14. Learning Under Knightian Uncertainty

In the statistics literature, Seidenfeld and Wasserman (1993) presented necessary and sufficient conditions that dilation of uncertainty (which corresponds to erosion of confidence discussed later) take place in the case of the “no-narrowing” Bayes rule if uncertainty is formulated as a set of distributions (that is, Knightian uncertainty). However, these conditions are hard to explain and thus they are difficult to apply in economic problems of our interest. The contribution of this chapter is, first, to show that confidence erosion can occur under relatively simple, not-so-implausible conditions in the case of \(\varepsilon \)-contamination; and second, to present sufficient conditions under which dilation still occurs in the “range-narrowing” maximum-likelihood rule.

The concept of \(\varepsilon \)-contamination defined in this chapter is used in Nishimura and Ozaki (2004) who examine search behavior under Knightian uncertainty. (See Chap. 9 of this book).

Nishimura and Ozaki (2006) show that if economic agents’ behaviors are in accordance with several axioms, then their perceived uncertainty can be characterized as \(\varepsilon \)-contamination of confidence. (See Chap. 12 of this book.) Their axioms are not at all singular. Thus, their results suggest that \(\varepsilon \) -contamination of confidence may commonly be observed.

We consider the maximum-likelihood rule and a multi-prior Bayesian rule because they are intuitive and sensible. After a new observation, the maximum-likelihood rule chooses, among all distributions in the set characterizing Knightian uncertainty, those that put the highest probability on the occurrence of an actual observation, and updates the chosen distributions by using the Bayes rule. The multi-prior Bayesian rule updates all distributions in the set by using the Bayes rule. Both rules are based on Bayesian ideas.

See Theorem 14.5.2 below. The exact meaning of “informational value” will be clarified later in this chapter. The result is surprising particularly in the case of the maximum-likelihood “update” rule, in which substantial “narrowing” of the range of probability charges seems to occur after obtaining a new observation through the maximum-likelihood principle.

In fact, to our knowledge, there is no other update rule that has been discussed as widely and intensively as these rules in the literature.

Rothschild (1974) considers an infinite horizon. We deviate from his work in this respect to make our argument simple and transparent.

Letting \(E[\cdot |w_{i}]\) be the posterior mean, (14.1) and the paragraph containing (14.2) imply that \((\forall j\ne i)\;E[p_{j}|w_{i}]=\alpha _{j}/(\sum _{\ell =1}^{k}\alpha _{\ell }+1)\) and \(E\left[ p_{i}|w_{i}\right] =(\alpha _{i}+1)/(\sum _{\ell =1}^{k}\alpha _{\ell }+1)\).

[Nishimura and Ozaki 2006, and Chap. 12 of this book] show that, if the decision-maker’s behavior is consistent with certain plausible axioms, her decision-making is characterized as maximizing the minimum of her expected utility over multiple priors that are characterized by \(\varepsilon \)-contamination of confidence explained in the text. The set of axioms they presented are an extension of Schmeidler’s (1982, 1989) axioms.

The \(\varepsilon \)-contamination has been widely used in statistics literature to specify a set of charges (see, for example, Berger 1985). There, the sensitivity of an estimator to the assumed prior distribution (\(( \varvec{p}^{0},\varvec{p}^{0})\) in the text) is the main concern in the context of Bayesian estimation problems. While we also specify a set of charges or Knightian uncertainty by \(\varepsilon \)-contamination, our main concern is not robustness of a specfic estimator but the set itself, which reflects the decision-maker’s lack of confidence.

In this section, we restrict contamination, \(({\varvec{q}},{\varvec{q}^{\prime }})\), to be a product probability charge to make a proof simple and intuitive. However, in general, contamination is not restricted to a product probability charge, but it is allowed to be any probability charge defined over the product space. We consider these general cases in the formal analysis of Sects. 14.3, 14.4 and 14.5. See in particular Eq. (14.14) in Sect. 14.5.

In other words, \(\mathcal {P}\times \mathcal {P}\) is the set of all product charges of the form: \(\varvec{p}\otimes \varvec{p^{\prime }}\) when we regard \(\varvec{p}\) and \(\varvec{p^{\prime }}\) as probability charges on W. In the text, we denote \(\varvec{p}\otimes \varvec{ p^{\prime }}\) by \((\varvec{p},\varvec{p^{\prime }})\).

The case of the maximum-likelihood rule is discussed in Sects. 14.3 and 14.4. Here we analyze the Bayesian rule because it is more tractable than the maximum-likelihood rule.

If the wage distribution of the first period is perfectly correlated with that of the second period, then we cannot have confidence erosion. A perfect correlation means that if the decision-maker gets wage \(w_{i}\) in the first period then she gets \(w_{i}\) in the second period. In this case, uncertainty is completely resolved in the first period. However, so long as the correlation is not perfect, there is a possibility of confidence erosion.

These updating rules of multiple probability charges correspond to those of some convex probability capacities. See Sect. 2.​3.​5.

In the statistics literature, the dilation is defined with respect to lower- and upper-probabilities. To be more precise, let \(\mathcal {P}\subseteq \mathcal {M}(\Omega ,\mathcal {F}_{2})\) and let \(B\in \mathcal {F}_{2}\) be such that \((\forall p\in \mathcal {P})\;p(B)>0\). Then, we define the lower-probability, denoted \(\underline{\mathcal {P}}\), by \((\forall A\in \mathcal {F}_{2})\;\underline{\mathcal {P}}(A):=\inf _{p\in \mathcal {P}}p(A)\) and define the conditional lower-probability, denoted \(\underline{ \mathcal {P}}(\cdot |B)\), by \((\forall A\in \mathcal {F}_{2})\;\underline{\mathcal {P}} (A|B):=\inf _{p\in \mathcal {P}}p(A\cap B)/p(B)\). The upper-probability \(\overline{\mathcal {P}}\) and the conditional upper-probability \(\overline{\mathcal {P}}(\cdot |B)\) are defined symmetrically. Each of these “probabilities” turns out to be a nonadditive probability charge, or probability capacity. It is said that B dilates A if the following holds:

For this concept of dilation and a study of its properties, see Seidenfeld and Wasserman (1993). Herron et al. (1997) contains some additional analysis. Walley (1991) extensively studies the lower- and upper-probabilities.

In particular, Seidenfeld and Wasserman (1993) derives a necessary and sufficient condition for the dilation to take place in the sense of (14.10), for cases including the \(\varepsilon \)-contamination. Their condition, however, is based on a particular eventA (not on a set of charges) so that its application to economic models is rather difficult if not impossible.

In Sect. 14.5, we derive a sufficient condition for the dilation to take place for the \(\varepsilon \)-contamination in the sense defined in the text. Our definition is more general than (14.10) because it is applied directly to a set of charges and not to a particular event A. We consider the maximum-likelihood updating rule as well as the generalized Bayesian updating rule (see the next section) while (14.10) is related only to the generalized Bayesian rule. Furthermore, we consider the dynamic nature of Knightian uncertainty explicitly to derive the economic intuition behind the dilation.

See Dempster (1967, 1968); Shafer (1976); Fagin and Halpern (1990); Gilboa and Schmeidler (1993); and Denneberg (1994).

The generalized Bayesian rule was originally proposed as an update rule for a capacity. More precisely, the rule was developed for \(\mathcal {P} \), which is characterized as the core of a convex capacity (Fagin and Halpern 1990; Denneberg 1994). The text use of the rule is its natural extension to the case of a more general \(\mathcal {P}\). See Sect. 2.​3.​5.

The maximum-likelihood rule was originally proposed as an updating rule for a capacity (Dempster 1967, 1968; Shafer 1976). Later, Gilboa and Schmeidler (1993) showed that this rule is identical to the maximum-likelihood updating rule, which we extend to the case of a more general \(\mathcal {P}\) in the text. See Sect. 2.​3.​5.

For a related work that provides some axiomatic foundation to the ML rule, see Gilboa and Schmeidler (1993).