are not Borel measurable, and the description of such “pathological” sets invariably tends to be highly involved and non-constructive.

  10. Note that for a given decision problem \({\mathscr {D}}\equiv (A,{\mathscr {A}},u)\), the action space is endowed with the same \(\sigma \)-algebra \({\mathscr {A}}\) (as defined by the description of \({\mathscr {D}}\)) for any adapted action policy. Intuitively, even if there are cognitive limits to the decision maker’s understanding of the structure of uncertainty, no technological-capacity issues arise as to what she can do.

  11. That \({\mathscr {M}}_3\subseteq {\mathscr {M}}_4\) holds is because any countable subset of the unit interval has Lebesgue measure zero, and the inclusion is strict because there exist uncountable Borel-measurable subsets of the unit interval with Lebesgue measure zero—most notably, the Cantor set (Billingsley 1995, pp. 16, 50).

  12. This result strikes me as intuitively quite clear and simple—after all, if the decision maker’s mind is capable of processing only very simple events, then one expects also her actions conditional on those events to be very simple. Yet, somewhat ironically, I have found that the proof of Lemma 1 requires the use of heavy-duty measure-theoretic machinery. But then again, that is an occupational hazard researchers doing work involving probability theory have to live with…

  13. This means that \({\mathbb {P}}(E)\) is either 1 or 0 for every \(E\in {\mathscr {F}}\) according as E contains or fails to contain the state \(\omega _0\).

  14. More precisely, \(\succsim _{{\mathscr {M}}}\) can be identified with the family of those ordered pairs \(({\mathscr {P}},{\mathscr {Q}})\) of partitions that satisfy \({\mathscr {P}}\succsim _{{\mathscr {M}}}{\mathscr {Q}}\). Actually, this formulation corresponds to the rigorous order-theoretic definition of binary relations.

  15. That is, \(\omega \in G\triangle H\) means that \(\omega \) is in precisely one of G or H, but not in both. Formally,

    $$\begin{aligned} G\triangle H\equiv (G\cap H^{{\textsf{c}}})\cup (G^{{\textsf{c}}}\cap H). \end{aligned}$$
  16. This means that \(\mathfrak {d}\) satisfies all the metric axioms except that it is possible for two \(\sigma \)-subalgebras \({\mathscr {G}}\) and \({\mathscr {H}}\) to satisfy \(\mathfrak {d}({\mathscr {G}},{\mathscr {H}})=0\) even when \({\mathscr {G}}\ne {\mathscr {H}}\).

  17. There is a vast literature on the theory of unawareness. Important contributions include Fagin and Halpern (1987), Geanakoplos (1989, 2021), Samet (1990), Brandenburger et al. (1992), Shin (1993), Modica and Rustichini (1994, 1999), Morris (1996), Morris and Shin (1997), Dekel et al. (1998), Halpern (2001), Fagin et al. (2003, Section 9.5), Heifetz et al. (2006, 2008, 2013a, 2013b), Halpern and Rêgo (2008, 2009, 2014), Li (2009), Galanis (2011, 2013, 2015), Grant and Quiggin (2013a, 2013b, 2015), Karni and Vierø (2013, 2017), Schipper (2014, 2015), Walker (2014), Quiggin (2016), Kochov (2018), and Fukuda (2021).

  18. Cf. Dekel et al. (1998, p. 165, n. 9), who “use the word ‘unawareness’ to mean that the agent fails to foresee a possibility, not that he knows of but does not understand some possibility” (emphasis added).

  19. In fact, standard state-space models with partitional information, on which the methodological foundations of the instant paper rest, are inconsistent with the possibility of unawareness. This is because

    1. (i)

      if an agent knows something, then she knows that she knows it (positive introspection)—see (1.8) in Aumann (1999a, p. 270); and

    2. (ii)

      if an agent does not know something, then she knows that she does not know it (negative introspection)—see (1.33) in Aumann (1999a, p. 267).

    See also Tóbiás (2021a) for the subtleties involved in the use of partitional information to model interactive knowledge. Dekel et al. (1998) criticize standard state-space models more generally, arguing that such models are ill equipped to capture non-trivial unawareness in a satisfactory manner even with information that may take a non-partitional form.

  20. This distinction between unawareness and cognitive limits ties in with the following argument made by Li (2009, p. 978): “If the agent is unaware of a question, then a message reminding the agent of the question itself must be informative.”

  21. While the two papers by Blackwell (1951, 1953) have been extremely influential and entire new strands of literature sprang up from them both in statistics and in the economics of information, they are not an especially easy read. For a particularly simple and transparent exposition of Blackwell’s theory, see, for example, de Oliveira (2018).

  22. Formally, if \({\mathscr {P}}\) is a partition and \(\pi (\cdot |{\mathscr {P}}):\Omega \rightarrow {\mathscr {P}}\) is the quotient map assigning each state of the world the unique partition cell to which the state belongs, then the information content of \({\mathscr {P}}\) can be represented as a family of degenerate probability distributions over \({\mathscr {P}}\) (viewed as a signal space) such that, conditional on each \(\omega \in \Omega \), the decision maker observes the signal \(\pi (\omega |{\mathscr {P}})\) with certainty.

  23. Relatedly, Frankel and Kamenica (2019) take an ex post perspective and define the value of a piece of information that induces a Bayesian decision maker to update her prior belief about the state of the world as the expected incremental value, evaluated under the posterior, of the resulting superior action choice. Frankel and Kamenica (2019) then provide an axiomatic characterization of those real-valued functions over the set of all possible prior–posterior pairs that measure the value of information assessed by some Bayesian decision maker.

  24. One possible way of doing this is supplementing the “base” state space \(\Omega \) with an auxiliary state space capturing “noise,” and representing a given stochastic information structure as a partition of the resulting product space along with a suitable product probability measure. This approach is advocated by, among others, Green and Stokey (1978), Allen (1983, Section 15), Gentzkow and Kamenica (2017), and Frankel and Kamenica (2019).

  25. The proof of this lemma could be somewhat simplified under certain topological assumptions imposed on the action space (for instance, if \({\mathscr {A}}\) were the Borel \(\sigma \)-algebra generated by a separable and metrizable topology on A). I am grateful to the pseudonymous user D_809 on the MathOverflow website for suggesting ideas for the proof of the general case, as well as Lemma 2. See https://mathoverflow.net/a/348850/55976; date of access: June 29, 2022.

  26. From a pedantic perspective, it would be more appropriate to replace h in the definitions of \(\psi _h\) and \(\Theta (h)\) in (6) and (7), respectively, with \(\left( I,(\alpha _i)_{i=1}^I,(H_i)_{i=1}^I\right) \) to reflect the possibility that the simple function h may admit multiple representations of the form (5). This would unnecessarily complicate the notation without affecting the substance and validity of the arguments that follow, which rely only on the existence of any one representation of the form (5).

  27. Stinchcombe (1990) and Hervés-Beloso and Monteiro (2013) define two \(\sigma \)-subalgebras \({\mathscr {G}}\) and \({\mathscr {H}}\) of \({\mathscr {F}}\) to be equivalent if condition 2. holds.

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Acknowledgements

I am grateful to Salvatore Greco (Editor-in-Chief of this journal) and a referee for their constructive comments and suggestions, which have led to a substantial improvement of this paper. In addition, I thank Ildikó Magyari and Alex Smolin for insightful conversations and advice. All remaining errors and omissions are mine.

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Appendix

Appendix

This appendix contains the proofs of the results presented in the main text, interspersed with auxiliary lemmata aiding those proofs.

Lemma 1

Let \({\mathscr {M}}\) be an almost-trivial technological capacity. Let, moreover, \({\mathscr {D}}\equiv (A,{\mathscr {A}},u)\) be any decision problem and \(f:\Omega \rightarrow A\) an \({\mathscr {M}}/{\mathscr {A}}\)-measurable action policy. Then, there exists some \(a_0\in A\) such that

$$\begin{aligned} u(f(\omega ),\omega )=u(a_0,\omega )\quad \text {for } {\mathbb {P}}\text {-almost every } \omega \in \Omega . \end{aligned}$$

Proof

Footnote 25 Define the \({\mathscr {F}}/({\mathscr {A}}\otimes {\mathscr {F}})\)-measurable map \(T:\Omega \rightarrow A\times \Omega \) as

$$\begin{aligned} T(\omega )\equiv (f(\omega ),\omega )\quad \text {for every } \omega \in \Omega , \end{aligned}$$

and the push-forward probability measures

$$\begin{aligned} \mu (B)&\equiv {\mathbb {P}}(f^{-1}(B)) \quad \text {for every } B\in {\mathscr {A}},\\ \nu (H)&\equiv {\mathbb {P}}(T^{-1}(H))\quad \text {for every } H\in {\mathscr {A}}\otimes {\mathscr {F}}, \end{aligned}$$

on the measurable spaces \((A,{\mathscr {A}})\) and \((A\times \Omega ,{\mathscr {A}}\otimes {\mathscr {F}})\), respectively. Note that since the technological capacity \({\mathscr {M}}\) is almost trivial and the action policy f is \({\mathscr {M}}/{\mathscr {A}}\)-measurable, one has \(\mu (B)\in \{0,1\}\) for every \(B\in {\mathscr {A}}\). This implies that for every \({\mathscr {A}}\otimes {\mathscr {F}}\)-measurable rectangle of the form \(B\times E\), where \(B\in {\mathscr {A}}\) and \(E\in {\mathscr {F}}\), the following holds:

$$\begin{aligned} \nu (B\times E)={\mathbb {P}}(T^{-1}(B\times E))={\mathbb {P}}(f^{-1}(B)\cap E)={\mathbb {P}}(f^{-1}(B)){\mathbb {P}}(E)=\mu (B){\mathbb {P}}(E), \end{aligned}$$

so that \(\nu \) coincides with the product measure \(\mu \times {\mathbb {P}}\) on \({\mathscr {A}}\otimes {\mathscr {F}}\).

For any \(H\in {\mathscr {A}}\otimes {\mathscr {F}}\), define

$$\begin{aligned} \Psi (H)\equiv \left\{ \omega \in \Omega \,|\,\mu \left( \{a\in A\,|\,(a,\omega )\in H\}\right) =1\right\} \end{aligned}$$

and notice that

$$\begin{aligned} (\Psi (H))^{\textsf {c}}=\left\{ \omega \in \Omega \,|\,\mu \left( \{a\in A\,|\,(a,\omega )\in H\}\right) =0\right\} , \end{aligned}$$

given that the probability measure \(\mu \) admits no values other than 0 or 1. By the argument presented in Billingsley (1995, p. 232), the set \(\Psi (H)\) is \({\mathscr {F}}\)-measurable. By Tonelli’s theorem,

$$\begin{aligned} \nu (H)&=\int _{\Omega }\mu \left( \{a\in A\,|\,(a,\omega )\in H\}\right) \,\text {d}{\mathbb {P}}(\omega )\nonumber \\&=\int _{\Psi (H)}\mu \left( \{a\in A\,|\,(a,\omega )\in H\}\right) \,\text {d}{\mathbb {P}}(\omega )\nonumber \\&=\int _{\Psi (H)}\int _A{\textbf {{1}}} _H(a,\omega )\,\text {d}\mu (a)\,\text {d}{\mathbb {P}}(\omega )\nonumber \\&=\int _{A\times \Psi (H)}{\textbf {{1}}} _H(a,\omega )\,\text {d}(\mu \times {\mathbb {P}})(a,\omega )\nonumber \\&=\nu (H\cap (A\times \Psi (H))), \end{aligned}$$
(3)

where \({\textbf {{1}}} _H:A\times \Omega \rightarrow \{0,1\}\) is the indicator function of the set H. Similarly,

$$\begin{aligned} \nu (H^{{\textsf{c}}})=\nu (H^{{\textsf{c}}}\cap (A\times (\Psi (H))^{{\textsf{c}}})). \end{aligned}$$
(4)

Define the set

$$\begin{aligned} \Theta (H)\equiv \{(a,\omega )\in A\times \Omega \,|\,{\textbf {{1}}} _H(a,\omega )={\textbf {{1}}} _{\Psi (H)}(\omega )\}, \end{aligned}$$

which is easily seen to be \({\mathscr {A}}\otimes {\mathscr {F}}\)-measurable. Then, for any \((a,\omega )\in A\times \Omega \), one has \((a,\omega )\in \Theta (H)\) if and only if either \((a,\omega )\in H\) and \(\omega \in \Psi (H)\), or \((a,\omega )\in H^{{\textsf{c}}}\) and \(\omega \in (\Psi (H))^{{\textsf{c}}}\). Therefore, (3) and (4) imply that

$$\begin{aligned} \nu (\Theta (H))=\nu (H\cap (A\times \Psi (H)))+\nu (H^{{\textsf{c}}}\cap (A\times (\Psi (H))^{{\textsf{c}}}))=\nu (H)+\nu (H^{\textsf {c}})=1. \end{aligned}$$

In words: the indicator function of any \({\mathscr {A}}\otimes {\mathscr {F}}\)-measurable subset of \(A\times \Omega \) is \(\nu \)-almost surely equal to the indicator function of a suitably constructed \({\mathscr {F}}\)-measurable subset of \(\Omega \).

Let \(h:A\times \Omega \rightarrow {\mathbb {R}}\) be any arbitrary measurable simple function of the form

$$\begin{aligned} h(a,\omega )\equiv \sum _{i=1}^I\alpha _i{\textbf {{1}}} _{H_i}(a,\omega )\quad \text {for every } (a,\omega )\in A\times \Omega , \end{aligned}$$
(5)

where \(I\in {\mathbb {N}}\) and, for each \(i\in \{1,\ldots ,I\}\), \(\alpha _i\in {\mathbb {R}}\) and \(H_i\in {\mathscr {A}}\otimes {\mathscr {F}}\). Define the measurable simple function \(\psi _h:\Omega \rightarrow {\mathbb {R}}\) asFootnote 26

$$\begin{aligned} \psi _h(\omega )\equiv \sum _{i=1}^I\alpha _i{\textbf {{1}}} _{\Psi (H_i)}(\omega )\quad \text {for every } \omega \in \Omega . \end{aligned}$$
(6)

Defining the set

$$\begin{aligned} \Theta (h)\equiv \{(a,\omega )\in A\times \Omega \,|\,h(a,\omega )=\psi _h(\omega )\}, \end{aligned}$$
(7)

one can conclude based on the foregoing results that

$$\begin{aligned}\nu (\Theta (h))=1. \end{aligned}$$

By Theorem 13.5 in Billingsley (1995, p. 185), the utility function u can be approximated pointwise by a sequence of measurable simple functions. That is, there exists a sequence \((u_n)_{n\in {\mathbb {N}}}\) such that \(u_n:A\times \Omega \rightarrow {\mathbb {R}}\) is an \(({\mathscr {A}}\otimes {\mathscr {F}})/{\mathscr {B}}\)-measurable simple function for each \(n\in {\mathbb {N}}\) and

$$\begin{aligned} \lim _{n\rightarrow \infty }u_n(a,\omega )=u(a,\omega )\quad \text {for each } (a,\omega )\in A\times \Omega . \end{aligned}$$
(8)

Let

$$\begin{aligned} \Theta _{\infty }\equiv \bigcap _{n\in {\mathbb {N}}}\Theta (u_n). \end{aligned}$$

Since \(\nu (\Theta (u_n))=1\) for each \(n\in {\mathbb {N}}\), it follows that \(\nu (\Theta _{\infty })=1\). By Tonelli’s theorem again, one has

$$\begin{aligned} 1=\nu (\Theta _{\infty })=\int _A{\mathbb {P}}\left( \{\omega \in \Omega \,|\,(a,\omega )\in \Theta _{\infty }\}\right) \,\text {d}\mu (a), \end{aligned}$$

which implies that

$$\begin{aligned} {\mathbb {P}}\left( \{\omega \in \Omega \,|\,(a,\omega )\in \Theta _{\infty }\}\right) =1\quad \text {for } \mu \text {-almost every }a\in A. \end{aligned}$$

Take any such \(a_0\in A\), so that

$$\begin{aligned} {\mathbb {P}}\left( \{\omega \in \Omega \,|\,(a_0,\omega )\in \Theta _{\infty }\}\right) =1. \end{aligned}$$

Note also that

$$\begin{aligned} {\mathbb {P}}\left( \{\omega \in \Omega \,|\,(f(\omega ),\omega )\in \Theta _{\infty }\}\right) ={\mathbb {P}}(T^{-1}(\Theta _{\infty }))=\nu (\Theta _{\infty })=1. \end{aligned}$$

Therefore, for \({\mathbb {P}}\)-almost every \(\omega \in \Omega \), one has that

$$\begin{aligned} \{(a_0,\omega ),(f(\omega ),\omega )\}\subseteq \Theta _{\infty }=\bigcap _{n\in {\mathbb {N}}}\Theta (u_n), \end{aligned}$$

and for any such \(\omega \), it follows from (7) and (8) that

$$\begin{aligned} u(f(\omega ),\omega )=\lim _{n\rightarrow \infty }u_n(f(\omega ),\omega )=\lim _{n\rightarrow \infty }\psi _{u_n}(\omega )=\lim _{n\rightarrow \infty }u_n(a_0,\omega )=u(a_0,\omega ). \end{aligned}$$

The proof is complete. \(\square \)

Proof of Proposition 1

Consider necessity (“only if”) first. Suppose that \({\mathscr {M}}\) is an almost-trivial technological capacity and take an arbitrary decision problem \({\mathscr {D}}\equiv (A,{\mathscr {A}},u)\). Clearly, if \(\overline{{\mathscr {P}}}\) is the discrete partition, \(\underline{{\mathscr {P}}}\) is the trivial partition, and \({\mathscr {P}}\) is an arbitrary partition, then one has

$$\begin{aligned} V(\overline{{\mathscr {P}}}|{\mathscr {D}},{\mathscr {M}})\ge V({\mathscr {P}}|{\mathscr {D}},{\mathscr {M}})\ge V(\underline{{\mathscr {P}}}|{\mathscr {D}},{\mathscr {M}}). \end{aligned}$$

Therefore, it will be enough to show that

$$\begin{aligned} V(\overline{{\mathscr {P}}}|{\mathscr {D}},{\mathscr {M}})=V(\underline{{\mathscr {P}}}|{\mathscr {D}},{\mathscr {M}}) \end{aligned}$$
(9)

to conclude that \({\mathscr {P}}\sim _{{\mathscr {M}}}{\mathscr {Q}}\) for any two partitions \({\mathscr {P}}\) and \({\mathscr {Q}}\). It is easy to verify that

$$\begin{aligned} \Sigma (\overline{{\mathscr {P}}}|{\mathscr {M}})&={\mathscr {M}},\\ \Sigma (\underline{{\mathscr {P}}}|{\mathscr {M}})&=\{\varnothing ,\Omega \}. \end{aligned}$$

Since both of these \(\sigma \)-algebras are almost trivial, Lemma 1 yields that the value of each of the partitions \(\overline{{\mathscr {P}}}\) and \(\underline{{\mathscr {P}}}\) can be approximated by constant action policies. Formally:

$$\begin{aligned} V(\overline{{\mathscr {P}}}|{\mathscr {D}},{\mathscr {M}})&=\sup \left\{ \int _{\Omega }u(f(\omega ),\omega )\,\text {d}{\mathbb {P}}(\omega )\, |\,f\in F(\overline{{\mathscr {P}}}|{\mathscr {D}},{\mathscr {M}})\right\} \\&=\sup \left\{ \int _{\Omega }u(a_0,\omega )\,\text {d}{\mathbb {P}}(\omega )\,|\,a_0\in A\right\} \\&=\sup \left\{ \int _{\Omega }u(f(\omega ),\omega )\,\text {d}{\mathbb {P}}(\omega )\,|\,f\in F(\underline{{\mathscr {P}}}|{\mathscr {D}},{\mathscr {M}})\right\} \\&=V(\underline{{\mathscr {P}}}|{\mathscr {D}},{\mathscr {M}}), \end{aligned}$$

establishing (9).

As for sufficiency (“if”), suppose that \({\mathscr {P}}\sim _{{\mathscr {M}}}{\mathscr {Q}}\) for any two partitions \({\mathscr {P}}\) and \({\mathscr {Q}}\). Take any \(E\in {\mathscr {M}}\). If \(E\in \{\varnothing ,\Omega \}\), then \({\mathbb {P}}(E)\in \{0,1\}\). If \(E\notin \{\varnothing ,\Omega \}\), then

$$\begin{aligned}{\mathscr {P}}\equiv \{E,E^{{\textsf{c}}}\} \end{aligned}$$

is a partition. Define the decision problem \({\mathscr {D}}\equiv (A,{\mathscr {A}},u)\) by \(A\equiv \{a_1,a_2\}\), \({\mathscr {A}}\equiv 2^A\), and

$$\begin{aligned} u(a,\omega )\equiv {\left\{ \begin{array}{ll}1&{}\text {if } a=a_1\, \text { and } \omega \in E,\\ 1&{}\text {if } a=a_2 \,\text { and}\, \omega \in E^{{\textsf{c}}},\\ 0&{}\text {otherwise,}\end{array}\right. } \end{aligned}$$

for every \((a,\omega )\in A\times \Omega \). Given that \({\mathscr {P}}\sim _{{\mathscr {M}}}\underline{{\mathscr {P}}}\), where \(\underline{{\mathscr {P}}}\) is the trivial partition, one has that

$$\begin{aligned} V({\mathscr {P}}|{\mathscr {D}},{\mathscr {M}})=V(\underline{{\mathscr {P}}}|{\mathscr {D}},{\mathscr {M}}). \end{aligned}$$
(10)

On the one hand, since

$$\begin{aligned} \Sigma ({\mathscr {P}}|{\mathscr {M}})=\{\varnothing ,E,E^{{\textsf{c}}},\Omega \}, \end{aligned}$$

the action policy \(f^*:\Omega \rightarrow A\) defined as

$$\begin{aligned} f^*(\omega )\equiv {\left\{ \begin{array}{ll}a_1&{}\text {if } \omega \in E,\\ a_2&{}\text {if } \omega \in E^{{\textsf{c}}},\end{array}\right. } \end{aligned}$$

maximizes the decision maker’s expected utility among all action policies in \(F({\mathscr {P}}|{\mathscr {D}},{\mathscr {M}})\), from which it follows that

$$\begin{aligned}V({\mathscr {P}}|{\mathscr {D}},{\mathscr {M}})=1. \end{aligned}$$

On the other hand, since \(F(\underline{{\mathscr {P}}}|{\mathscr {D}},{\mathscr {M}})\) consists of the constant action policies, it follows that

$$\begin{aligned} V(\underline{{\mathscr {P}}}|{\mathscr {D}},{\mathscr {M}})=\max \{{\mathbb {P}}(E),{\mathbb {P}}(E^{{\textsf{c}}})\}=\max \{{\mathbb {P}}(E),1-{\mathbb {P}}(E)\}. \end{aligned}$$

Then, the requirement (10) yields that

$$\begin{aligned}\max \{{\mathbb {P}}(E),1-{\mathbb {P}}(E)\}=1, \end{aligned}$$

which implies that \({\mathbb {P}}(E)\in \{0,1\}\). Hence, the technological capacity \({\mathscr {M}}\) is almost trivial, and the proof is complete. \(\square \)

Lemma 2

Let \({\mathscr {M}}_1\) and \({\mathscr {M}}_2\) be two technological capacities, and let \({\mathscr {M}}_1\triangle {\mathscr {M}}_2\) denote the symmetric difference between them. If \(E\in {\mathscr {M}}_1\triangle {\mathscr {M}}_2\) implies that \({\mathbb {P}}(E)\in \{0,1\}\), then either

  1. (i)

    \({\mathscr {M}}_1={\mathscr {M}}_2\) (in which case \({\mathscr {M}}_1\triangle {\mathscr {M}}_2\) is actually empty); or

  2. (ii)

    both \({\mathscr {M}}_1\) and \({\mathscr {M}}_2\) are almost trivial.

Proof

Suppose that \({\mathscr {M}}_1\ne {\mathscr {M}}_2\). Then, without loss of generality, there exists some \(E\in {\mathscr {M}}_1\) such that \(E\notin {\mathscr {M}}_2\). Given the premise, this set satisfies \({\mathbb {P}}(E)\in \{0,1\}\). Again, without loss of generality, one can assume that \({\mathbb {P}}(E)=1\); the case in which \({\mathbb {P}}(E)=0\) could be handled by replacing E with \(E^{{\textsf{c}}}\).

Take an arbitrary \(M\in {\mathscr {M}}_1\). Then, since

$$\begin{aligned} E=(E\cap M)\cup (E\cap M^{{\textsf{c}}})\notin {\mathscr {M}}_2, \end{aligned}$$

it must be the case that either \(E\cap M\notin {\mathscr {M}}_2\) or \(E\cap M^{{\textsf{c}}}\notin {\mathscr {M}}_2\). If, on the one hand, \(E\cap M\notin {\mathscr {M}}_2\), then

$$\begin{aligned} E\cap M\in {\mathscr {M}}_1{\setminus }{\mathscr {M}}_2\subseteq {\mathscr {M}}_1\triangle {\mathscr {M}}_2, \end{aligned}$$

which implies that \({\mathbb {P}}(E\cap M)\in \{0,1\}\). But since \({\mathbb {P}}(E)=1\), it follows that \(P(M)\in \{0,1\}\). If, on the other hand, \(E\cap M^{{\textsf{c}}}\notin {\mathscr {M}}_2\), then

$$\begin{aligned} E\cap M^{{\textsf{c}}}\in {\mathscr {M}}_1{\setminus }{\mathscr {M}}_2\subseteq {\mathscr {M}}_1\triangle {\mathscr {M}}_2, \end{aligned}$$

which implies that \({\mathbb {P}}(E\cap M^{{\textsf{c}}})\in \{0,1\}\). But since \({\mathbb {P}}(E)=1\), it follows that \(P(M^{{\textsf{c}}})\in \{0,1\}\), which entails \(P(M)\in \{0,1\}\) as well. In conclusion, the technological capacity \({\mathscr {M}}_1\) is almost trivial.

If \({\mathscr {M}}_2\subseteq {\mathscr {M}}_1\), then \({\mathscr {M}}_2\) is obviously almost trivial, too. If \({\mathscr {M}}_2\not \subseteq {\mathscr {M}}_1\), then there exists some \(F\in {\mathscr {M}}_2\) such that \(F\notin {\mathscr {M}}_1\), and one can show using the same arguments as above that \({\mathscr {M}}_2\) is almost trivial in this case as well. \(\square \)

Proof of Proposition 2

Let \({\mathscr {M}}\) and \({\mathscr {M}}'\) be two technological capacities with \({\succsim _{{\mathscr {M}}}}={\succsim _{{\mathscr {M}}'}}\) and suppose that it is not the case that \({\mathscr {P}}\sim _{{\mathscr {M}}}{\mathscr {Q}}\) for all pairs of partitions \({\mathscr {P}}\) and \({\mathscr {Q}}\). By Proposition 1, neither \({\mathscr {M}}\) nor \({\mathscr {M}}'\) is almost trivial. For the sake of contradiction, assume that \({\mathscr {M}}\ne {\mathscr {M}}'\). By Lemma 2, there must exist some \(E\in {\mathscr {M}}\triangle {\mathscr {M}}'\) with \(0<{\mathbb {P}}(E)<1\).

Without loss of generality, assume that \(E\in {\mathscr {M}}\) but \(E\notin {\mathscr {M}}'\). Then, \(E\notin \{\varnothing ,\Omega \}\), so that

$$\begin{aligned} {\mathscr {P}}\equiv \{E,E^{{\textsf{c}}}\} \end{aligned}$$

is a partition. Let \(\underline{{\mathscr {P}}}\) denote the trivial partition. Since \(E\notin {\mathscr {M}}'\), it follows that

$$\begin{aligned} \Sigma ({\mathscr {P}}|{\mathscr {M}}')=\{\varnothing ,\Omega \}=\Sigma (\underline{{\mathscr {P}}}|{\mathscr {M}}'), \end{aligned}$$

so that \(F({\mathscr {P}}|{\mathscr {D}},{\mathscr {M}}')=F(\underline{{\mathscr {P}}}|{\mathscr {D}},{\mathscr {M}}')\) and

$$\begin{aligned} V({\mathscr {P}}|{\mathscr {D}},{\mathscr {M}}')=V(\underline{{\mathscr {P}}}|{\mathscr {D}},{\mathscr {M}}') \end{aligned}$$

for any decision problem \({\mathscr {D}}\). Therefore, \({\mathscr {P}}\sim _{{\mathscr {M}}'}\underline{{\mathscr {P}}}\).

Define the decision problem \({\mathscr {D}}\equiv (A,{\mathscr {A}},u)\) by \(A\equiv \{a_1,a_2\}\), \({\mathscr {A}}\equiv 2^A\), and

$$\begin{aligned} u(a,\omega )\equiv {\left\{ \begin{array}{ll}1&{}\text {if } a=a_1 \text { and } \omega \in E,\\ 1&{}\text {if } a=a_2 \text { and }\omega \in E^{{\textsf{c}}},\\ 0&{}\text {otherwise,}\end{array}\right. } \end{aligned}$$

for every \((a,\omega )\in A\times \Omega \). On the one hand, since

$$\begin{aligned}\Sigma ({\mathscr {P}}|{\mathscr {M}})=\{\varnothing ,E,E^{{\textsf{c}}},\Omega \}, \end{aligned}$$

the action policy \(f^*:\Omega \rightarrow A\) defined as

$$\begin{aligned} f^*(\omega )\equiv {\left\{ \begin{array}{ll}a_1&{}\text {if } \omega \in E,\\ a_2&{}\text {if } \omega \in E^{{\textsf{c}}},\end{array}\right. } \end{aligned}$$

maximizes the decision maker’s expected utility among all action policies in \(F({\mathscr {P}}|{\mathscr {D}},{\mathscr {M}})\), from which it follows that

$$\begin{aligned}V({\mathscr {P}}|{\mathscr {D}},{\mathscr {M}})=1. \end{aligned}$$

On the other hand, since \(F(\underline{{\mathscr {P}}}|{\mathscr {D}},{\mathscr {M}})\) consists of the constant action policies, it follows that

$$\begin{aligned} V(\underline{{\mathscr {P}}}|{\mathscr {D}},{\mathscr {M}}) =\max \{{\mathbb {P}}(E),{\mathbb {P}}(E^{{\textsf{c}}})\} =\max \{{\mathbb {P}}(E),1-{\mathbb {P}}(E)\}<1=V({\mathscr {P}}|{\mathscr {D}},{\mathscr {M}}), \end{aligned}$$

where the inequality follows from \({\mathbb {P}}(E)\in (0,1)\). Therefore, it cannot be the case that \({\mathscr {P}}\sim _{{\mathscr {M}}}\underline{{\mathscr {P}}}\), which contradicts the assumption that \({\succsim _{{\mathscr {M}}}}={\succsim _{{\mathscr {M}}'}}\). The only way to resolve this contradiction is to conclude that \({\mathscr {M}}={\mathscr {M}}'\). \(\square \)

Lemma 3

Let \({\mathscr {G}}\) and \({\mathscr {H}}\) be two \(\sigma \)-subalgebras of \({\mathscr {F}}\) and consider the pseudometric defined in (1). Then, the following conditions are equivalent:

  1. 1.

    The distance between \({\mathscr {G}}\) and \({\mathscr {H}}\) vanishes: \(\mathfrak {d}({\mathscr {G}},{\mathscr {H}})=0\).

  2. 2.

    There exist,

    1. (i)

      for every \(G\in {\mathscr {G}}\), some \(H\in {\mathscr {H}}\) such that \({\mathbb {P}}(G\triangle H)=0\); and

    2. (ii)

      for every \(H\in {\mathscr {H}}\), some \(G\in {\mathscr {G}}\) such that \({\mathbb {P}}(G\triangle H)=0\).Footnote 27

Proof

Suppose that 1. holds and fix \(G\in {\mathscr {G}}\). The condition \(\mathfrak {d}({\mathscr {G}},{\mathscr {H}})=0\) implies, in particular, that

$$\begin{aligned}\inf _{H\in {\mathscr {H}}}{\mathbb {P}}(G\triangle H)=0, \end{aligned}$$

so that one can find a sequence \((H_n)_{n\in {\mathbb {N}}}\) in \({\mathscr {H}}\) such that

$$\begin{aligned} {\mathbb {P}}(G\triangle H_n)={\mathbb {P}}(G\cap H_n^{\textsf {c}})+{\mathbb {P}}(G^{{\textsf{c}}}\cap H_n)<\frac{1}{2^n}\quad \text {for every } n\in {\mathbb {N}}. \end{aligned}$$
(11)

Let

$$\begin{aligned}H\equiv \bigcup _{m=1}^{\infty }\bigcap _{n=m}^{\infty }H_n \end{aligned}$$

denote the set-theoretic limit inferior of the sequence \((H_n)_{n\in {\mathbb {N}}}\) (Billingsley 1995, p. 52). Since the intersections and unions are countable, one has \(H\in {\mathscr {H}}\). Moreover, De Morgan’s laws imply that

$$\begin{aligned} {\mathbb {P}}(G\triangle H)={\mathbb {P}}\left( \bigcap _{m=1}^{\infty }\bigcup _{n=m}^{\infty }(G\cap H_n^{\textsf {c}})\right) +{\mathbb {P}}\left( \bigcup _{m=1}^{\infty }\bigcap _{n=m}^{\infty }(G^{\textsf {c}}\cap H_n)\right) . \end{aligned}$$
(12)

As for the first term on the right-hand side of (12), the following holds for any given \(M\in {\mathbb {N}}\):

$$\begin{aligned} {\mathbb {P}}\left( \bigcap _{m=1}^{\infty }\bigcup _{n=m}^{\infty }(G\cap H_n^{\textsf {c}})\right) \le {\mathbb {P}}\left( \bigcup _{n=M}^{\infty }(G\cap H_n^{\textsf {c}})\right) \le \sum _{n=M}^{\infty }{\mathbb {P}}(G\cap H_n^{\textsf {c}})<\sum _{n=M}^{\infty }\frac{1}{2^n}=\frac{1}{2^{M-1}}, \end{aligned}$$

where the strict inequality follows from (11). Letting \(M\rightarrow \infty \), one can conclude that

$$\begin{aligned} {\mathbb {P}}\left( \bigcap _{m=1}^{\infty }\bigcup _{n=m}^{\infty }(G\cap H_n^{\textsf {c}})\right) =0. \end{aligned}$$

As for the second term on the right-hand side of (12), the following holds for any fixed non-negative \(K\in {\mathbb {Z}}_+\):

$$\begin{aligned} {\mathbb {P}}\left( \bigcup _{m=1}^{\infty }\bigcap _{n=m}^{\infty }(G^{\textsf {c}}\cap H_n)\right)&\le \sum _{m=1}^{\infty }{\mathbb {P}}\left( \bigcap _{n=m}^{\infty }(G^{\textsf {c}}\cap H_n)\right) \le \sum _{m=1}^{\infty }{\mathbb {P}}(G^{\textsf {c}}\cap H_{m+K})<\sum _{m=1}^{\infty }\frac{1}{2^{m+K}}\\&=\frac{1}{2^K}, \end{aligned}$$

where the strict inequality follows again from (11). Letting \(K\rightarrow \infty \), one can conclude that

$$\begin{aligned}{\mathbb {P}}\left( \bigcup _{m=1}^{\infty }\bigcap _{n=m}^{\infty }(G^{\textsf {c}}\cap H_n)\right) =0. \end{aligned}$$

Therefore, \({\mathbb {P}}(G\triangle H)=0\) and 2.(i) is satisfied. That 2.(ii), too, holds can be proven in an analogous manner.

Conversely, suppose that 2. holds. For any fixed \(G_0\in {\mathscr {G}}\), 2.(i) implies that

$$\begin{aligned} \inf _{H\in {\mathscr {H}}}{\mathbb {P}}(G_0\triangle H)=0. \end{aligned}$$

Since \(G_0\in {\mathscr {G}}\) can be arbitrary, it follows that

$$\begin{aligned} \sup _{G\in {\mathscr {G}}}\inf _{H\in {\mathscr {H}}}{\mathbb {P}}(G\triangle H)=0. \end{aligned}$$

Similarly, 2.(ii) entails that

$$\begin{aligned} \sup _{H\in {\mathscr {H}}}\inf _{G\in {\mathscr {G}}}{\mathbb {P}}(G\triangle H)=0. \end{aligned}$$

In conclusion, \(\mathfrak {d}({\mathscr {G}},{\mathscr {H}})=0\). \(\square \)

Proof of Proposition 3

Suppose that \({\succsim _{{\mathscr {M}}_1}}={\succsim _{{\mathscr {M}}_2}}\). By Propositions 1 and 2, either \({\mathscr {M}}_1={\mathscr {M}}_2\), or both \({\mathscr {M}}_1\) and \({\mathscr {M}}_2\) are almost trivial. If \({\mathscr {M}}_1={\mathscr {M}}_2\), then (2) trivially holds. If both \({\mathscr {M}}_1\) and \({\mathscr {M}}_2\) are almost trivial, then so are \(\Sigma ({\mathscr {P}}|{\mathscr {M}}_1)\) and \(\Sigma ({\mathscr {P}}|{\mathscr {M}}_2)\) for any partition \({\mathscr {P}}\). Take any \(M_1\in \Sigma ({\mathscr {P}}|{\mathscr {M}}_1)\). If \({\mathbb {P}}(M_1)=0\), then \(\varnothing \in \Sigma ({\mathscr {P}}|{\mathscr {M}}_2)\) and \({\mathbb {P}}(M_1\triangle \varnothing )=0\). If \({\mathbb {P}}(M_1)=1\), then \(\Omega \in \Sigma ({\mathscr {P}}|{\mathscr {M}}_2)\) and \({\mathbb {P}}(M_1\triangle \Omega )=0\). Similarly, one can find for every \(M_2\in \Sigma ({\mathscr {P}}|{\mathscr {M}}_2)\) some \(E\in \{\varnothing ,\Omega \}\subseteq \Sigma ({\mathscr {P}}|{\mathscr {M}}_1)\) such that \({\mathbb {P}}(E\triangle M_2)=0\). Therefore, (2) follows from Lemma 3.

Conversely, suppose that (2) holds. I will show that \(E\in {\mathscr {M}}_1\triangle {\mathscr {M}}_2\) implies that \({\mathbb {P}}(E)\in \{0,1\}\). Then, Lemma 2 will yield that either \({\mathscr {M}}_1={\mathscr {M}}_2\) (in which case \({\succsim _{{\mathscr {M}}_1}}={\succsim _{{\mathscr {M}}_2}}\) trivially holds), or both \({\mathscr {M}}_1\) and \({\mathscr {M}}_2\) are almost trivial (in which case \({\succsim _{{\mathscr {M}}_1}}={\succsim _{{\mathscr {M}}_2}}\) follows from Proposition 1). Take any \(E\in {\mathscr {M}}_1\triangle {\mathscr {M}}_2\). Without loss of generality, suppose that \(E\in {\mathscr {M}}_1\) and \(E\notin {\mathscr {M}}_2\). Since \(E\notin \{\varnothing ,\Omega \}\),

$$\begin{aligned}{\mathscr {P}}\equiv \{E,E^{\textsf {c}}\} \end{aligned}$$

is a partition, and

$$\begin{aligned} \Sigma ({\mathscr {P}}|{\mathscr {M}}_1)&=\{\varnothing ,E,E^{{\textsf{c}}},\Omega \},\\ \Sigma ({\mathscr {P}}|{\mathscr {M}}_2)&=\{\varnothing ,\Omega \}. \end{aligned}$$

Given that \(\mathfrak {d}(\Sigma ({\mathscr {P}}|{\mathscr {M}}_1),\Sigma ({\mathscr {P}}|{\mathscr {M}}_2))=0\), that \(E\in \Sigma ({\mathscr {P}}|{\mathscr {M}}_1)\), and that \(\Sigma ({\mathscr {P}}|{\mathscr {M}}_2)\) consists of only \(\varnothing \) and \(\Omega \), Lemma 3 yields that either \({\mathbb {P}}(E\triangle \varnothing )=0\) or \({\mathbb {P}}(E\triangle \Omega )=0\). Either possibility implies that \({\mathbb {P}}(E)\in \{0,1\}\), as claimed. \(\square \)

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Tóbiás, Á. Cognitive limits and preferences for information. Decisions Econ Finan 46, 221–253 (2023). https://doi.org/10.1007/s10203-022-00376-9

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