Keywords: Ambiguity, multiple priors, Aumann plausibility, Blackwell order, partial identification, robust Bayesian inference, Gamma-minimax, persuasion.

1 Introduction↩︎

The data we encounter in the world seldom allow us to make precise inferences. Blood tests come back positive without distinguishing between two related variants of the same disease; forensic analyses reveal a suspect’s involvement in a crime without indicating whether the act was intentional or accidental; and policy evaluations show that schooling subsidies change average outcomes without revealing what would have happened in the absence of the policy, leaving the average treatment effect indeterminate. In each case, the “experiments" the decision maker (DM) has at her disposal are partially identified, and the data narrows down a set of plausible states without uniquely identifying the true state of the world.

This paper investigates the connection between the ambiguity a DM perceives over the state space and her beliefs about the partially identified nature of experiments. In the multiple priors framework, we refer to the resulting form of ambiguity as partially identified ambiguity. Our approach closely relates to the literature on robust Bayesian inference under partial identification, which takes partially identified experiments as the primitive and “backs out” a class of priors for Bayesian inference. In particular, [1] shows that performing Bayesian inference with the class of partially identified prior sets we consider reconciles the well-known inconsistencies between frequentist and Bayesian inference under partial identification [2]. From the perspective of axiomatic decision theory, whether a DM has beliefs compatible with partially identified ambiguity can be determined from preferences over state-contingent outcomes. More specifically, partially identified ambiguity is a special case of the ambiguity that arises from generalizing subjective expected utility (SEU) theory to “smaller" domains, the axiomatic foundations of which have been studied extensively.² To the best of our knowledge, we are the first to formalize the connection between these two distinct literatures.

We begin by offering a novel interpretation of the multiple priors model as imposing a consistency constraint on the collection of experiments the DM believes to be conceivable. Similar in spirit to the martingale property of beliefs, this restriction captures the intuition that the DM ought not to have been ignorant ex ante about the data-generating process (beyond its membership in the prior set) if it were possible to distinguish between priors based on the realization of some unambiguous sample space. At the level of posterior beliefs, we introduce Aumann plausibility as the generalization of Bayes plausibility to the setting of ambiguity. The condition imposes a set-valued version of the martingale property under Aumann expectations, requiring that DMs who perceive more ambiguity ex ante continue to do so, on average, after the realization of experiments as well.

Our first main result shows that, under prior-by-prior updating, consistent experiments induce Aumann-plausible distributions over posterior beliefs if and only if the prior set is partially identified. Recall that in the Bayesian setting with a single prior, all experiments induce Bayes-plausible distributions over posteriors, regardless of the prior.³ Our result thus illustrates that the martingale property of beliefs only extends to the set-valued setting under specific structural assumptions about the experiment and prior set. It turns out that this structure greatly simplifies the analysis of learning under ambiguity. For instance, we show that the equivalence between the more informative and more valuable orders of [8], [9] can be easily extended to the setting of partially identified ambiguity when the value function takes the maxmin expected utility (MEU) form of [10].

We also contribute to the econometrics literature on Bayesian decision-making under partial identification, where it is typically assumed that DMs with partially identified prior sets perform robust Bayesian inference à la [1].⁴ Within this framework, two notions of optimality arise: the Gamma-minimax criterion chooses the ex ante optimal sample-contingent decision rule, and the conditional Gamma-minimax criterion chooses the optimal action conditional on the realization of the sample space. Unless the prior set is a singleton, these criteria do not generally coincide, and there is no clear consensus on which criterion should be used [14]. Using our characterization of Aumann plausible distributions, we address this issue by showing that the time consistent ex ante criterion, termed Gamma\({}^*\)-minimax, retains the desirable properties of the Gamma-minimax criterion under partial identification. Because Gamma\({}^*\)-minimax also coincides with the conditional Gamma-minimax action, we interpret our results as presenting an argument in favor of the latter criterion.

Our second main result characterizes the conditions under which consistent experiments and Aumann-plausible distributions are fully equivalent: consistent experiments not only induce Aumann-plausible distributions, but every Aumann-plausible distribution can also be generated by a consistent experiment. The condition, which we refer to as maximal partial identification, requires the prior set to admit a Minkowski decomposition into extreme sets of priors with disjoint support. It nests the class of prior sets used in robust Bayesian inference as a special case, where the decomposition consists of lower-dimensional simplices supported on the cells of some partition of the state space. Our result can be interpreted as extending the “splitting" characterization of [15] to the multiple priors setting under Aumann plausibility.⁵

This paper also contributes to the recent literature on information design with ambiguous beliefs by examining the implications of maximal partial identification in a persuasion game. We show that both consistency and Aumann plausibility can be interpreted as a constraint on the set of strategies available to Sender. Consider, for instance, the classic example of a persuasion game in which a prosecutor (Sender) seeks to persuade a judge (Receiver) to convict a defendant. Legal and technological constraints limit the kinds of “data" the prosecutor can generate: forensic tests can establish whether the defendant was involved in a crime, but they cannot speak to his mental state or the degree of malice, even though such factors are central to the judge’s decision. Our theory of partially identified persuasion illustrates how such limitations influence both the beliefs and strategies of agents, yielding a more realistic description of communication under ambiguity. Moreover, the equivalence result between consistent experiments and Aumann-plausible distributions allows us to apply the standard concavification logic of [17] to solve for equilibrium.

1.1 Related Literature↩︎

A growing literature incorporates ambiguity into classical problems in economic theory. In such settings, ambiguity can either enter the model exogenously [18]–[20] or arise endogenously from the use of ambiguous communication devices [21]–[23]. We provide an alternative justification for the presence of ambiguity that is grounded in the DM’s beliefs about the collection of conceivable data, in connection with the econometrics literature on Bayesian inference under partial identification [1], [13], [24]. As such, our results contribute to the literature at the intersection of decision theory and partial identification [25]–[27].

Taken literally, our DM can be viewed as an econometrician who performs robust Bayesian inference with a set-identified model à la [28] and [1]. More generally, we model settings in which the DM holds probabilistic beliefs over a “smaller" collection of events, generated by the collection of identified sets over the state space. Because our theory of partially identified ambiguity is inherently dynamic, our results connect the literature on dynamic ambiguity [29]–[33] with the literature on ambiguity with probabilistic beliefs over a smaller domain of”unambiguous" events [3]–[7].

We also contribute to the robust Bayesian analysis literature on optimal statistical decision-making under partial identification [11]–[13], [34]. Our theory of partially identified ambiguity sheds light on the well-known dynamic inconsistency of the Gamma-minimax rule [35], [36], and the Gamma\({}^*\)-minimax criterion we propose coincides with the conditional Gamma-minimax criterion when the DM performs robust Bayesian inference à la [1] with a set-identified model.

Our model of partially identified persuasion contributes to the recent literature that generalizes the persuasion game of [17] to permit the presence of ambiguity [16], [19], [21], [37], [38]. To the best our knowledge, we are the first to interpret ex ante ambiguity as imposing a constraint on the collection of Sender’s strategies. Moreover, Aumann plausibility restricts agents’ perception of ambiguity throughout the game as a function of the amount of ambiguity they perceive ex ante. This differs significantly from the model of ambiguous persuasion developed in [21] and subsequent work, where the probabilistic beliefs of agents are dilated via model uncertainty.

The remainder of the paper is organized as follows. Section 2 develops our theory of partially identified ambiguity, and Section 3 presents our two main equivalence results. Sections 4 and 5 apply our first characterization result to Blackwell’s theorem and the Gamma minimax problem, respectively. Section 6 applies our second characterization result to a persuasion game. Section 7 discusses the relation between partially identified ambiguity and utility representations, and Section 8 concludes. All proofs are in the Appendix, unless stated otherwise.

2 Model↩︎

2.1 Partially Identified Prior Sets↩︎

Consider some finite state space \(\Theta\) and partition \(\Phi\), with \(\phi(\theta)\) denoting the partition cell that each \(\theta \in \Theta\) belongs to. The sets \(\Delta(\Theta)\) and \(\Delta(\Phi)\) correspond to the collection of probability mass functions supported on \(\Theta\) and \(\Phi\), respectively. The space of closed and convex subsets of \(\Delta(\Theta)\) is denoted \(\mathcal{K}(\Delta \Theta)\), and we endow it with the Hausdorff topology, corresponding Borel \(\sigma\)-algebra, and the mixture operation given by Minkowski addition. The measures \(p_0 \in \Delta(\Theta)\) and sets of measures \(\mathcal{P}_0 \in \mathcal{K}(\Delta \Theta)\) are called priors and prior sets, respectively. Following the partial identification literature in econometrics, we refer to \(\phi \in \Phi\) as the reduced-form parameter and \(\tau \in \Delta(\Phi)\) as the reduced-form prior.

The partition \(\Phi\) can be interpreted as the collection of events the DM perceives to be “identifiable" in the state space \(\Theta\), as the existence of a reduced-form prior \(\tau \in \Delta(\Phi)\) implies that the DM has precise probabilistic beliefs over \(\Phi\). Ambiguity thus arises whenever \(\Phi\) is coarser than the discrete partition \(\{\theta\}_{\theta \in \Theta}\). Let \(\Delta(\phi) \subseteq \Delta(\Theta)\) denote the collection of measures supported on \(\phi\). The following definition motivates the relationship between the DM’s perceived ambiguity, as captured by the prior set, and the reduced-form prior.

The simplex \(\Delta(\Theta)\) is partially identified by the trivial partition \(\Phi = \{\Theta \}\) and reduced-form prior \(\tau(\Theta) = 1\). At the other extreme, singleton prior sets \(\mathcal{P}_0 = \{p_0\}\) are partially identified by the discrete partition \(\Phi = \{\theta\}_{\theta \in \Theta}\) and \(\tau = p_0\). We say that the prior set is trivially identified in the former scenario; in the latter, we say it is uniquely identified. More generally, the definition of partially identified sets imply that \(\dim(\mathcal{P}_0) = |\Theta| - |\Phi|\) because each partition cell corresponds to a linear constraint on \(\Delta(\Theta)\).⁷ Note that we can assume \(\mathop{\mathrm{supp}}(\tau) = \Phi\) without loss, redefining \(\Theta\) as the union of the sets in \(\mathop{\mathrm{supp}}(\tau)\) whenever \(\mathop{\mathrm{supp}}(\tau) \neq \Phi\).

The following familiar example considers an intermediate case where the prior set is neither trivially nor uniquely identified.

The partially identified sets we have considered thus far share an additional special feature: they are generated by setting \(\mathcal{P}_\phi = \Delta(\phi)\) for all \(\phi \in \Phi\). Such sets play an important role in the Bayesian econometrics literature. In particular, [1] show that performing robust Bayesian inference with \(\mathcal{P}_0 = \sum_\phi \tau(\phi) \Delta(\phi)\) as the prior set reconciles the asymptotic disagreement between Bayesian and frequentist inference under set-identified models. This is often referred to as the “full ambiguity" approach to robust Bayesian inference, and \(\sum_\phi \tau(\phi) \Delta(\phi)\) contains every set partially identified by \((\tau, \Phi)\) as a subset [13].

Of course, not all partially identified sets reflect full ambiguity. Partially identified sets that incorporate additional information about the distribution over \(\Theta\) often admit the representation \(\mathcal{P} = \sum_\phi \tau(\phi) \mathcal{P}_\phi\) with \(\mathcal{P}_\phi \subsetneq \Delta(\phi)\) for some \(\phi \in \Phi\). The following simple example returns to the case of the Ellsberg urn, but with additional knowledge about the relative composition of green and blue balls.

An extensive literature in axiomatic decision theory studies preferences that are consistent with partially identified beliefs [3]–[7]. The main focus of these papers is to identify a collection of events that are subjectively “unambiguous" to each DM based on preferences. Prior sets partially identified by \((\tau, \Phi)\) thus correspond to the special case where the \(\sigma\)-algebra generated by \(\Phi\) is the set of unambiguous events, capturing the complete lack of quantifiable knowledge about the data-generating process over \(\Theta\) beyond what is encoded in the reduced-form prior. We provide a more comprehensive discussion of the connection between this literature and partially identified ambiguity in Section 7.

2.2 Consistent Experiments↩︎

For some finite sample space \(Y\), let \(\Delta(Y)\) denote the collection of probability mass functions supported on \(Y\). A Blackwell experiment, i.e., likelihood function, is a mapping \(\pi: \Theta \to \Delta(Y)\) that specifies the distribution over \(Y\) conditional on each state \(\theta \in \Theta\). We assume that a DM with the prior set \(\mathcal{P}_0\) updates her beliefs prior-by-prior according to Bayes rule after observing the realization of \(Y\), such that each outcome \(y \in Y\) induces a posterior set \(\mathcal{P}_y \in \mathcal{K}(\Delta \Theta)\). An experiment \(\pi\) is non-trivial if \(\pi(\cdot|\theta)\) is non-constant in \(\theta\).

Under the multiple priors assumption, ambiguity corresponds to the absence of quantifiable knowledge about the data-generating process over \(\Theta\) beyond its membership in the prior set. For this reason, we argue that experiments which enable the DM to distinguish between priors in \(\mathcal{P}_0\) are inconsistent with the multiple priors model. As in the martingale property of beliefs, the DM ought not to have been completely ignorant ex ante if it were possible to distinguish between priors based on the realization of some unambiguous sample space. This intuition motivates the following definition of consistent experiments, which holds vacuously for singleton \(\mathcal{P}\).

If an experiment is consistent with the prior set \(\mathcal{P}_0\), every prior \(p_0 \in \mathcal{P}_0\) induces the same prior predictive, i.e., marginal, distribution over the sample space \(Y\). It is then without loss to define \(\mu(\cdot) \equiv \sum_\theta \pi(\cdot| \theta) p_0(\theta) \in \Delta(Y)\) using any \(p_0 \in \mathcal{P}_0\) and assume that it has full support. Note that \(\mu\) can equivalently be interpreted as a measure over the collection of posterior sets \(\{ \mathcal{P}_y\}_{y \in Y} \subseteq \mathcal{K}(\Delta \Theta)\). For this reason, we refer to \(\mu\) as the information structure induced by the experiment \(\pi\). We say \(\mu\) is precise if it is supported on singleton posterior sets, and coarse otherwise.

Observe that every trivial experiment \(\pi(\cdot|\theta)\) constant in \(\theta\) is consistent with any \(\mathcal{P} \in \mathcal{K}(\Delta \Theta)\). The following result shows that the collection of non-trivial consistent experiments is non-empty if and only if the prior set has strictly lower dimension than the probability simplex.

Consistency also has an intuitive interpretation in the context of partial identification. The following lemma illustrates that for partially identified prior sets, consistency coincides precisely with the classical notion of set-identification in econometrics: the likelihood \(\pi(\cdot|\theta)\) being flat over cells of the partition \(\Phi\).¹⁰ To avoid confusion with our definition of partially identified prior sets, we say that \(\pi\) is \(\Phi\)-measurable when this property holds.

We can then interpret partially identified prior sets as encoding the DM’s beliefs about the type of data, i.e., Blackwell experiments, she believes to be conceivable. Relative to the full ambiguity benchmark, if the DM’s prior set reflects additional knowledge about the data-generating process while remaining compatible with the corresponding reduced-form parameters, then this additional information is independent of her beliefs about the collection of conceivable experiments.

Under complete prior ambiguity \(\mathcal{P}_0 = \Delta(\Theta)\), only trivial experiments are consistent with \(\mathcal{P}_0\), and the DM does not expect to learn anything beyond the information contained in the trivial partition. If \(\mathcal{P}_0\) is a singleton, all experiments are consistent, and the DM believes that she can learn the true state \(\theta \in \Theta\) given sufficiently informative experimentation over the sample space. In the previous Ellsberg urn examples, an experiment is consistent with the prior set if and only if \(\pi(\cdot | G) = \pi(\cdot |B)\), and realizations of the sample space allow the DM to learn about the relative likelihoods of the reduced-form parameters \(\{R\}\) and \(\{G, B\}\).

2.3 Updating Beliefs↩︎

In this section, we consider the implications of partial identification and consistency in the context of learning under ambiguity. To illustrate the key ideas, we first revisit the case of the Ellsberg urn from Example 1.

In the preceding example, we see that the prior-by-prior update of a set partially identified by \((\tau, \Phi)\) given the realization of \(y \in Y\) can be obtained equivalently by considering the set partially identified by the reduced-form posterior \(\tau_y\) over the same partition. The following lemma establishes that this property holds in generality with consistent experiments when the prior set is partially identified.

For any \(\phi \in \Phi\), let \(p_0(\cdot | \phi) \equiv p_0(\cdot) \mathbb{1}_{ \{ \, \cdot \, \in \phi\} } / \tau_0(\phi)\) denote the conditional distribution over \(\Theta\) given \(\phi\), such that \(p_0 (\theta) = p_0(\theta|\phi) \tau_0(\phi)\) for every \(\theta \in \phi\) and \(p_0 \in \mathcal{P}_0\). An important consequence of the previous result is that the priors in \(\mathcal{P}_0\) are only updated through the reduced-form beliefs. More precisely, let \(p_y(\cdot)\) be the posterior and \(\tau_y(\cdot)\) the reduced-form posterior corresponding to \(p_0(\cdot)\) and \(\tau_0(\cdot)\), respectively. The posterior distribution over \(\Theta\) conditional on \(\phi\) is \[\begin{align} p_y(\cdot | \phi) \equiv \frac{ p_y(\cdot)}{\tau_y(\phi)} \mathbb{1}_{ \{ \, \cdot \, \in \phi\} } = \frac{\pi(y|\phi ) p_0(\cdot) / \mu(y)}{ \pi(y|\phi) \tau_0(\phi) / \mu(y)} \mathbb{1}_{ \{ \, \cdot \, \in \phi\} } = p_0(\cdot | \phi), \nonumber \end{align}\] which implies that \(p_y(\theta) = p_0(\theta|\phi(\theta)) \tau_y(\phi(\theta))\), i.e., realizations of consistent experiments only affect beliefs through the reduced-form prior.¹¹

3 Main Equivalence Characterizations↩︎

3.1 Aumann Plausibility↩︎

In the Bayesian setting, the equivalence between experiments and information structures is well-known: given some prior, experiments induce Bayes-plausible information structures over posteriors under Bayesian updating, and any Bayes-plausible information structure can be induced by an experiment over a sufficiently rich sample space. This section presents a generalization of this equivalence result to the setting of ambiguity.

We first introduce Aumann plausibility as the set-valued generalization of Bayes plausibility. It naturally extends the intuition of the martingale property of beliefs to the multiple priors model, requiring the expected “amount" of ambiguity after the realization of the experiment to coincide with the amount of ambiguity present ex ante.

In other words, \(\mu\) is Aumann-plausible with respect to \(\mathcal{P}_0\) if and only if the \(\mu\)-Aumann expectation of the posterior set \(\mathcal{P}_Y\) equals \(\mathcal{P}_0\). The following remark presents another interpretation of Aumann-plausibility in the context of rectangularity à la [30] and the timing of resolution of uncertainty.

When the prior set is a singleton, Aumann plausibility reduces to Bayes plausibility, and all Aumann-plausible information structures must be supported on singleton posterior sets. Once we consider non-singleton prior sets, neither direction of the equivalence result in the Bayesian setting holds when Bayes-plausible, precise information structures are generalized to Aumann-plausible, coarse information structures. The subsequent sections show that restricting our attention to consistent experiments and partially identified prior sets restores this equivalence.

3.2 Consistent Experiments to Aumann-Plausible Information Structures↩︎

Our first characterization result proves that a prior set \(\mathcal{P}_0\) is partially identified if and only if every experiment consistent with \(\mathcal{P}_0\) induces an Aumann-plausible information structure. We emphasize that, taken individually, neither the consistency of experiments nor the Aumann plausibility of information structures relate to the partially identified nature of the underlying prior set. The connection arises only when the two properties are considered jointly under prior-by-prior updating.

Because any full-dimensional prior set is partially identified by the trivial partition \(\Phi = \{\Theta\}\) and corresponding reduced-form prior, we restrict our attention to prior sets partially identified by some non-trivial \((\tau, \Phi)\). In the special case where the prior set \(\mathcal{P}_0 = \{p_0\}\) is a singleton, \(\mathcal{P}_0\) is partially identified by \((p_0, \{\theta\}_{\theta \in \Theta})\) and our result reduces to the standard fact that experiments induce Bayes-plausible distributions over posteriors under Bayesian updating.

It is remarkable that consistent experiments induce Aumann-plausible information structures only if the underlying prior set is partially identified, as the latter condition substantially restricts the collection of permissible prior sets. We interpret this result as highlighting the fundamental role that well-defined probabilistic beliefs—as captured by the reduced-form prior—play in sustaining the logic of the martingale property, even when that logic is extended to its set-valued analog. The significance of our characterization result is further discussed in Sections 4 and 5.

The proof of necessity relies on first establishing that the collection of unambiguous events is non-trivial, defines \(\Phi\) as the corresponding collection of atoms, then uses the partition-revealing experiment \(\pi: \Theta \to \Delta(\Phi)\), \(\pi(\phi|\theta) = \mathbb{1}_{ \{ \theta \in \phi\} }\) and Aumann plausibility to conclude that \(\mathcal{P}_0 = \sum_{\phi \in \Phi} \tau(\phi) \mathcal{P}_\phi\). The proof of sufficency follows more naturally. Recall from Lemma 3 that updating \(\mathcal{P}_0 = \sum_\phi \tau(\phi) \mathcal{P}_\phi\) prior-by-prior is equivalent to first obtaining the reduced-form posterior \(\tau_y\), then taking the Minkowski sum over \(\mathcal{P}_\phi\) using the new weights given by \(\tau_y\). The Aumann plausibility of the information structure induced by \(\pi\) is then a consequence of the fact that \(\pi\) induces a Bayes-plausible information structure over the reduced-form posteriors \(\{\tau_y\}_{y \in Y}\).

3.3 Aumann-Plausible Information Structures to Consistent Experiments↩︎

Given the equivalence result presented in Theorem 1, a natural question is whether imposing the full ambiguity condition on partially identified prior sets is necessary and/or sufficient for the generalization of both equivalence results in the Bayesian setting to hold. That is, in addition to every consistent experiment inducing an Aumann-plausible information structure, we now consider the case where every Aumann-plausible information structure can be induced by a consistent experiment. It turns out that full ambiguity is a sufficient, but not necessary condition for the full equivalence characterization.

Recall that \(\mathcal{P}_0\) is partially identified by \((\tau, \Phi)\) if it admits the decomposition \(\mathcal{P}_0 = \sum_\phi \tau(\phi) \mathcal{P}_\phi\) for \(\mathcal{P}_\phi \subseteq \Delta(\phi)\) and \(\dim(\mathcal{P}_\phi) = |\phi|- 1\). The set \(\mathcal{P}_\phi\) is extreme in \(\mathcal{K}(\Delta \Theta)\) if it cannot be written as a non-trivial convex combination, i.e., weighted Minkowski sum, of other closed and convex sets in \(\mathcal{K}(\Delta \Theta)\). That is, if \(\mathcal{P}_\phi\) is extreme and \(\alpha \mathcal{Q} + (1-\alpha) \mathcal{Q}' = \mathcal{P}_\phi\) for some \(\alpha \in [0,1]\) and \(\mathcal{Q}, \mathcal{Q}' \in \mathcal{K}(\Delta \Theta)\), then \(\mathcal{Q} = \mathcal{Q}' = \mathcal{P}_\phi\). This captures the intuition that extreme sets in \(\mathcal{K}(\Delta \Theta)\) cannot be “split" into different sets in \(\mathcal{K}(\Delta \Theta)\). We say that a prior set \(\mathcal{P}_0 = \sum_\phi \tau(\phi) \mathcal{P}_\phi\) is maximally partially identified if every \(\mathcal{P}_\phi\) is extreme.

It is easy to see that sub-simplices of the form \(\Delta(\phi)\) are extreme in \(\mathcal{K}(\Delta \Theta)\). As a result, full ambiguity prior sets \(\mathcal{P}_0 = \sum_\phi \tau(\phi) \Delta(\phi)\) are maximally partially identified by \((\tau, \Phi)\). In the special case where \(|\phi| \leq 2\) for every \(\phi \in \Phi\), the following lemma illustrates that maximal partial identification is equivalent to imposing full ambiguity.

An immediate corollary is that under (non-trivial) partial identification, maximality and full ambiguity are equivalent whenever \(|\Theta| \leq 3\). However, as illustrated in the following example, this equivalence does not extend to general state spaces with \(|\Theta| > 3\).

The following theorem proves that maximal partial identification of the underlying prior set is the necessary and sufficient condition for consistent experiments and Aumann-plausible information structures to be fully equivalent. Because the collection of full ambiguity prior sets is a strict subset of the collection of maximally partially identified sets, it follows immediately that full ambiguity is a sufficient, but not necessary condition except in the special case where \(|\phi| \leq 2\) for every \(\phi \in \Phi\).

Recall from Theorem 1 that \(\mathcal{P}_0\) being partially identified implies property 2(a). The intuition behind the sufficiency of maximal partial identification for property 2(b) follows from the fact that \(\{\mathcal{P}_\phi\}_{\phi \in \Phi}\) forms a “basis" in the space of sets contained in the support of information structures that are Aumann-plausible with respect to \(\mathcal{P}_0\). That is, if \(\mathcal{P}_y \in \mathop{\mathrm{supp}}(\mu)\) and \(\mu\) is Aumann-plausible with respect to some maximal \(\mathcal{P}_0 = \sum_\phi \tau(\phi) \mathcal{P}_\phi\), then every \(\mathcal{P}_y\) admits the decomposition \(\mathcal{P}_y = \sum_\phi \tau_y(\phi) \mathcal{P}_\phi\) for some \(\tau_y \in \Delta(\Phi)\). It turns out that we can interpret \(\mu\) as a Bayes-plausible information structure over such \(\{\tau_y\}_{y \in Y}\), in which case the experiment that induces \(\mu\) can be recovered from the version of property 2(b) in the Bayesian setting.

As for the necessity of maximal partial identification, we already know from Theorem 1 that 2(a) implies \(\mathcal{P}_0\) is partially identified by some \((\tau, \Phi)\). Now suppose, for a contradiction, that \(\mathcal{P}_0\) is not maximal, in which case there exists some \(\mathcal{P}_\phi\) that is not extreme in \(\mathcal{K}(\Delta \Theta)\). We can then construct an Aumann-plausible information structure with support points \(\mathcal{Q}, \mathcal{Q}' \subseteq \Delta(\phi)\) that average out to \(\mathcal{P}_\phi\). Similar in spirit to our discussion of conditional measures in Section 2.3, it can be shown that no consistent experiment can induce such \(\mathcal{Q}, \mathcal{Q}'\) under prior-by-prior updating, in contradiction of property 2(b).

In Section 6, we illustrate how Theorem 2 greatly simplifies the analysis of persuasion games where agents have prior sets that are maximally partially identified. The intuition mirrors that in the standard Bayesian setting, where recasting the problem in terms of the Aumann-plausible information structures allows for the application of the concavification technique à la [17].

4 Value of Information↩︎

4.1 Preliminaries↩︎

In this section, we investigate the value of information, in the sense of [8], [9], under partially identified ambiguity.¹⁵ Our main result illustrates that many of the insights from Blackwell’s original setting apply directly when ambiguity arises from partial identification, where the key step is to reformulate the problem in the standard Bayesian setting using the reduced-form parameters. As a result, our results admit relatively simple proofs and are different in spirit to much of the existing work that extends Blackwell’s original theorem of informativeness to permit the presence of ambiguity.¹⁶

Given some prior set \(\mathcal{P}_0 \in \mathcal{K}(\Delta \Theta)\) and experiment \(\pi: \Theta \to \Delta(Y)\) consistent with \(\mathcal{P}_0\), let \(\mathcal{P}_y^\pi\) denote the posterior set obtained via prior-by-prior updating and \(\mu^\pi \in \Delta(Y)\) the induced information structure. For some action space \(\mathcal{A}\), let \(A \subseteq \mathcal{A}\) be a finite action set. The DM’s utility over \(A\) is permitted to be state-dependent, such that \(u : A \times \Theta \to \mathbb{R}\). Assuming that the DM makes decisions according to the maxmin criterion, the value of the information captured in the experiment \(\pi\) can be defined as follows.

Note that the value function corresponds to the usual value of information whenever the prior set \(\mathcal{P}_0\), and consequently the posterior sets \(\mathcal{P}_y^\pi\), are singletons. Next, we define the garbling order over the collection of Blackwell experiments. The definition is identical to that in the standard Bayesian setting, capturing the intuition that any information in \(\pi_2\) can be generated by adding noise to \(\pi_1\) whenever \(\pi_2\) is a garbling of \(\pi_1\).

We say that \(\pi_1\) is more informative than \(\pi_2\), denoted \(\pi_1 \triangleright \pi_2\), if \(\pi_2\) is a garbling of \(\pi_1\). In the case where the prior set \(\mathcal{P}_0\) is a singleton, recall that Blackwell’s theorem of informativeness says that the more informative and value function orders are equivalent. The same result holds even if we permit the use of mixed strategies \(\sigma \in \Delta(A)\) when defining the value function, since \(\mathop{\mathrm{arg\,max}}_{\sigma \in \Delta(A)} \sum_{\theta, a } u(a,\theta) \sigma(a) p^\pi_y(\theta)\) always contains a pure strategy.

4.2 Comparing Experiments↩︎

We now generalize Blackwell’s original result to permit the presence of partially identified ambiguity. Let \(\mathcal{K}_{PI}\) be the collection of prior sets partially identified by some \((\tau_0, \Phi)\), where \(\Phi\) is any partition of \(\Theta\) and \(\tau_0 \in \Delta(\Phi)\) is any corresponding reduced-form prior. Since every singleton prior set \(\mathcal{P}_0 = \{p_0\}\) is partially identified by \((\{\theta\}_{\theta \in \Theta}, p_0)\), we know that \(\mathcal{K}_{PI} \supseteq \Delta(\Theta)\). For some experiment \(\pi: \Theta \to \Delta(Y)\) consistent with \(\mathcal{P}_0 = \sum_\phi \tau_0(\phi) \mathcal{P}_\phi \in \mathcal{K}_{PI}\), let \(\tau_y^\pi\) be the corresponding reduced-form posterior. Recall from Lemma 3 that each posterior set \(\mathcal{P}_y^\pi\) is partially identified by \((\tau_y^\pi, \Phi)\).

Define \(\underline{u} : A \times \Phi \to \mathbb{R}\) as \(\underline{u}( a, \phi) \equiv \min_{ p \in \mathcal{P}_\phi} \sum_{\theta} u(a, \theta) p(\theta)\), in which case the value function can be rewritten as \[\begin{align} V_{\mathcal{P}_0, A, u}(\pi) & = \mathbb{E}_{\mu^\pi} \left[ \max_{a \in A } \sum_{\phi} \tau_y^\pi(\phi) \min_{p \in \mathcal{P}_\phi} \sum_{\theta} u(a, \theta) p (\theta) \right] \nonumber \\ & = \mathbb{E}_{\mu^\pi} \left[ \max_{a \in A} \sum_{\phi} \tau_y^\pi(\phi) \underline{u}(a , \phi) \right], \nonumber \end{align}\] now interpreting \(\mu^\pi \in \Delta(Y)\) as a distribution over the reduced-form posteriors \(\{\tau_y^\pi\}_{y \in Y} \subseteq \Delta(\Phi)\). By Lemma 2, we know that \(\pi(\cdot|\theta)\) is constant on the cells of \(\Phi\). We can then reformulate the problem by taking \(\Phi\) as the state space, such that the above value function coincides with \(V_{ \{\tau_0\}, A, \underline{u}}(\pi)\).

Since Blackwell’s original theorem tells us that \(\pi_1 \triangleright \pi_2\) implies \(V_{ \{\tau_0\}, A, \underline{u}}(\pi_1) \geq V_{ \{\tau_0\}, A, \underline{u}}(\pi_2)\) for all \((\tau_0, A, \underline{u})\), the preceding discussion proves the forward direction of the following lemma. The converse also follows from Blackwell’s original theorem because \(\pi_1, \pi_2\) are consistent with any singleton \(\mathcal{P}_0\).

Recall that for singleton \(\mathcal{P}_0\), the value function remains the same regardless of whether we permit the DM to use mixed strategies \(\sigma \in \Delta(A)\). For non-singleton \(\mathcal{P}_0\), it is no longer without loss to restrict our attention to pure strategies, as there may exist strict benefits from hedging with a value function of the form \[\begin{align} W_{\mathcal{P}_0, A, u} (\pi) \equiv \mathbb{E}_{\mu^\pi} \left[ \max_{ \sigma \in \Delta(A)} \min_{p^\pi_y \in \mathcal{P}_y^\pi} \sum_{\theta, a } u(a ,\theta) \sigma(a) p^\pi_y(\theta) \right]. \nonumber \end{align}\]

Nonetheless, repeating a slightly modified version of the preceding exercise allows us to prove the analog to Lemma 5 using \(W_{\mathcal{P}_0, A, u} (\cdot)\) as the value function.¹⁷ This implies that even though \(V_{\mathcal{P}_0, A, u}(\cdot) \neq W_{\mathcal{P}_0, A, u}(\cdot)\), the order over experiments induced by the two value functions are equivalent.

5 Application: Robust Bayesian Analysis↩︎

5.1 Preliminaries↩︎

The robust Bayesian approach to statistical decision theory has recently gained attention in the econometrics literature as a tool for analyzing decision-making under set-identified models. Problems of this form arise commonly across many applied settings, such as pricing with unobserved heterogeneity and treatment choice under partial identification of the treatment effect [11], [12], [14], [36]. Unlike in the Bayesian case, decision rules that attain the robust Bayesian analog of minimum risk do not in general coincide with actions that minimize expected posterior loss [13], [35]. In this section, we provide a detailed analysis of this phenomenon and propose an alternative optimality criterion that retains the time consistency of Bayesian decision-making.

Continuing with the notation of the previous section, we define a decision rule as the sample-contingent action given by the mapping \(\delta: Y \to A\) for some finite action set \(A\). We take \(\mathcal{D}\) to be the collection of all decision rules and \(\mathcal{D}_{A}\) as the special class of constant rules that map deterministically to \(A\). The loss function is a mapping \(L: \Theta \times A \to \mathbb{R}\), where \(L(\theta, a)\) is interpreted as the loss, i.e., disutility, the DM incurs when the true state is \(\theta\) and she takes action \(a\). Given some Blackwell experiment \(\pi : \Theta \to \Delta(Y)\), the risk \(R(\theta, \delta)\) of the decision rule \(\delta\) is the expected loss at \(\theta\), such that \(R(\theta, \delta) \equiv \sum_{y} \pi(y|\theta) L(\theta, \delta(y))\).

For some prior \(p_0 \in \Delta(\Theta)\), the Bayes risk of a decision rule is \[\begin{align} r(p_0, \delta) \equiv \sum_\theta R(\theta, \delta) p_0(\theta), \end{align}\] which is the \(p_0\)-expectation of \(R(\theta, \delta)\). Given the realization of some \(y \in Y\), the posterior expected loss of the action \(a \in A\) is defined as \(\rho_y(p_0, a) \equiv \sum_\theta L(\theta,a) p_y^\pi(\theta)\). The Bayes rule \(\delta^* = \mathop{\mathrm{arginf}}_{\delta \in \mathcal{D}} r(p_0, \delta)\) minimizes \(\rho_y(p_0,a)\) at every \(y \in Y\), as \[\begin{align} \inf_{\delta \in \mathcal{D}} r(p_0, \delta) & = \inf_{\delta \in \mathcal{D}} \sum_{\theta, y} L(\theta, \delta(y)) \pi(y|\theta) p_0(\theta) \nonumber \\ & = \inf_{\delta \in \mathcal{D}} \sum_{y} \mu^\pi(y) \sum_\theta L(\theta, \delta(y)) p_y^\pi(\theta)\nonumber \\ & = \sum_y \mu^\pi(y) \inf_{a \in A} \sum_\theta L(\theta, a) p_y^\pi(\theta), \nonumber \end{align}\] where the second equality holds because \(\pi(y|\theta)p_0(\theta) = \mu^\pi(y) p_y^\pi(\theta)\) for all \((y,\theta)\).

5.2 Gamma-minimax and Conditional Gamma-minimax↩︎

In the setting with ambiguous prior beliefs described by some prior set \(\mathcal{P}_0 \in \mathcal{K}(\Delta \Theta)\), two notions of optimality exist. One takes an ex ante perspective and evaluates decision rules according to the minimax analog of Bayes risk, while the other takes an interim perspective and evaluates actions conditional on the realization of the sample space. We follow [35] and [13] by referring to them as the Gamma-minimax and conditional Gamma-minimax criteria, respectively.

Note that the Gamma-minimax rule coincides with the Bayes rule whenever \(\mathcal{P}_0\) is a singleton. However, for non-singleton \(\mathcal{P}_0\), the Gamma-minimax rule does not inherit the time consistency of the Bayes rule [13], [35], [36]. In the context of our previous discussion of rectangularity in Remark 3, observe that the Gamma-minimax rule can be rewritten as \[\begin{align} \delta^* & = \mathop{\mathrm{arginf}}_{\delta \in \mathcal{D}} \sup_{ \gamma \in \mathcal{P}_0 \otimes \pi } \sum_{\theta, y} L(\theta, \delta(y)) \gamma(\theta, y). \nonumber \end{align}\] Since the conditional Gamma-minimax criterion corresponds to a DM who makes choices after observing the realization of the sample space \(Y\), the dynamic inconsistency of \(\delta^*\) then follows directly as a consequence of the fact that the joint prior set \(\mathcal{P}_0 \otimes \pi\) is not \(\{\Theta \times \{y\}\}_{y\in Y}\)-rectangular for non-singleton \(\mathcal{P}_0\).¹⁸

In principle, the prior set \(\mathcal{P}_0 \in \mathcal{K}(\Delta \Theta)\) and experiment \(\pi : \Theta \to \Delta(Y)\) impose no structure on the manner in which the joint prior set over \(\Theta \times Y\) ought to be created. That is, time consistency of the optimal decision rule can be restored by simply redefining the optimality criterion and taking the maximum expected loss over \(\gamma \in \mathcal{P}_Y^\pi \otimes \mu^\pi\), the joint prior set rectangular with respect to the partition \(\{ \Theta \times \{y\}\}_{y \in Y}\). However, as alluded to in Remark 3 and illustrated in the next section, the ex ante valuation of actions under this optimality criterion suffers from a different shortcoming—except when the prior set is partially identified.

5.3 New Optimality Criterion Under Partial Identification↩︎

Suppose the DM has access to a \(\Phi\)-measurable experiment \(\pi : \Theta \to \Delta(Y)\) that set-identifies \(\Theta\) up to the reduced-form parameters \(\Phi\). She performs robust Bayesian inference à la [1], such that her prior beliefs are described by the prior set \(\mathcal{P}_0 = \sum_\phi \tau_0(\phi) \mathcal{P}_\phi\) for some reduced-form prior \(\tau_0 \in \Delta(\Phi)\) and \(\mathcal{P}_\phi \subseteq \Delta(\phi)\), \(\dim(\mathcal{P}_\phi)=|\phi|-1\) for all \(\phi \in \Phi\).¹⁹ Recall that Lemma 2 then implies that the experiment \(\pi\) is consistent with \(\mathcal{P}_0\).

As discussed in the previous section, the following criterion of ex ante optimality restores the time consistency of Bayes rules to the robust Bayesian setting. The definition only modifies the Gamma-minimax criterion by changing how the joint prior set is constructed over the extended space \(\Theta \times Y\), and it reduces to Bayes rule for singleton \(\mathcal{P}_0\).

Although the Gamma\({}^*\)-minimax rule coincides with the conditional Gamma minimax action, it suffers a different inconsistency: the ex ante valuation of constant decision rules, i.e., actions, do not coincide with the maximum Bayes risk over \(\mathcal{P}_0\). This is because \[\begin{align} \inf_{\delta \in \mathcal{D}_A} \sup_{\gamma \in \mathcal{P}_Y^\pi \otimes \mu^\pi} \sum_{\theta, y} L(\theta, \delta(y)) \gamma(\theta,y ) & = \inf_{a \in A} \sup_{\gamma \in \mathcal{P}_Y^\pi \otimes \mu^\pi} \sum_{\theta} L(\theta, a) \sum_{y} \gamma(\theta,y ) \nonumber \\ & = \inf_{a \in A} \sup_{p \in \sum_y \mu^\pi(y) \mathcal{P}_y^\pi } \sum_{\theta} L(\theta, a) p(\theta) \nonumber \\ & \geq \inf_{a \in A} \sup_{p_0 \in \mathcal{P}_0 } \sum_{\theta} L(\theta, a) p_0(\theta), \nonumber \end{align}\] where the last inequality holds because \(\sum_y \mu^\pi(y) \mathcal{P}_y^\pi \supseteq \mathcal{P}_0\). Aumann plausibility of \(\mu^\pi\) is then precisely the condition that requires this relation to hold with equality.

Recall from Theorem 1 that the information structure induced by any consistent experiment is Aumann-plausible if and only if the underlying prior set is partially identified. The Gamma\({}^*\)-minimax criterion thus restores the time consistency of Bayesian decision-making, while remaining compatible with the prior set \(\mathcal{P}_0\), in the robust Bayesian paradigm under partial identification.²⁰

6 Application: Partially Identified Persuasion↩︎

6.1 Preliminaries↩︎

Receiver agrees to be “persuaded" by Sender in a persuasion game à la [17] because any non-trivial experiment is more informative, and thus more valuable, than the trivial experiment that leads to no updating of the prior. We believe this reflects two important features that microfound persuasion:

1. Receiver’s expected payoff from choosing her action after the realization of the experiment is (weakly) greater than that her expected payoff at the prior;

2. Receiver’s valuation of actions, i.e, singleton action sets, is the same regardless of whether she agrees to be persuaded.

The main contribution of this section is to develop a theory of persuasion that retains both properties, while allowing for the presence of ambiguity that can be rationalized by restrictions on the conceivable collection of available data.

Continuing with the notation of the previous sections, suppose that both Sender and Receiver share a common prior set \(\mathcal{P}_0 \in \mathcal{K}(\Delta \Theta)\) that is maximally partially identified by some non-trivial \((\tau, \Phi)\).²¹ Sender and Receiver have utility functions \(v: A \times \Theta \to \mathbb{R}\) and \(u: A \times \Theta \to \mathbb{R}\), respectively, that depend on the realization of \(\Theta\) and the action \(a \in A\) that Receiver takes, and we assume that both agents are ambiguity averse à la [10]. Before the realization of \(\Theta\), Sender can choose an experiment \(\pi : \Theta \to \Delta(Y)\) consistent with \(\mathcal{P}_0\) to persuade Receiver. After observing the signal realization \(y \in Y\), Receiver takes an action from the set \(a^*(\mathcal{P}_y) = \mathop{\mathrm{arg\,max}}_{a \in A} \min_{p \in \mathcal{P}_y} \mathbb{E}_{p}[u(a,\theta)]\), where \(\mathcal{P}_y\) is the prior-by-prior update of \(\mathcal{P}_0\). As in [17], our solution concept is Sender-preferred subgame perfect equilibrium.

Suppose Sender chooses some experiment \(\pi\), and let \(\mu \in \Delta(\mathcal{K}(\Delta \Theta))\) be the corresponding information structure induced over the posterior sets \(\{\mathcal{P}_y\}_{y \in Y}\). The expected payoffs of Sender and Receiver are, respectively, \[\begin{align} V(A, \mu) & \equiv \mathbb{E}_{\mu} \left[ \min_{p \in \mathcal{P}_y} \mathbb{E}_p[ v(\hat{a} (\mathcal{P}_y) , \theta) ] \right] \nonumber \\ U(A, \mu) & \equiv \mathbb{E}_\mu\left[ \min_{p \in \mathcal{P}_y} \mathbb{E}_p[ u(\hat{a} (\mathcal{P}_y) , \theta)] \right], \nonumber \end{align}\] where \(\hat{a}( \mathcal{P} )\) denotes Receiver’s equilibrium action at the belief \(\mathcal{P} \in \mathcal{K}(\Delta \Theta)\). Let \(\mu_0\) denote the degenerate information structure that maps deterministically to \(\mathcal{P}_0\). In symbols, properties (i) and (ii) then translate to \(U(A,\mu) \geq U(A,\mu_0)\) and \(U(\{a\}, \mu) = U(\{a\}, \mu_0)\) for every action set \(A\) and action \(a\). The following result shows that the Aumann plausibility of \(\mu\) is equivalent to these two properties.

Because \(\mathcal{P}_0\) is maximally partially identified, the equivalence characterization between consistent experiments and Aumann-plausible information structures from Theorem 2 applies. Thus, restricting Sender to the collection of consistent experiments is equivalent to requiring that properties (i) and (ii) are satisfied, and Receiver’s decision to participate in the persuasion game can be microfounded.²²

6.2 Reducing to Bayesian Persuasion↩︎

As in Section 4, the main strategy for solving the persuasion game will be to express the problem in terms of the reduced-form parameters. Because the prior set \(\mathcal{P}_0\) is maximally partially identified by \((\tau_0, \Phi)\), Lemma 3 implies that the posterior sets \(\mathcal{P}_y\) are partially identified by the reduced-form posteriors \(\tau_y\) over the same partition. Sender’s expected payoff can then be written as \[\begin{align} V(A,\mu) & = \mathbb{E}_\mu \left[ \sum_{\phi} \tau_y(\phi) \min_{p \in \mathcal{P}_\phi} \mathbb{E}_p[v( \hat{a}(\tau_y), \theta)] \right], \nonumber \end{align}\] where \(\hat{a}(\tau_y) \equiv \hat{a}(\mathcal{P}_y)\) and \(\mu\) is now interpreted as an information structure over the reduced-form posteriors. Receiver’s expected payoff is expressed analogously.

Define \(\underline{v}( a, \phi) \equiv \min_{p \in \mathcal{P}_\phi} \mathbb{E}_p[v( a, \theta)]\) and \(\underline{u}(a, \phi) \equiv \min_{p \in \mathcal{P}_\phi} \mathbb{E}_p[u( a, \theta)]\), and interpret these as Sender and Receiver’s utility functions, respectively, over \(A \times \Phi\). Observe that our formulation of the persuasion game can now be entirely expressed using \(\Phi\) and the reduced-form measures \(\tau_0, \tau_y \in \Delta(\Phi)\). Take \(\hat{v}( \tau) \equiv \sum_\phi \tau(\phi) \underline{v} (\hat{a}(\tau), \phi)\), such that Sender’s expected payoff is \(V(A,\mu) = \mathbb{E}_\mu[ \hat{v}(\tau_y) ]\). The characterization of Sender’s problem in the following proposition then corresponds to that in the standard setting of Bayesian persuasion.

Define the curve \(K = \{ (\tau, \hat{v}(\tau)) : \tau \in \Delta(\Phi)\}\). The concavification of \(\hat{v}\) over \(\Delta(\Phi)\) is \(\overline{V}(\tau) \equiv \sup \{ z : (\tau, z) \in \mathop{\mathrm{co}}(K) \}\), and it follows directly from [17] that Sender’s optimal payoff is \[\begin{align} \overline{V}(\tau_0) = \sup \{ z: (\tau_0, z) \in \text{co}(K) \}. \nonumber \end{align}\] All existing equilibrium characterizations in the Bayesian setting also apply directly to our framework, once everything is interpreted over \(\Phi\).

7 Discussion↩︎

In this section, we discuss the connection between partially identified ambiguity and the axiomatic literature on identifying such beliefs from preferences. For simplicity, we restrict our discussion to the case of full ambiguity, such that prior sets admit the decomposition \(\mathcal{P}_0 = \sum_\phi \tau(\phi) \Delta(\phi)\).

Let \(\mathcal{F}\) be the collection of acts \(f: \Theta \to X\) mapping to some outcome space \(X\). The DM’s preferences over \(\mathcal{F}\) is defined by the binary relation \(\succsim\), and we assume that the utility function \(u: X \to \mathbb{R}\) represents her preferences over the collection of constant acts that deterministically map to some \(x \in X\). For \(f,g \in \mathcal{F}\), we write \(f \geq g\) if \(f(\theta) \succsim g(\theta)\) for all \(\theta \in \Theta\).²³ If we assume that DMs with the prior set \(\mathcal{P}_0\) evaluate alternatives according to the maxmin criterion, the corresponding utility representation for \(\succsim\) is \[\begin{align} U_{MEU(\tau, \Phi)}(f) & \equiv \min_{p \in \mathcal{P}_0} \mathbb{E}_p[u(f)] = \sum_{\phi \in \Phi} \tau(\phi) \min_{ \theta \in \phi} u(f(\theta)). \nonumber \end{align}\]

Partially Specified Probabilities. In the Anscombe-Aumann setting (i.e., \(X\) is the collection of lotteries over some finite prize space), [6] axiomatizes the utility representation \[\begin{align} U_{Lehrer}(f) & = \sup\left\{ \sum_{i \in I} \lambda_i \mathbb{E}_p[u(g_i)] : \sum_{i \in I} \lambda_i g_i \leq f, \,\, \lambda_i \in \mathbb{R}\right\} , \nonumber \end{align}\] where \(p \in \Delta(\Theta)\) and \(\mathcal{G} = \{g_i\}_{i \in I}\) spans the collection of acts for which the DM knows the value of \(\mathbb{E}_p[u(g_i)]\). Let \(\mathbb{1}_\phi \equiv \mathbb{1}_{ \{\cdot \, \in \phi\} }\) be the indicator variable corresponding to each reduced-form parameter \(\phi \in \Phi\), in which case \(U_{Lehrer}(\cdot) = U_{MEU(\tau, \Phi)}(\cdot)\) when \(\mathcal{G} = \{ \mathbb{1}_\phi\}_{\phi \in \Phi}\).

Choquet expected utility (CEU) with inner probabilities. In the Savage setting, [4] axiomatizes a special case of the CEU representation of [47] where the corresponding capacity is an inner measure. A subjective \(\lambda\)-system \(\mathcal{L}\) of “unambiguous" events is recovered from preferences, and the DM has SEU preferences over the collection of \(\mathcal{L}\)-measurable acts \(\mathcal{F}^{\mathcal{L}}\). Letting \(p\) be the corresponding SEU measure, the representation takes the form \[\begin{align} U_{Zhang}(f) = \sup \left\{ \int g \, \, \mathrm{d}p : g \leq f, \, \, g \in \mathcal{F}^{\mathcal{L}} \right\}. \nonumber \end{align}\] In the special case where \(\mathcal{L}\) is the \(\sigma\)-algebra generated by the reduced-form parameters \(\Phi\), it follows that \(U_{MEU(\tau, \Phi)}(\cdot) = U_{Zhang}(\cdot)\).²⁴

Finally, we conclude by moving to a richer domain of acts in which the DM’s prior set can be explicitly interpreted as the Aumann expectation of the posterior sets obtained from fully informative consistent experiments.

Set-identifiable smooth representation. Suppose the DM envisages infinitely many independent realizations from the consistent experiment \(\pi : \Theta \to \Delta(Y)\), such that the joint space is \(\Theta \times Y^\infty\). By the strong law of large numbers, the realization of \(y^\infty\) identifies the reduced form parameter \(\phi_{y^\infty} \in \Phi\), and thus the true marginal over the data space \(\pi(\cdot|\phi_{y^\infty})\). In the language of [27], the realization of \((\theta, y^\infty)\) identifies the set of joint laws \[\begin{align} \Gamma_{(\theta, y^\infty)} = \left\{ \gamma(\cdot, \ast) = p(\cdot) \pi(\ast | \phi_{y^\infty})^{\otimes \infty} : p \in \Delta(\phi_{y^\infty}) \right\}, \end{align}\] and the corresponding collection of marginals over \(\Theta\) is \(\int_{Y^\infty} \Gamma_{(\theta, y^\infty)} \equiv \Delta(\phi_{y^\infty})\).

Let \(\mathfrak{G} \equiv \{ \Gamma_{(\theta, y^\infty)}\}_{ (\theta, y^\infty) \in \Theta \times Y^\infty}\) be the collection of identified sets. Taking preferences over \(\Theta \times Y^\infty\)-measurable acts as the primitive, [27] axiomatizes the set-identifiable smooth representation given by \[\begin{align} U_{DP}(f) = \int_{ \mathfrak{G}} \psi \left( \min_{\gamma \in \Gamma } \int_{ \Theta \times Y^\infty } u(f) \, \, \mathrm{d}\gamma \right) \, \mathrm{d}\mu(\Gamma), \end{align}\] where \(\psi : \mathbb{R}\to \mathbb{R}\) is continuous and strictly increasing, and \(\mu\) is a non-atomic measure over \(\mathfrak{G}\).²⁵ If \(\psi(\cdot)\) is affine and \(f\) is \(\Theta\)-measurable, we have \[\begin{align} \int_{\mathfrak{G}} \left( \min_{\gamma \in \Gamma } \int_{ \Theta \times Y^\infty } u(f) \, \, \mathrm{d}\gamma \right) \, \mathrm{d}\mu(\Gamma) & = \int_{\mathfrak{G}} \left( \min_{ p \in \int_{Y^\infty} \Gamma } \int_{ \Theta} u(f) \, \, \mathrm{d}p \right) \, \mathrm{d}\mu(\Gamma) \\ & = \sum_\phi \tau(\phi) \min_{p \in \Delta(\phi)} \int_\Theta u(f) \, \, \mathrm{d}p \end{align}\] for \(\tau(\phi) \equiv \mu ( \{ \Gamma_{(\theta, y^\infty)} \in \mathfrak{G} : \phi_{y^\infty} = \phi \} )\). In this special case, we then see that \(U_{MEU(\tau ,\Phi)}(\cdot) = U_{DP}(\cdot)\) over the collection of \(\Theta\)-measurable acts.

8 Conclusion↩︎

This paper presents a theory of partially identified ambiguity, in connection with the econometrics literature on robust Bayesian inference under partial identification. We propose a novel interpretation of the multiple priors model as imposing a consistency constraint on the collection of conceivable experiments, and introduce Aumann plausibility as a set-valued generalization of the martingale property of beliefs. Within this framework, we extend the standard equivalence between Blackwell experiments and Bayes-plausible distributions over posterior beliefs to the setting of ambiguity, deriving a sharp characterization of the necessary and sufficient conditions on the class of prior sets under which this equivalence holds. We also show that the more valuable order induced by maxmin expected utility under partially identified ambiguity corresponds to the more informative order over Blackwell experiments. Finally, we highlight the value of our results across various domains by applying our theory of partially identified ambiguity to the Gamma minimax problem in robust Bayesian analysis, as well as an information design problem in the context of a persuasion game.

9 Omitted Proofs↩︎

For any partition \(\Phi\) of \(\Theta\), recall that \(\Delta(\phi) \subseteq \Delta(\Theta)\) denotes the collection of measures that are supported on the partition cell \(\phi \in \Phi\). If an experiment \(\pi: \Theta \to \Delta(Y)\) is \(\Phi\)-measurable, we write \(\pi(\cdot| \phi)\) to denote \(\pi(\cdot|\theta)\) for any \(\theta \in \phi\) without loss. The following preliminary result will be invoked multiple times throughout the remainder of this section.

9.1 Proof of Lemma 1↩︎

Define \(D \equiv \text{span}\{ p - p' : p, p' \in \mathcal{P}\}\) and let \(V \equiv D^\perp\) denote its orthogonal complement. Suppose \(\pi: \Theta \to \Delta(Y)\) is a non-trivial experiment consistent with \(\mathcal{P}\). The definition of consistency implies \(\sum_{\theta} \pi(y|\theta) [p(\theta) - p'(\theta) ] = 0\) for every \(p, p' \in \mathcal{P}\) and \(y \in Y\). In other words, \(V \supseteq \text{span} \{\boldsymbol{1}, \{\pi(y|\cdot)\}_{y \in Y} \}\), and we know that \(\dim(V) \geq 2\) because \(\{\pi(y|\cdot)\}_{y \in Y}\) is not co-linear with the constant vector \(\boldsymbol{1} \in \mathbb{R}^\Theta\) for at least one \(y \in Y\) by non-triviality. The desired claim then follows from noting that \(\dim(\mathcal{P}) = \dim(D) = \dim(\Theta) - \dim(V) = \dim(\Delta \Theta) - \dim(V) + 1\).

Next, fix \(Y = \{y_1, y_2\}\) and suppose \(\dim(\mathcal{P}) \leq \dim(\Delta \Theta) - 1\), such that \(\dim(V) \geq 2\). Choose some \(e \in V\) that is not co-linear with \(\boldsymbol{1}\). Define \[\begin{align} \pi(y_1|\theta) \equiv \frac{1}{2} \boldsymbol{1} + \epsilon e(\theta),\pi(y_2|\theta) \equiv \frac{1}{2} \boldsymbol{1} - \epsilon e(\theta), \nonumber \end{align}\] for some \(\epsilon > 0\) small enough, such that \(\pi(\cdot | \theta) \in \Delta(Y)\) for every \(\theta \in \Theta\). By construction, \(\pi\) is a non-trivial experiment. That \(\pi\) is consistent with \(\mathcal{P}\) follows from observing that for any \(p, p' \in \mathcal{P}\), \[\begin{align} \sum_{\theta} \pi(y_i|\theta) [p(\theta) -p'(\theta)] & = \sum_{\theta} \left( \frac{1}{2} \boldsymbol{1} \pm \epsilon e(\theta) \right) [p(\theta) -p'(\theta)] \nonumber \\ & = \frac{1}{2} \sum_{\theta} \boldsymbol{1} [p(\theta) -p'(\theta)] \pm \epsilon \sum_{\theta} e(\theta) [p(\theta) -p'(\theta)] = 0, \nonumber \end{align}\] where the last equality follows from noting that \(\boldsymbol{1}, e \in V\) by construction. \(\blacksquare\)

9.2 Proof of Lemma 2↩︎

2 \(\Rightarrow\) 1. Immediate from the second statement of Lemma 6.

1 \(\Rightarrow\) 2. Suppose \(\pi : \Theta \to \Delta(Y)\) is consistent with \(\mathcal{P}\). Since \(\mathcal{P}\) is partially identified by \((\tau, \Phi)\), it admits the decomposition \(\mathcal{P} = \sum_\phi \tau(\phi) \mathcal{P}_\phi\). Fix \(\phi \in \Phi\) and choose any \(p_\phi, \tilde{p}_\phi \in \mathcal{P}_\phi\) and \(\{p_{\phi'}\}_{\phi' \neq \phi} \in \{ \mathcal{P}_{\phi'} \}_{\phi' \neq \phi}\). Define \[\begin{align} p (\cdot) & = \tau(\phi) p_\phi(\cdot) + \sum_{\phi' \neq \phi} \tau(\phi') p_{\phi'} (\cdot),\tilde{p} (\cdot) = \tau(\phi) \tilde{p}_\phi (\cdot) + \sum_{\phi' \neq \phi} \tau(\phi') p_{\phi'} (\cdot), \nonumber \end{align}\] which are both elements of \(\mathcal{P}\). Consistency of \(\pi\) then implies \[\begin{align} 0 & = \sum_{\theta \in \Theta} \pi(\cdot | \theta) [p(\theta) - \tilde{p}(\theta)] = \tau(\phi) \sum_{\theta \in \phi} \pi(\cdot | \theta) [ p_\phi(\theta) - \tilde{p}_\phi(\theta)]. \nonumber \end{align}\] Since \(\tau\) has full support, it follows that \(\sum_{\theta \in \phi} \pi(\cdot | \theta) [ p_\phi(\theta) - \tilde{p}_\phi(\theta)] = 0\).

For \(\theta', \theta'' \in \phi\), define \(d_{\theta', \theta''}(\cdot) = \mathbb{1}_{ \{\cdot \, = \theta'\} } - \mathbb{1}_{ \{\cdot \, = \theta''\} }\). Because \(\dim(\mathcal{P}_\phi) = \dim(\Delta(\phi))\), we can express \(d_{\theta', \theta''}(\cdot)\) as a linear combination of the elements in \(\{p - p' : p,p' \in \mathcal{P}_\phi\}\), which implies \[\begin{align} 0 & = \sum_{\theta \in \phi} \pi(\cdot | \theta) d_{\theta', \theta''}(\theta) = \pi(\cdot | \theta') - \pi(\cdot | \theta''). \nonumber \end{align}\] This proves that \(\pi(\cdot | \theta)\) is constant over \(\theta \in \phi\) for every \(\phi \in \Phi\), as desired. \(\blacksquare\)

9.3 Proof of Lemma 3↩︎

Let \(\mu \in \Delta(Y)\) be the information structure induced by \(\pi\), and define \(\tilde{\mathcal{P}}_y \equiv \sum_{\phi} \tau_y(\phi) \mathcal{P}_\phi\). Since \(\pi\) is consistent with \(\mathcal{P}_0\), Lemma 2 implies that it is \(\Phi\)-measurable. Then, \(\mu(y) \equiv \sum_\theta \pi(y|\theta) p_0(\theta) = \sum_{\phi} \pi(y|\phi) \tau_0(\phi)\) by Lemma 6. For any \(p_y \in \mathcal{P}_y\), there exists some \(p_0 \in \mathcal{P}_0\) such that \(p_y(\cdot) = \frac{\pi(y|\phi(\cdot)) p_0(\cdot)}{\mu(y)}\). Since \(\mathcal{P}_0\) is partially identified by \((\tau_0, \Phi)\), we have \(p_0(\cdot) = \sum_\phi \tau_0(\phi) p_\phi(\cdot) = \tau_0(\phi(\cdot)) p_{ \phi(\cdot)}(\cdot)\) for some \(\{p_\phi\}_{\phi \in \Phi} \in \{\mathcal{P}_\phi\}_{\phi \in \Phi}\). Simplifying, we obtain \[\begin{align} p_y(\cdot) = \frac{\pi(y|\phi(\cdot)) p_0(\cdot)}{\mu(y)} = \frac{\pi(y|\phi(\cdot)) \tau_0(\phi(\cdot))}{\sum_\phi \pi(y|\phi) \tau_0(\phi)} p_{ \phi(\cdot) }(\cdot) = \tau_y(\phi(\cdot)) p_{\phi(\cdot)}(\cdot). \nonumber \end{align}\] This implies \(p_y = \sum_\phi \tau_y(\phi) p_\phi \in \tilde{\mathcal{P}}_y\), proving that \(\mathcal{P}_y \subseteq \tilde{\mathcal{P}}_y\).

Now suppose \(p_y \in \tilde{\mathcal{P}}_y\), such that there exists some \(\{p_\phi\}_{\phi \in \Phi} \in \{\mathcal{P}_\phi\}_{\phi \in \Phi}\) satisfying \(p_y(\cdot) = \sum_\phi \tau_y(\phi) p_\phi(\cdot) = \tau_y(\phi(\cdot)) p_{\phi (\cdot)}(\cdot)\). The same string of equalities as above implies the existence of some \(p_0 \in \mathcal{P}_0\) such that \(p_y\) is the prior-by-prior update of \(p_0\) under \(\pi\), proving that \(\mathcal{P}_y \supseteq \tilde{\mathcal{P}}_y\), as desired. \(\blacksquare\)

9.4 Proof of Theorem 1↩︎

1 \(\Rightarrow\) 2. Suppose \(\pi: \Theta \to \Delta(Y)\) is consistent with \(\mathcal{P}_0\), such that we can define \(\mu(\cdot) \equiv \sum_{\theta} \pi(\cdot | \theta)p_0(\theta)\) for some \(p_0 \in \mathcal{P}_0\) without loss. By Lemma 3, the prior-by-prior update of \(\mathcal{P}_0\) takes the form \(\mathcal{P}_y = \sum_\phi \tau_y(\phi) \mathcal{P}_\phi\) for every \(y \in Y\). Then, \[\begin{align} \sum_y \mu(y) \mathcal{P}_y = \sum_y \mu(y) \sum_\phi \tau_y(\phi) \mathcal{P}_\phi & = \sum_\phi \sum_y \mu(y) \tau_y(\phi) \mathcal{P}_\phi \nonumber \\ & = \sum_\phi \left( \sum_y \mu(y) \tau_y(\phi) \right) \mathcal{P}_\phi _\phi} \\ & = \sum_\phi \left( \sum_y \pi(y|\phi) \tau_0(\phi) \right) \mathcal{P}_\phi = \sum_{\phi} \tau_0(\phi) \mathcal{P}_\phi = \mathcal{P}_0, \nonumber \end{align}\] as desired. The third equality is due to the standard convex analysis fact that if \(A\) is a convex set and \(\alpha, \beta \geq 0\), then \(\alpha A + \beta A = (\alpha + \beta) A\). The existence of a non-trivial consistent experiment follows from observing that the partition-revealing experiment \(\pi : \Theta \to \Delta(\Phi)\), \(\pi(\phi|\theta) = \mathbb{1}_{ \{\theta \in \phi\}}\) is consistent with \(\mathcal{P}_0\).

2 \(\Rightarrow\) 1. Let \(D \equiv \text{span}\{ p - p' : p,p' \in \mathcal{P}_0\}\) and take \(V \equiv D^\perp\) as the orthogonal complement of \(D\). For any experiment \(\pi: \Theta \to \Delta(Y)\) consistent with \(\mathcal{P}_0\), we have \(\sum_\theta \pi(y | \theta) [ p_0(\theta) - p_0'(\theta)] = 0\) for every \(p_0, p_0' \in \mathcal{P}_0\). This implies \(\pi(y|\cdot) \in V\) for every \(y \in Y\).

We have by assumption that every information structure \(\mu\) induced by a consistent experiment \(\pi\) is Aumann-plausible. For any \(\{p_0^y\}_{y \in Y} \subseteq \mathcal{P}_0\), posterior measures of the form \(\frac{\pi(y|\cdot)p_0^y(\cdot)}{\mu(y)}\) are contained in the posterior set \(\mathcal{P}_y\). Aumann plausibility then implies \[\begin{align} \sum_y \mu(y) \frac{\pi(y|\cdot)p_0^y(\cdot)}{\mu(y)} = \sum_y \pi(y|\cdot) p_0^y(\cdot) \in \mathcal{P}_0 \end{align}\] for every \(\{p_0^y\}_{y \in Y} \subseteq \mathcal{P}_0\).

An important corollary is that the collection \(V\) of linear functionals constant over \(\mathcal{P}_0\) is closed under multiplication.

Define the equivalence relation \[\begin{align} \theta \sim \theta'\Leftrightarrowv(\theta) = v(\theta') \,\,\, \text{ for all } v \in V \nonumber \end{align}\] over \(\Theta\), and let \(\Phi\) be the partition over \(\Theta\) induced by the equivalence classes given by \(\sim\). For any \(v \in V\), let \(v(\phi)\) denote the value of \(v(\theta)\) for any \(\theta \in \phi\). Fix \(\phi \in \Phi\), and observe that for every \(\phi' \in \Phi \setminus \{\phi\}\), there exists some \(v_{\phi,\phi'} \in V\) such that \(v_{\phi,\phi'}(\phi) \neq v_{\phi,\phi'}(\phi')\) by the definition of \(\Phi\).

Since \(\frac{v_{\phi,\phi'}(\theta) - v_{\phi,\phi'}(\phi') }{ v_{\phi,\phi'}(\phi) - v_{\phi,\phi'}(\phi')} = 1\) for \(\theta \in \phi\) and \(\frac{v_{\phi,\phi'}(\theta) - v_{\phi,\phi'}(\phi') }{ v_{\phi,\phi'}(\phi) - v_{\phi,\phi'}(\phi')} = 0\) for \(\theta \in \phi'\), we can construct the Lagrange basis polynomial \[\begin{align} \xi_\phi(\cdot) \equiv \prod_{\phi' \in \Phi \setminus \{\phi\} } \frac{v_{\phi,\phi'}(\cdot) - v_{\phi,\phi'}(\phi') }{ v_{\phi,\phi'}(\phi) - v_{\phi,\phi'}(\phi')} = \mathbb{1}_{ \{\cdot \, \in \phi\}}, \nonumber \end{align}\] which serves as the indicator function for the partition cell \(\phi\). Lemma 8 implies \(\xi_\phi \in V\), and repeating the argument for every \(\phi \in \Phi\) yields \(\text{span}\{ \mathbb{1}_{ \{\cdot \, \in \phi \} } \}_{\phi \in \Phi} \subseteq V\). Since every \(v \in V\) is constant over the partition cells in \(\Phi\), we also have \(\text{span}\{ \mathbb{1}_{ \{\cdot \, \in \phi \} } \}_{\phi \in \Phi} \supseteq V\). As a result, we have \(\text{span}\{ \mathbb{1}_{ \{\cdot \, \in \phi \} } \}_{\phi \in \Phi} = V\), and \[\begin{align} 0 & = \sum_{\theta} \mathbb{1}_{ \{ \theta \in \phi \} } [ p(\theta) - p'(\theta)] = \sum_{\theta \in \phi} p(\theta) - \sum_{\theta \in \phi} p'(\theta). \nonumber \end{align}\] for every \(p,p' \in \mathcal{P}_0\) and \(\phi \in \Phi\).

Define the reduced form prior \(\tau \in \Delta(\Phi)\) by taking \(\tau(\phi) = \sum_{\theta \in \phi} p(\theta)\) for any \(p \in \mathcal{P}_0\) without loss. The previous lemma implies that the partition-revealing experiment \(\pi: \Theta \to \Delta(\Phi)\), \(\pi(\phi|\theta) = \mathbb{1}_{ \{ \theta \in \phi \} }\) is non-trivial. Observe that \(\pi\) is consistent with \(\mathcal{P}_0\) because \(\sum_\theta \mathbb{1}_{ \{ \theta \in \phi \} } p(\theta) = \tau(\phi)\) for any \(p \in \mathcal{P}_0\). The reduced-form prior \(\tau\) can then be interpreted as the information structure induced by \(\pi\), in which case Aumann plausibility implies \[\begin{align} \mathcal{P}_0 = \sum_{\phi \in \Phi} \tau(\phi) \mathcal{P}_\phi, \nonumber \end{align}\] where \(\mathcal{P}_\phi \in \mathcal{K}(\Delta \Theta)\) is the prior-by-prior update of \(\mathcal{P}_0\) given \(\phi\). That \(\mathcal{P}_\phi \subseteq \Delta(\phi)\) follows from noting that every \(p(\cdot|\phi) \in \mathcal{P}_\phi\) takes the form \(p(\cdot| \phi) = \frac{ p(\cdot) \mathbb{1}\{\cdot \in \phi\} }{\tau(\phi)}\) for some \(p \in \mathcal{P}_0\).

It remains to show that \(\dim(\mathcal{P}_\phi) = \dim(\Delta(\phi))\) for every \(\phi \in \Phi\). This holds immediately when \(\phi\) is a singleton, in which case \(\dim(\mathcal{P}_\phi) = \dim(\Delta(\phi)) = 0\). We henceforth assume \(|\phi| \geq 2\).

Since \(\mathcal{P}_\phi \subseteq \Delta(\phi)\), we also have \(\dim(\mathcal{P}_\phi) \leq \dim(\Delta(\phi))\). Combined with the previous lemma, we can thus conclude that \(\dim(\mathcal{P}_\phi) = \dim(\Delta(\phi))\), as desired. \(\blacksquare\)

9.5 Proof of Lemma 4↩︎

Since all full ambiguity sets partially identified by \((\tau, \Phi)\) are maximally partially identified by \((\tau, \Phi)\), it remains to show the converse also holds when \(|\phi| \leq 2\). Suppose \(\mathcal{P} \in \mathcal{K}(\Delta \Theta)\) is maximally partially identified by \((\tau, \Phi)\), such that \(\mathcal{P} = \sum_\phi \tau(\phi) \mathcal{P}_\phi\). It suffices to show that if \(\mathcal{P}_\phi\) is extreme in \(\mathcal{K}(\Delta \Theta)\), then \(\mathcal{P}_\phi = \Delta(\phi)\).

Since \(\mathcal{P}_\phi \subseteq \Delta(\phi)\), the desired claim follows immediately if \(|\phi| = 1\). Now assume \(|\phi| = 2\) and associate \(\mathcal{P}_\phi\) with the range of values it assigns to \(\theta \in \{\theta, \theta'\} \equiv \phi\). Suppose, for a contradiction, that \(\mathcal{P}_\phi = [a,b] \neq [0,1]\). If \(a \neq 0\), there exists some \(\epsilon > 0\) small enough such that the \(\epsilon\)-ball around \(a\) is contained in \([0,1]\). We can then define the intervals \(\mathcal{Q} = [a-\epsilon, b]\), \(\mathcal{Q}' = [a+ \epsilon, b]\) such that \(\frac{1}{2} \mathcal{Q} + \frac{1}{2} \mathcal{Q}' = \mathcal{P}_\phi\), a contradiction. The same argument holds for the case where \(b \neq 1\). \(\blacksquare\)

9.6 Proof of Theorem 2↩︎

1 \(\Rightarrow\) 2. Suppose \(\mathcal{P}_0\) is maximally partially identified by some non-trivial \((\tau, \Phi)\). That any experiment consistent with \(\mathcal{P}_0\) induces an Aumann-plausible information structure follows immediately from the sufficiency of partial identification in Theorem 1. It remains to show that any Aumann-plausible information structure can be induced by a consistent experiment.

Let \(\mu\) be an Aumann-plausible information structure with \(\mathop{\mathrm{supp}}(\mu) = \{ \mathcal{P}_y\}_{y \in Y}\). The following preliminary result will be useful in our analysis.

Consider the linear functional \(f_\phi: \mathbb{R}^\Theta \to \mathbb{R}\) defined by \(f_\phi(p) = \sum_{\theta \in \phi} p(\theta)\). Since \(\mathcal{P}_0\) is partially identified by \((\tau, \Phi)\), Lemma 6 implies \(f_\phi(p_0) = \tau(\phi)\) for every \(p_0 \in \mathcal{P}_0\) and \(\phi \in \Phi\). From Lemma 11, it follows that \[\begin{align} \{ \tau(\phi)\} = f_\phi \left( \mathcal{P}_0 \right) & = f_\phi \left( \sum_y \mu(y) \mathcal{P}_y \right) \nonumber \\ & = \left[ \sum_y \mu(y) \min_{p \in \mathcal{P}_y} f_\phi(p), \, \sum_y \mu(y) \max_{p \in \mathcal{P}_y} f_\phi(p) \right], \nonumber \end{align}\] in which case \(\min_{p \in \mathcal{P}_y} f_\phi(p) = \max_{p \in \mathcal{P}_y} f_\phi(p)\) and thus \(f_\phi(p) = f_\phi(p')\) for every \(p,p' \in \mathcal{P}_y\) and \(y \in Y\). We can then define \(\tau_y(\phi) \equiv \sum_{\theta \in \phi} p(\theta)\) for any \(p \in \mathcal{P}_y\) without loss, and it follows that \[\begin{align} \tau(\phi) & = \sum_y \mu(y) \tau_y(\phi). \end{align}\] In other words, \(\mu\) can be interpreted as a distribution over \(\{\tau_y\}_{y \in Y}\) that is Bayes-plausible with respect to \(\tau\).

The partition-revealing experiment is consistent with \(\mathcal{P}_y\) as \(\sum_{\theta} \mathbb{1}_{ \{\theta \in \phi\} } p_y(\theta) = \sum_{\theta \in \phi} p_y(\theta) = \tau_y(\phi)\) for every \(\phi \in \Phi\). Moreover, the previous lemma implies that it induces the information structure \(\tau_y \in \Delta(\Phi)\) over \(\{\mathcal{P}_\phi\}_{\phi \in \Phi}\). We now show that \(\tau_y\) is Aumann-plausible with respect to \(\mathcal{P}_y\).

Recall from \((\ast\ast)\) that \(\mu\) can be interpreted as a Bayes-plausible information structure over the collection of reduced-form posteriors \(\{\tau_y\}_{y \in Y}\). The equivalence between Bayes-plausible information structures and experiments in the Bayesian setting then implies the existence of some experiment \(\pi_r : \Theta \to \Delta(Y)\) that induces \(\mu\). By Lemma 3, the prior-by-prior update of \(\mathcal{P}_0\) under \(\pi_r\) takes the form \(\mathcal{P}_{y}^{\pi_r} = \sum_{\phi} \tau_y(\phi) \mathcal{P}_\phi\). Since we know from Lemma 13 that \(\mathcal{P}_y = \sum_\phi \tau_y(\phi) \mathcal{P}_\phi\) for every \(\mathcal{P}_y \in \mathop{\mathrm{supp}}(\mu)\), it follows that \(\mathcal{P}_y = \mathcal{P}_{y}^{\pi_r}\) for all \(y \in Y\). In other words, we can conclude that \(\mu\) is supported on the posterior sets induced by the experiment \(\pi_r\) under prior-by-prior updating, as desired.

2 \(\Rightarrow\) 1. The necessity of partial identification in Theorem 1 implies \(\mathcal{P}_0 = \sum_\phi \tau(\phi) \mathcal{P}_\phi\), where \(\mathcal{P}_\phi \subseteq \Delta(\phi)\) and \(\dim(\mathcal{P}_\phi) = |\phi| - 1\) for every \(\phi \in \Phi\). Suppose, for a contradiction, \(\mathcal{P}_\phi\) is not maximal for some \(\phi \in \Phi\), i.e., there exists a collection of sets \(\{ \mathcal{Q}_i \}_{i \in I} \subseteq \mathcal{K}(\Delta \Theta)\), \(\mathcal{Q}_i \neq \mathcal{P}_\phi\) and \(\nu \in \Delta(I)\) such that \(\mathcal{P}_\phi = \sum_i \nu(i) \mathcal{Q}_i\).

Next, we show that no \(\Phi\)-measurable experiment can induce two different posterior sets \(\mathcal{Q} \neq \mathcal{Q}'\) that are subsets of the same \(\Delta(\phi)\). The result follows from our discussion of the invariance of \(\phi\)-conditional measures in Section 2.3.

Now define \(Y \equiv (\Phi \setminus \{\phi\}) \cup I\) and consider the information structure \(\mu \in \Delta(Y)\), \[\begin{align} \mu(y) & = \begin{cases} \tau(y) & \text{if } y = \phi' \in \Phi \setminus \{\phi\} \\ \tau(\phi) \nu(y) & \text{if } y \in I, \end{cases} \nonumber \end{align}\] with \(\mathop{\mathrm{supp}}(\mu) = \{ \mathcal{P}_{\phi'}\}_{\phi' \in \Phi \setminus \{\phi\} } \cup \{ \mathcal{Q}_i\}_{i \in I}\). Observe that \[\begin{align} \sum_{y \in Y} \mu(y) \mathcal{P}_y & = \sum_{\phi' \neq \phi} \tau(\phi') \mathcal{P}_{\phi'} + \tau(\phi) \sum_{i \in I} \nu(i) \mathcal{Q}_i = \sum_{\phi' \neq \phi} \tau(\phi') \mathcal{P}_{\phi'} + \tau(\phi) \mathcal{P}_\phi = \mathcal{P}_0, \nonumber \end{align}\] implying that \(\mu\) is Aumann-plausible with respect to \(\mathcal{P}_0\). By assumption, there then exists an experiment \(\pi: \Theta \to \Delta(Y)\) consistent with \(\mathcal{P}_0\) that induces \(\mu\). We then see that Lemma 15 yields a contradiction, as \(\mathcal{Q}_i \subseteq \Delta(\phi)\) by Lemma 14, yet we have \(\{\mathcal{Q}_i\}_{i \in I} \subseteq \mathop{\mathrm{supp}}(\mu)\) and \(\mathcal{Q}_{i} \neq \mathcal{Q}_{i'}\) for \(i,i' \in I\). \(\blacksquare\)

9.7 Proof of Theorem 3↩︎

Lemma 5 proves the equivalence between first and second statements. That 3 implies 1 follows immediately from noting that \(\pi_1,\pi_2\) are consistent with any singleton prior set, and \(W_{\mathcal{P}_0, A, u}(\cdot) = V_{\mathcal{P}_0, A, u}(\cdot)\) for singleton \(\mathcal{P}_0\). To see that 1 implies 3, define \(\underline{u}(\sigma, \phi) \equiv \min_{p \in \mathcal{P}_\phi} \sum_{\theta, a } u(a ,\theta) \sigma(a) p(\theta)\) and observe that the value function can be rewritten as \[\begin{align} W_{\mathcal{P}_0, A, u}(\pi) & = \mathbb{E}_{\mu^\pi} \left[ \max_{ \sigma \in \Delta(A)} \sum_{\phi} \tau_y^\pi(\phi) \min_{p \in \mathcal{P}_\phi} \sum_{\theta, a } u(a ,\theta) \sigma(a) p(\theta) \right] \nonumber \\ & = \mathbb{E}_{\mu^\pi} \left[ \max_{ \sigma \in \Delta(A)} \sum_{\phi} \tau_y^\pi(\phi) \underline{u}(\sigma,\phi) \right]. \nonumber \end{align}\] The above expression corresponds to the value of \(\pi\) when \(\Delta(A)\) is the action set and \(\Phi\) is the state space, with \(\mu^\pi\) interpreted as the information structure over the reduced-form posteriors \(\{\tau_y^\pi\}_{y \in Y}\). Because \(\underline{u}(\cdot,\phi)\) is bounded over \(\Delta(A)\) and Blackwell’s theorem of informativeness extends to the case where the action set is a Polish space [48], the desired result follows. \(\blacksquare\)

9.8 Proof of Proposition 1↩︎

1 \(\Rightarrow\) 2. Let \(\mathop{\mathrm{supp}}(\mu) = \{ \mathcal{P}_y\}_{y \in Y} \subseteq \mathcal{K}(\Delta \Theta)\). Property (ii) implies that \[\begin{align} \min _{p \in \mathcal{P}_0} \mathbb{E}_p[u(a, \theta)] & =\sum_{y \in Y} \mu(y) \min _{p \in \mathcal{P}_y} \mathbb{E}_p[u(a, \theta)] =\min _{p \in \sum_{y \in Y} \mu(y) \mathcal{P}_y} \mathbb{E}_p[u(a, \theta)] \nonumber \end{align}\] for every \(u: A \times \Theta \rightarrow \mathbb{R}\). Since \(\mathcal{P}_0\) and \(\sum_{y \in Y} \mu(y) \mathcal{P}_y\) are both convex, invoking the separating hyperplane theorem over \(\mathbb{R}^{\Theta}\) allows us to conclude that \(\mathcal{P}_0=\sum_{y \in Y} \mu(y) \mathcal{P}_y\), as desired.

2 \(\Rightarrow\) 1. Property (i) follows easily from observing that for any \(u: A \times \Theta \to \mathbb{R}\), \[\begin{align} U(A, \mu_0) = \max_{a \in A} \min_{p \in \mathcal{P}_0} \mathbb{E}_{p} \left[ u(a,\theta) \right] & = \max_{a \in A} \mathbb{E}_{\mu} \left[ \min_{p \in \mathcal{P}_y} \mathbb{E}_{p} \left[ u(a,\theta) \right] \right] \nonumber \\ & \leq \mathbb{E}_{\mu} \left[ \max_{a \in A} \min_{p \in \mathcal{P}_y} \mathbb{E}_{p} \left[ u(a,\theta) \right] \right] = U(A, \mu), \nonumber \end{align}\] where the second equality holds by Aumann plausibility. Property (ii) follows similarly, where we see that \[\begin{align} U(\{a\}, \mu_0) & = \min_{p \in \mathcal{P}_0} \mathbb{E}_p [u(a,\theta)] = \mathbb{E}_{\mu} \left[ \min_{p \in \mathcal{P}_y} \mathbb{E}_{p} \left[ u(a,\theta) \right] \right] = U(\{a\}, \mu), \nonumber \end{align}\] as desired. \(\blacksquare\)

9.9 Proof of Proposition 2↩︎

Suppose \(\mathbb{E}_{\mu^*}[\hat{v}(\tau)] = v^*\) for some \(\mu^* \in \Delta(\Delta \Phi)\). Caratheodory’s theorem implies the existence of a Bayes-plausible information structure \(\mu \in \Delta(\Delta \Phi)\) with finite support such that \(\mathbb{E}_{\mu}[\hat{v}(\tau)] = v^*\), in which case it is without loss to impose the additional constraint that \(\mu\) be finitely supported. Let \(\mathcal{K}_\Phi \equiv \{ \mathcal{P} \in \mathcal{K}(\Delta \Theta) : \mathcal{P} = \sum_\phi \tau(\phi) \mathcal{P}_\phi \text{ for some } \tau \in \Delta(\Phi) \}\), and observe that \[\begin{align} \sup_{ \substack{\mu \in \Delta(\Delta \Phi) \\ \text{s.t. } \mathbb{E}_\mu[\tau] = \tau_0 }} \mathbb{E}_\mu [\hat{v}(\tau)] & = \sup_{ \substack{\mu \in \Delta(\Delta \Phi) \\ \text{s.t. } \mathbb{E}_\mu[\tau] = \tau_0 } } \mathbb{E}_\mu \left[ \sum_\phi \tau(\phi) \min_{p \in \mathcal{P}_\phi} \mathbb{E}_p[v (\hat{a}(\tau), \theta)] \right] \nonumber \\ & = \sup_{ \substack{\mu \in \Delta(\Delta \Phi) \\ \text{s.t. } \mathbb{E}_\mu[ \sum_\phi \tau(\phi) \mathcal{P}_\phi ] = \mathcal{P}_0 } } \mathbb{E}_\mu \left[ \min_{p \in \sum_\phi \tau(\phi) \mathcal{P}_\phi} \mathbb{E}_p[v (\hat{a}(\tau), \theta)] \right] \nonumber \\ & = \sup_{ \substack{\mu \in \Delta(\mathcal{K}_\Phi) \, : \, \mathbb{E}_\mu[ \mathcal{P} ] = \mathcal{P}_0 } } \mathbb{E}_\mu \left[ \min_{p \in \mathcal{P}} \mathbb{E}_p[v (\hat{a}(\mathcal{P}), \theta)] \right]. \nonumber \end{align}\] The second equality holds because \(\mathbb{E}_\mu[\tau] = \tau_0\) if and only if \(\mathbb{E}_\mu[ \sum_\phi \tau(\phi) \mathcal{P}_\phi ] = \mathcal{P}_0\) when \(\mathcal{P}_0 = \sum_\phi \tau_0(\phi) \mathcal{P}_\phi\) is maximally partially identified by \((\tau_0, \Phi)\), since we have \(\mathbb{E}_\mu[ \sum_\phi \tau(\phi) \mathcal{P}_\phi ] = \sum_\phi \mathbb{E}_\mu[\tau(\phi)] \mathcal{P}_\phi\) by convexity of \(\mathcal{P}_\phi\).

Recall from Lemma 13 in the proof of sufficiency in Theorem 2 that every set \(\mathcal{P}\) in the support of an Aumann-plausible information structure admits the decomposition \(\mathcal{P} = \sum_\phi \tau(\phi) \mathcal{P}_\phi\) for some \(\tau \in \Delta(\Phi)\) when the underlying prior set \(\mathcal{P}_0 = \sum_\phi \tau_0(\phi) \mathcal{P}_\phi\) is maximally partially identified. This implies \(\{ \mu \in \Delta(\mathcal{K}_\Phi) : \mathbb{E}_\mu[\mathcal{P}] = \mathcal{P}_0 \} = \{ \mu \in \Delta(\mathcal{K}(\Delta \Theta)) : \mathbb{E}_\mu[\mathcal{P}] = \mathcal{P}_0 \}\), and we can write \[\begin{align} \sup_{ \substack{\mu \in \Delta(\Delta \Phi) \\ \text{s.t. } \mathbb{E}_\mu[\tau] = \tau_0 }} \mathbb{E}_\mu [\hat{v}(\tau)] & = \sup_{ \mu \in \Delta(\mathcal{K}(\Delta \Theta)) \, : \, \mathbb{E}_\mu [\mathcal{P}] = \mathcal{P}_0 } \mathbb{E}_\mu \left[ \min_{p \in \mathcal{P}} \mathbb{E}_p[v (\hat{a}(\mathcal{P}), \theta)] \right] \nonumber \\ & = \sup_{ \mu \in \Delta(\mathcal{K}(\Delta \Theta)) \, : \, \mathbb{E}_\mu [\mathcal{P}] = \mathcal{P}_0 } V(A,\mu). \nonumber \end{align}\] Because Theorem 2 implies Aumann-plausible information structures are equivalent to consistent experiments when the prior set is maximally partially identified, the expression on the right corresponds to Sender’s problem, as desired. \(\blacksquare\)

References↩︎

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1. Department of Economics, Harvard University. I am indebted to Tomasz Strzalecki and Davide Viviano for detailed comments that significantly improved earlier versions of this draft. I also thank Aislinn Bohren, Xiaoyu Cheng, Tommaso Denti, Francesco Fabbri, Brian Hill, Peter Klibanoff, Shengwu Li, Yechan Park, Matthew Rabin, Elie Tamer, Vitalii Tubdenov, Andrei Savochkin, Weijie Zhong, and all seminar participants at Harvard University and RUD 2026 for their insightful comments and suggestions.↩︎

2. An incomplete list of such axiomatic work include [3]–[7].↩︎

3. Our consistency condition holds vacuously with singleton prior sets.↩︎

4. See, for instance, [11]–[13].↩︎

5. [16] show that a prior-by-prior version of Bayes plausibility and an additional likelihood ratio constraint characterizes the profile of posterior beliefs that can be induced by experiments. Despite the superficial similarity, our results are quite distinct as the Aumann expectation of the convex hull of a profile of posteriors does not in general correspond to the convex hull of the profile of priors.↩︎

6. Convexity of \(\mathcal{P}\) implies that we can assume \(\mathcal{P}_\phi\) is convex without loss. This is because the convex hull operation commutes with Minkowski addition, i.e., \(\mathcal{P} = \mathop{\mathrm{co}}(\mathcal{P}) = \mathop{\mathrm{co}}( \sum_\phi \tau(\phi) \mathcal{P}_\phi ) = \sum_\phi \tau(\phi) \mathop{\mathrm{co}}(\mathcal{P}_\phi)\).↩︎

7. To be precise, \(\dim(\Delta(\Theta)) = |\Theta| - 1\), and the partition \(\Phi\) corresponds to \(|\Phi| - 1\) linear constraints of the form \(\sum_{\theta \in \phi}p(\theta) = \tau(\phi)\).↩︎

8. \(\Delta(\mathcal{P}_0)\) denotes the collection of Borel measures over \(\Delta(\Theta)\) with support contained in \(\mathcal{P}_0\).↩︎

9. [33] argues that decisions made under ambiguity must be assessed relative to the DM’s subjective decision tree, which captures the contingencies (i.e., types of data) the DM envisages. In this context, our consistency condition can be interpreted as placing structure on the collection of admissible subjective trees as a function of the DM’s prior ambiguity.↩︎

10. If we drop the \(\dim(\mathcal{P}_\phi) = \dim(\Delta(\phi))\) condition in the definition of partially identified sets, the consistency of \(\pi\) is a necessary, but insufficient condition for \(\Phi\)-measurability.↩︎

11. For this reason, [1] refer to the reduced-form prior \(\tau_0\) as the revisable prior knowledge and the collection of conditional priors \(\{p_0(\cdot | \phi) \}_{\phi \in \Phi}\) as the unrevisable prior knowledge in the context of full ambiguity sets that are partially identified. Lemma 3 slightly extends this observation to all partially identified prior sets.↩︎

12. As shown by [40], only singleton sets are rectangular with respect to both \(\{ \{\theta\} \times Y\}_{\theta \in \Theta}\) and \(\{\Theta \times \{y\}\}_{y \in Y}\).↩︎

13. Section 5 contains a more detailed discussion of this interpretation in the context of statistical decision theory. Proposition 1 in Section 6 illustrates how Aumann plausibility can also be interpreted as a participation constraint on Sender’s problem in a persuasion game.↩︎

14. The set \(\mathcal{P}'\) is positively homothetic to \(\mathcal{P}\) if \(\mathcal{P}' = \alpha \mathcal{P}\) for some \(\alpha \geq 0\). See Chapter 15 of [41] for a classic reference on Minkowski decompositions.↩︎

15. We do not require maximal partial identification because our results only depend on consistent experiments inducing Aumann plausible information structures; see Theorem 1.↩︎

16. See, for instance, [42]–[45].↩︎

17. As in the proof of Lemma 5, the key step is to define \(\underline{u}: \Delta(A) \times \Phi \to \mathbb{R}\), \(\underline{u}(\sigma,\phi) = \min_{p \in \mathcal{P}_\phi} \sum_{\theta, a} u(a,\theta) \sigma(a) p(\theta)\) and interpret \(\Phi\) as the state space.↩︎

18. Recall that this is because \(\mathcal{P}_0 \otimes \pi\) is \(\{ \{\theta\} \times Y\}_{\theta \in \Theta}\)-rectangular, and only singletons are rectangular with respect to both partitions [40].↩︎

19. Technically, this setup is slightly more general than that of [1] because we do not impose the full ambiguity assumption.↩︎

20. As the arrival of information is the only source of dynamic inconsistency, Corollary 3 of [46] implies that the same result holds for the corresponding minimax regret criterion.↩︎

21. When \(\mathcal{P}_0 = \{p_0\}\), this implies that \(p_0\) is in the interior of \(\Delta(\Theta)\) because \(\tau\) has full support.↩︎

22. Properties (i) and (ii) can be also shown by invoking Theorem 3 to compare the trivial experiment to the one chosen by Sender. Proposition 1 proves a more general result, as information structures can be Aumann-plausible with respect to prior sets that are not partially identified.↩︎

23. We use the standard abuse of notation here by writing \(x \succsim y\) for \(x,y \in X\) to denote preferences between constant acts that map deterministically to outcomes.↩︎

24. Strictly speaking, [4] requires the SEU measure \(p\) on \(\mathcal{L}\) to be convex-ranged, which cannot hold when \(\mathcal{L}\) is the \(\sigma\)-algebra generated by a finite \(\Phi\). This can be addressed by passing to an atomless refinement: replace each \(\phi\) with \(\phi\times[0,1]\), let \(\lambda\) be Lebesgue measure, and take \(p(\phi,t)=\tau(\phi)\lambda(t)\). Then \(p\) is convex-ranged, and for \(\Theta\)-measurable acts, \(U_{MEU(\tau,\Phi)}(\cdot) = U_{Zhang}(\cdot)\). This resembles the finite-partition reduction in [7].↩︎

25. Although \(\mathfrak{G}\) is finite when \(\Phi\) is finite, we can address the issue the same way as in the case of \(U_{Zhang}(\cdot)\) by enriching the state space.↩︎