1 Introduction↩︎

Social welfare analysis under uncertainty has two distinct tasks: it aggregates how individuals value policies, and it also aggregates their beliefs to determine which probabilistic descriptions of the world should be used to evaluate policies. We study the second task, which recent work emphasizes is an epistemic problem that should be separated from the aggregation of preferences [1], [2]. This separation is especially relevant when individuals have conflicting and imprecise beliefs about an unknown state of the world; imprecision is modeled using sets of beliefs to capture the fact that some individuals may be unable to specify a unique belief [3], [4]. Consider, for example, a financial market where individuals are trading an asset because they disagree about its future payoff. Each individual may expect to gain under his own beliefs, while a planner who lacks knowledge of the objective belief must decide whether the trade reflects socially valuable risk sharing or speculation due to belief disagreement. In these environments, [5] ask: “which belief should the planner use?” This raises the main question of our paper: how should the planner decide which collective beliefs are admissible when individual beliefs are formed through specific information-acquisition technologies?

We answer this question by developing an epistemic foundation for the intersection rule, which aggregates belief sets by retaining only shared (or common) beliefs. This rule is appealing at an institutional level because it is simple, conservative, and reflects a social consensus: a belief is collectively admissible only if it is admissible for every individual. It is widely used in many fields [6]–[12]; in public policy, [7] refers to the intersection as a “domain of consensus,” and it is referred to as “shared beliefs” in organization theory [13] and markets [14]. However, this appeal is incomplete without an account of why belief sets should be intersected in the first place. Our argument is that the appropriate belief-aggregation rule should be derived from the information technologies that generate individual beliefs. This has first-order welfare implications because different models of individual belief formation justify different collective beliefs, which in turn lead to distinct social welfare criteria and can therefore reverse the ranking of uncertain policies. In the trading example, failing to base collective beliefs on the underlying information technologies can lead the planner to endorse trades that generate aggregate financial losses. Our model clarifies the epistemic problem across two dimensions.

First, rational inattention and partial verifiability explain why collective admissibility of beliefs takes the form of an intersection. On one hand, each individual is interpreted as an information source with a reference belief, acquires costly information subject to a capacity constraint, observes a private signal, and reports posterior beliefs to a planner. Rational inattention delivers individual restrictions: a posterior is admissible for a source only if it can be produced within its capacity constraint. On the other hand, the planner is interpreted as a designer who uses the sources’ capacity constraints as evidentiary constraints on posterior reports in a mechanism with partially verifiable evidence. The implementable certification region consists of posteriors consistent with all these constraints. Thus, a posterior is admissible for the planner only if it satisfies every source’s capacity constraint simultaneously.

Second, information theory identifies a geometry for the belief sets being intersected: under standard information costs discussed below, the posteriors that can be produced from each source form a Bregman ball centered around its reference belief. Bregman balls are widely used in economics, statistics, and information geometry, and special cases are entropy balls [15]–[19]. Then, the main representation result is that the certification region generated by all individual capacity constraints is precisely the intersection of these Bregman balls.

Our assumption on information cost is standard. Before deciding how to trade, suppose each individual can run a Blackwell experiment about the asset payoff—an information structure that produces a signal [20]—and then updates his reference belief via Bayes rule. We assume the cost of producing posteriors is uniformly posterior-separable: the cost of an experiment is the expected “difficulty” of updating from the prior to the realized posteriors [21]. This family of cost functions is motivated by the experimental evidence of [22], the sequential-sampling foundations of [23] and [24], [25], and the related theoretical justifications of [26]. As in much of rational-inattention theory, the standard capacity constraint must be adapted to fit our institutional environment—[27]’s ([27]) canonical restriction is a channel-capacity constraint: the information system chosen by a single individual must transmit only a bounded amount of mutual information before a particular signal realization is observed. Later applications adapt the timing, state space, and object of attention to fit their specific environments [28]–[30]. Our adaptation is required because the planner aggregates posterior reports after individuals have acquired signals. Since uniformly posterior-separable costs satisfy [31]’s ([31]) constant-marginal-cost property, expected cost cannot distinguish genuinely cheap posteriors from rare expensive posteriors. That is, any posterior can be made arbitrarily “cheap” in expectation by simply making the corresponding signal realization sufficiently rare. To avoid this vacuity, an appropriate posterior-admissibility restriction must instead constrain the unit cost of producing a posterior. We show that this unit cost equals the Bregman divergence between the prior and the candidate posterior; this divergence is a measure of discrepancy between probability distributions that generalizes the Kullback-Leibler divergence and is widely used in economics [21], [24], [25], [32]–[34]. Thus, individuals are rationally inattentive in the sense that each one has a unit-cost budget; this aims to capture [35]’s ([35, p. 303]) sentiment that “The choice of information structures must be subject to some limits, otherwise, of course, each agent would simply observe the entire state of the world.”

The Bregman geometry is economically relevant because it is rich enough to nest many of the popular belief-aggregation rules used in economics and finance. The key is that Bregman balls embed an extra degree of freedom: the order of arguments in Bregman balls matters because Bregman divergences are typically asymmetric. The two orders are referred to as primal and dual, and are derived from different information technologies: uniformly posterior-separable costs deliver dual Bregman balls, whereas scoring-rule regrets deliver primal Bregman balls. These orders pin down different belief-aggregation rules. On one hand, primal Bregman balls can generate linear pooling of beliefs, so they recover [36]’s ([36]) linear-linear welfare criterion and the convex-hull belief set in [5]’s ([5]) belief-neutral welfare criterion. On the other hand, dual Bregman balls can generate geometric pooling of beliefs, so they recover [37]’s ([37]) linear-geometric welfare criterion; they can also generate the power pooling rules used in welfare economics for inequality measurement [38] and in finance for asset pricing [39]; we also show that they can recover the multiplicative pooling rule axiomatized by [40]. Then, we show that it is also possible to include all these familiar pooling rules in the same intersection, thereby allowing the planner to have access to all of them at once. Thus, the same trading environment can lead to different welfare-relevant collective beliefs depending on the information technology that constrains individual belief formation.

We return to the financial-market example in our application. We embed Bregman balls in the incomplete-preference exchange economy of [41], where individuals trade Arrow securities while holding belief sets. Although there is no “planner” explicitly choosing a collective belief in this economy, a full-insurance equilibrium converts any shared belief in the intersection of belief sets into a normalized Arrow price. This makes financial markets a useful test of the observable implications of the Bregman geometry because the belief selected by the intersection is capitalized into equilibrium prices. Specifically, when the intersection is a singleton, we show that primal entropy balls generate the same prices as a log-utility risk economy with linearly pooled beliefs; dual entropy balls generate the same prices as an exponential-utility risk economy with geometrically pooled beliefs; and other Bregman balls generate power-pooled prices corresponding to isoelastic risk economies. Thus, the Bregman geometry explains how conflicting beliefs are priced in financial markets. This illustrates the importance of forming collective beliefs based on individual information technologies, since doing otherwise can distort valuations and lead to trading losses.

— Related literature: [9], hereafter DGHT, provide a normative foundation for the intersection rule via welfare aggregation. They consider individuals with unambiguous preferences à la [42], where each individual has a set of beliefs. Their robust Pareto principle says that if all individuals unambiguously prefer a policy to another, then so should the planner. Under a restriction on utility functions called “c-diversity,” DGHT show that robust Pareto is equivalent to the collective utility function being a linear combination of individual utility functions and collective beliefs being a subset of the intersection of individual belief sets. While this result is elegant and explains why the intersection rule is normatively compelling, it does not explain why it is epistemically compelling. The reason is that DGHT’s joint aggregation relies on c-diversity to derive the intersection, meaning the formation of collective beliefs cannot be disentangled from utility functions. Their framework therefore does not specify the information technologies that generate individual belief sets, the epistemic basis for intersecting them, or the geometric structure these sets should exhibit based on information costs. As a result, the epistemic foundation of the intersection rule remains an open problem. We complement DGHT by supplying such a foundation.

This paper contributes to epistemic social choice theory, which is the study of collective decision problems in which disagreement concerns not only values or tastes but also facts, probabilities, and information. In environments where there are objective probabilistic facts, the correct way to aggregate individuals’ beliefs about these facts depends on specific assumptions on how they form these beliefs. In a survey, [2] argues that in such environments, epistemic social choice should start from a model of belief formation and then derive the appropriate aggregation rule from that model. In a related survey, [1] notes: “The aggregation of beliefs should be completely separated from, and be performed prior to, the aggregation of preferences.” These methodological points are reflected in [5]: they assume individual belief formation is distorted by psychological biases and propose their convex-hull aggregation rule to address this problem. Their approach illustrates that assumptions about belief formation naturally determine which belief-aggregation rule is appropriate. Our epistemic foundation follows these perspectives by deriving collective beliefs from individual information technologies, separate from preference aggregation. More broadly, our foundation contributes to [43]’s ([43, p. 175]) agenda pointing to “develop formal, explicit theories of the belief formation process.”

We also contribute to the literature on welfare aggregation under heterogeneous beliefs. Starting with [44], the classical approach derives utilitarian aggregation from the standard Pareto principle when uncertainty is objective. However, [45] shows that this approach is incompatible with subjective heterogeneous beliefs. The literature following [36] therefore weakens Pareto-style axioms to accommodate belief heterogeneity while still disciplining collective utility and collective beliefs jointly [9], [37], [46]–[50]. In contrast, a growing strand of this literature deliberately separates the ethical aggregation of utilities from the epistemic aggregation of beliefs. [51] distinguish objective from subjective uncertainty to limit where common beliefs are imposed. [52] uses accumulating evidence to obtain utilitarian social welfare without determining collective beliefs. [53] use almost-objective uncertainty to decouple utility aggregation from belief aggregation, which is then treated as a distinct epistemic problem. We complement this strand by providing a microfoundation for such epistemic problems. Specifically, we derive collective beliefs from individual information technologies as follows: rational inattention generates capacity constraints, certification under partial verifiability requires simultaneous satisfaction of these constraints, and information theory gives the resulting certification region a Bregman geometry. This yields an epistemic justification for the intersection rule and explains why linear, geometric, power, and multiplicative pooling arise from different models of belief formation.

The rest of the paper proceeds as follows. Section 2 presents a normative foundation, and Section 3 analyzes implications of our epistemic foundation, which is formalized in Section 4. Section 5 explores an application to financial markets and Section 6 concludes the paper.

2 Normative Foundation↩︎

We consider a society with individuals \(i\in I:=\{1,\dots,n\}\) and a planner indexed by \(0\). Let \(S\) be a finite set of states of the world with \(|S|\geq2\) and \(X\) be a convex set of outcomes. The planner evaluates acts (or policies), which are functions \(f:S\to X\), and let \(F\) denote the set of all acts. Let \(\Delta:=\Delta(S)\) be the space of all probability distributions over \(S\). A preference over acts is described by a binary relation \(\succsim\) on \(F\). We write \(f\succsim g\) when act \(f\) is weakly preferred to \(g\), while strict preference and indifference are denoted by \(\succ\) and \(\sim\), respectively.

Unambiguous preferences are also known as Bewley preferences, due to [42]. The key is that \(\succsim\) is incomplete unless \(P\) is a singleton. The set \(P\) is uniquely pinned down by \(\succsim\), while \(u\) is unique up to a positive affine transformation. Throughout, the planner and all individuals have unambiguous preferences \((\succsim_i)_{i=0}^n\) represented by pairs \(\{(u_i,P_i)\}_{i=0}^n\).

2.1 Robust Pareto principle↩︎

With unambiguous preferences in place, the next step is to identify a normative justification for the intersection rule. DGHT show that their robust Pareto principle has a clear implication: once individual tastes are sufficiently diverse, the planner’s admissible beliefs must be contained in the intersection of individual belief sets. This section recalls that result first and then isolates the epistemic question that remains open.

DGHT’s robust Pareto principle states that if all individuals unambiguously prefer a policy over another, then so should the planner. The next definition formalizes this principle.

The welfare aggregation result will use a diversity condition on individuals’ tastes. Specifically, for each individual, there must exist two outcomes between which an individual is the only one to have a strict preference whereas all other individuals are indifferent.

C-diversity, also known as “independent prospects” in the aggregation literature, is equivalent to the individual utility functions being linearly independent when \(X\) is at least \(n\)-dimensional. This condition will play a key role in DGHT’s welfare aggregation.

2.2 Welfare aggregation and open problem↩︎

The next result is DGHT’s Theorem 1, which shows the welfare implications of robust Pareto.

Observation 1 is the normative benchmark that disciplines both sides of the social preference. The utility side is Harsanyi-style: the collective utility function is a linear combination of individual utility functions. The belief side is set-valued: every collective belief lies in the intersection of individual belief sets. For this aggregation rule to make sense, we assume throughout that individual belief sets are compatible so that their intersection is nonempty.³ Shared beliefs are plausible in environments involving deep uncertainties [54], where most people (including experts) entertain wide ranges of beliefs; e.g., climate policy [55] and health policy [56].⁴

A key feature of Observation 1 is the joint aggregation of beliefs and utility functions. This forces the intersection to be dependent on c-diversity, and hence the set of collective beliefs in ?? cannot be disentangled from utility weights \(\theta\). In contrast, a primary objective of epistemic social choice theory is to understand the information-acquisition costs and processing constraints that shape individual belief formation and determine aggregation rules [2]. By relying on Pareto-style axioms, DGHT’s framework abstracts from these mechanisms. In fact, while referring to DGHT’s aggregation approach, [1] notes that: “it does not make sense to jointly aggregate beliefs and preferences [...] a rational evaluation would instead separate the formation of beliefs from the aggregation of preferences.” To operationalize this separation, welfare economics requires an epistemic foundation detailing how information technologies generate individual belief sets and how to aggregate these sets. The rest of our paper delivers such a foundation by answering the question: What theory of information-acquisition costs and information-processing constraints makes the intersection rule an epistemically grounded restriction on collective beliefs?

Section 4 answers this question by developing an epistemic foundation that combines ideas from rational inattention, posterior certification, and information theory. Before presenting this formal analysis, we first highlight its key economic implications in Section 3.

3 Implications of Epistemic Foundation↩︎

The purpose of this section is to abstract from the formalism of Section 4 and isolate the key economic content of our epistemic foundation. We start with a high-level summary of the foundation. Each individual is an information source who begins with a reference belief, acquires costly information subject to a capacity constraint, and reports posterior beliefs after observing a private signal. The planner evaluates beliefs at the reporting stage, so the relevant admissibility question is whether a posterior can be justified within an individual’s information technology. Formally, the planner designs a posterior-reporting mechanism that allows individuals to partially verify posteriors, which forms the certification region: a posterior is collectively admissible only if it satisfies all individual capacity constraints.

The main economic object delivered by our epistemic foundation is the Bregman geometry of individual belief sets. This geometric structure originates from a standard assumption on information-acquisition costs, and it determines how belief disagreement is aggregated in welfare analysis. In particular, different information costs generate different Bregman balls, which in turn induce different collective beliefs when intersected. This section therefore starts by studying the welfare implications of the Bregman geometry. We show that this geometry is useful because it is rich enough to recover several familiar belief-aggregation rules used in economics and finance, including linear, geometric, power, and multiplicative pooling.

3.1 Bregman balls↩︎

To define Bregman balls, we first introduce its main component: the Bregman divergence associated with a convex differentiable generator \(G\), defined as \[\label{eq:bregdiv} D_G(q\lVert p):=G(q)-G(p)-\langle \nabla G(p),q-p\rangle,\tag{1}\] for any \(q,p\in \Delta\) such that the expression is well-defined [59]. This divergence is nonnegative, equals zero when \(q=p\), and is popular in many fields, including information economics [21], [24], [32]–[34]. Bregman divergences are asymmetric,⁵ which affects Bregman balls, so we use one notation for each order of arguments in \(D_G(\cdot\lVert\cdot)\): \[\begin{align} \text{(\textit{primal} ball)}& \qquad B^G_{\eta_i}(p_i):=\{q\in\Delta:D_G(p_i\lVert q)\leq \eta_i\},\tag{2}\\ \text{(\textit{dual} ball)}& \qquad \hat{B}^G_{\eta_i}(p_i):=\{q\in\Delta:D_G(q\lVert p_i)\leq \eta_i\},\tag{3} \end{align}\] where \(p_i\in \Delta\) is \(i\)’s reference belief and \(\eta_i\geq0\) measures the imprecision around \(p_i\), or inversely, \(i\)’s confidence in \(p_i\). An interpretation is that, while \(i\) believes \(p_i\) is the “best” approximation of the objective belief, \(i\) still considers nearby models that are within \(\eta_i\)-range of \(p_i\) because such models may capture features of the objective belief that were missed by \(p_i\). [61] visualize differences of primal and dual balls.

When \(G(q)=\sum_{s\in S}q(s)\log q(s)\), we obtain the Kullback-Leibler (KL) divergence (or relative entropy) \(D_{\mathrm{KL}}(q\lVert p):=\sum_{s\in S}q(s)\log\frac{q(s)}{p(s)}\). The KL divergence gives rise to entropy balls, which are the most popular subclass of belief sets. A primal entropy ball is denoted \(B^{\text{\normalfont\scriptsize KL}}_{\eta_i}(p_i)\) and a dual entropy ball is denoted \(\hat{B}^{\text{\normalfont\scriptsize KL}}_{\eta_i}(p_i)\), and both are convex sets.⁶ For example, [15]’s ([15]) constraint preference uses a dual entropy ball as the set of subjective beliefs; [19] extends this preference by axiomatizing generalized dual entropy balls in the maxmin-expected-utility setting, where the radius can be endogenous.

3.2 Linear pooling↩︎

Linear pooling is the most prominent aggregation rule, dating back to [62]. However, [63] claim: “linear pooling (the weighted or unweighted linear averaging of probabilities) can be justified on procedural grounds but not on epistemic ones.” We show that our model provides an epistemic foundation for linear pooling. For all our aggregation results to be well-behaved, we assume some standard regularity conditions on the generator \(G\) and support restrictions on reference beliefs; we collect all these technical conditions in Appendix 9. The first result uses primal Bregman balls to partially identify linear pooling.

This result shows that if every individual has a primal ball, then the intersection of these balls must contain at least one linear pooling of their centers; here, \(\operatorname{co}(.)\) denotes the convex hull. The next two results build on this insight to recover familiar welfare criteria.

Corollary 1 explains how familiar welfare criteria based on individuals with subjective expected utility (SEU) preferences can arise within our unambiguous-preference framework. In those criteria, every individual SEU preference is represented by a unique prior belief; in our framework, the same role is played by the reference beliefs \((p_i)_{i\in I}\), while individual preferences are incomplete and represented by belief sets centered around them. Thus, when the Bregman radii \((\eta_i)_{i\in I}\) are large enough that the intersection contains \(\operatorname{co}(p_i)_{i\in I}\) and the subset \(P_0=\operatorname{co}(p_i)_{i\in I}\) is chosen in Observation 1, the resulting welfare criterion \((u_0,\operatorname{co}(p_i)_{i\in I})\) recovers [5]’s ([5]) belief-neutral criterion: the collective utility is a linear combination of individual utilities and the collective-belief set is the entire convex hull of reference beliefs. In related frameworks with SEU individuals, \(\operatorname{co}(p_i)_{i\in I}\) is the set of collective beliefs in [46], while any nonempty subset of \(\operatorname{co}(p_i)_{i\in I}\) is a valid set of collective beliefs in the welfare criterion of [46].

Notice that the social preference in Observation 1 is forced to be an SEU welfare criterion when the intersection of individual belief sets is a singleton; this is the most studied class of welfare criteria. Thus, for our remaining examples, we focus on singleton intersections to recover familiar SEU welfare criteria, then we will generalize the analysis to non-singleton cases in Section 3.5 and provide an institutional interpretation for the singleton case in Section 3.5.1. Corollary 2 will help to point-identify the most popular welfare criterion.

When the intersection of primal Bregman balls is a singleton, the unique point-identified belief in the intersection is always a linear pooling of reference beliefs—the model weights \((\mu_i)_{i\in I}\) depend on \(G\) but we always suppress that dependence to ease notation. Combined with Observation 1, Corollary 2 shows that the planner behaves identically to [36]’s ([36]) SEU planner with social preference \((u_0,q^*_0)\), which is known as a “utilitarian” preference or linear-linear aggregation rule. Thus, the primal Bregman geometry recovers the belief component of [36]’s ([36]) welfare criterion within an unambiguous-preference environment: the key is that the reference beliefs \((p_i)_{i\in I}\) play the role of individual SEU priors, and the singleton intersection selects their linear pool as the unique collective belief.

3.3 Geometric pooling↩︎

The previous section showed that the primal order mostly identifies linear pooling under the intersection rule. The rest of our examples build on the dual order because, as we shall see, it is more flexible. We start with the case where individuals have dual entropy balls \(\hat{B}^{\text{\normalfont\scriptsize KL}}_{\eta_i}(p_i)\).

When the intersection of dual entropy balls is a singleton, the unique point-identified belief is always a geometric (or logarithmic) pooling of reference beliefs. Combined with Observation 1, Proposition 2 shows that our planner behaves identically to [37]’s ([37]) SEU planner with social preference \((u_0,\hat{q}_0)\), which is known as a linear-geometric aggregation rule. [37] axiomatizes this rule and shows that, under basic conditions, it is the only preference that is dynamically consistent, i.e., it is compatible with Bayesian updating.

3.3.1 Comparing primal and dual orders↩︎

A comparison with Section 3.2 clarifies the economic content of the two Bregman orders. The main insight is that they stem from two distinct information technologies. A primal ball measures admissibility from the perspective of the individual reference belief: the collective belief must be a sufficiently good approximation to what the individual considers to be plausible. This produces linear pooling because the disagreement across individuals is resolved by averaging probability levels. In contrast, a dual entropy ball measures admissibility from the perspective of the candidate collective belief: the candidate must be sufficiently close to each reference belief when probabilities are evaluated in relative-entropy terms. This produces geometric pooling because disagreement is resolved by averaging log probabilities.

3.4 Power pooling and other aggregation rules↩︎

We now consider a broader class of dual Bregman balls to show that they can also rationalize more sophisticated pooling rules. Here, we will obtain different belief-aggregation rules by varying the generator \(G\) while holding the dual order fixed. To this end, let \(G\) be separable, in the sense that \(G(q)=\sum_{s\in S}\beta_s g(q(s)),\) where \(\beta_s>0\) for every \(s\in S\), and \(g\) is twice continuously differentiable and strictly convex on \((0,\infty)\), with \(g''>0\).

This formula is very flexible. If \(\beta_s\) is constant across states and \(g'(x)=x\), then we recover the linear pooling in Corollary 2, and \(g'(x)=\log x\) recovers the geometric pooling in Proposition 2. With general \(\beta_s\), if \(g'(x)=x^r/r\) for \(r\neq0\), then the formula becomes \[\begin{align} \label{eq:power} \tilde{q}_0(s)= \Big(\frac{ra}{\beta_s}+\sum_{i\in I}\mu_i p_i(s)^r\Big)^{1/r} \qquad \forall s\in S, \end{align}\tag{4}\] where \(a\in\mathbb{R}\) is a normalizing constant that ensures \(\tilde{q}_0\in\Delta\). Here, the parameter \(r\) controls how the planner treats disagreement across states. Larger \(r\) puts relatively more weight on reference beliefs that assign high probability to a state, whereas smaller \(r\) gives relatively more influence to reference beliefs that assign low probability to that state.

Notably, the expression 4 also nests the standard power pooling rule \[\tilde{q}_0(s)= \frac{\big(\sum_{i\in I}\mu_i p_i(s)^r\big)^{1/r}}{\sum_{t\in S}\big(\sum_{i\in I}\mu_i p_i(t)^r\big)^{1/r}} \qquad \forall s\in S.\] when \(a=(Z^{-r}-1)/r\), \(\beta_s=1/V_s\), \(Z:=\sum_{t\in S}V_t^{1/r}\), and \(V_s:=\sum_{i\in I}\mu_i p_i(s)^r\) in 4 . This rule is used in welfare economics to measure multi-dimensional inequality [38], and it is also used in finance for asset pricing [39].

The next result identifies another relevant aggregation rule related to geometric pooling.

This rule is known as multiplicative (or product) pooling and is axiomatized by [40], who relates it to aggregation of private information. Its key property is that it is parameter-free. However, unlike all previous results, Corollary 3 requires reference beliefs to satisfy a “compatibility” restriction because this rule does not satisfy the unanimity axiom.⁷

To summarize, all our previous examples have shown that changing the Bregman geometry changes the economic meaning of belief aggregation: it changes whether collective belief formation emphasizes optimistic beliefs, pessimistic beliefs, or more balanced beliefs.

3.5 Extension: beyond singleton condition↩︎

The previous results focused on singleton intersections to point-identify a unique collective belief. We now show that all previous pooling rules can be contained in the same intersection.

Observation 2 shows that it is possible to construct an intersection of dual entropy balls that contains all the previous aggregation rules simultaneously, so that a planner would have access to them all at once. This result therefore highlights that when the intersection is not a singleton, the Bregman aggregation problem becomes very flexible and can accommodate several combinations of belief-aggregation rules at once. Notably, allowing the planner to have access to many popular aggregation rules at once is economically relevant because it is known in the literature of epistemic social choice that different belief-aggregation rules are appropriate in different environments [2], [51]–[53], [63]. Thus, the Bregman geometry provides a tractable and unified framework to study the emergence of different pooling rules.

3.5.1 Interpretation of singleton condition↩︎

The singleton condition can be interpreted institutionally through the lens of social deliberation. [64] study deliberation as repeated interactions on a social network in which individuals exchange beliefs and iteratively update their beliefs according to some rule, such as [65]’s ([65]) rule. They show that deliberation can converge to a consensus belief that takes various forms depending on the updating rule, such as the linear pooling in Corollary 2 and the geometric pooling in Proposition 2. This suggests interpreting the singleton condition as a reduced-form outcome of a convergent deliberation process: individuals start with their own reference beliefs, agree to exchange beliefs with others, and update until a consensus belief is reached. When this deliberation process converges, the consensus belief can be interpreted as the unique collective belief in the singleton intersection.

4 Epistemic Foundation↩︎

[2] suggests the following outline for an epistemic foundation: “begin with a detailed model of how the voters formed their probabilistic beliefs, and from this model derive the optimal way to aggregate these beliefs.” [1] recommends a similar outline. The key point of the outline is the timing, suggesting that individuals’ beliefs must be formed before the planner evaluates which probability distributions are collectively admissible. We operationalize this by modeling each individual as a rationally inattentive information source with a reference belief, an information cost, and a capacity constraint. Then, the planner evaluates which posterior reports can be certified jointly by all sources.

4.1 Information technology↩︎

Let \(\Omega\) be a finite state space with \(|\Omega|=m\geq2\); we change the notation throughout so that the epistemic foundation is self-contained. A signal alphabet is a finite set \(Y\). A signal structure is a collection \(\sigma=\{\sigma_{\omega}\in\Delta(Y)\}_{\omega\in\Omega}\), where \(\sigma_{\omega}(y)\) is the probability of signal \(y\) in state \(\omega\). Equivalently, \(\sigma\) is a finite Blackwell experiment: before the state is observed, an individual with full-support prior \(q_0\in\Delta(\Omega)\) chooses a stochastic kernel from states to signals [20];⁸ after signal \(y\) is realized, \(q_0\) is updated to the posterior \(q_y(\sigma,q_0)\) via Bayes rule. Specifically, given \(q_0\in\Delta(\Omega)\), the unconditional probability of signal \(y\) is \(\pi_y(\sigma,q_0):=\sum_{\omega\in\Omega}q_0(\omega)\sigma_{\omega}(y),\) and the posterior after observing \(y\), when \(\pi_y(\sigma,q_0)>0\), is \[q_y(\sigma,q_0)(\omega):=\frac{q_0(\omega)\sigma_{\omega}(y)}{\pi_y(\sigma,q_0)}.\]

Fix a prior \(q_0\in\Delta(\Omega)\). An information cost function assigns to each finite signal structure \((Y,\sigma)\) a cost \(C(\sigma,q_0;Y)\in\mathbb{R}_+.\) The function \(C\) is interpreted as the resource cost required to implement \((Y,\sigma)\) given prior \(q_0\). Let \(\Delta^{\circ}(\Omega) := \{q\in\Delta(\Omega):q(\omega)>0 \forall \omega\in \Omega\}.\)

4.1.1 Uniform posterior separability↩︎

Motivated by the theoretical results of [24], [25]; [21]; and [26], and the experimental evidence of [22], we restrict attention to cost functions in the uniform posterior separability family.

Uniform posterior separability was introduced by [21]. It is justified in [24], [25] by a sequential-sampling foundation: optimal evidence accumulation can induce uniformly posterior-separable costs, so ?? can be viewed as a static reduced-form representation. ?? says that the total cost of an experiment is the average (over realized posteriors) of a local “difficulty” measure \(D_G\), where the function \(G\) summarizes the local discriminability properties of the underlying sampling process; the relevant belief domain of \(G\) is standard but technical, so it is stated in Appendix 9. For example, given any posterior \(q\), if we set \(G(q)=\sum_{\omega}q(\omega)\log q(\omega)\), then we get \(D_G(q\lVert q_0)=D_{\mathrm{KL}}(q\lVert q_0)\), which is the dual KL divergence and ?? becomes the mutual information used by [27].

4.2 Institutional environment↩︎

In our institutional setting, the planner’s problem is to determine collective beliefs after individuals have already acquired signals and formed their own beliefs. Notice that this timing differs from the timing in canonical rational-inattention theory, which typically studies a single agent’s ex-ante choice of an information strategy subject to a capacity constraint imposed before observing signals [27].⁹ Our posterior-reporting framework therefore requires adapting rational inattention to the object being evaluated by the planner.¹⁰ In our specific environment, the scarce information-processing resource constrains posterior reports, so the collective admissibility question is therefore not only how costly an entire signal structure is in expectation, but which posterior reports can be justified as outcomes of an individual’s information technology. To answer this question, we first show that uniformly posterior-separable costs have a relevant property: their marginal costs are constant.

4.2.1 Constant marginal cost↩︎

Fix any signal structure \((Y,\sigma)\) and let \(o\notin Y\) be a null signal realization. For \(\alpha\in[0,1]\), define the dilution \(\alpha\cdot(Y,\sigma)\) as the signal structure on \(Y\cup\{o\}\) given by \[\sigma_{\omega}^\alpha(y):=\alpha\sigma_{\omega}(y)\qquad\forall y\in Y; \qquad \sigma_{\omega}^\alpha(o):=1-\alpha.\] The diluted Blackwell experiment first randomizes between activating the original experiment with probability \(\alpha\) and producing an uninformative null signal with probability \(1-\alpha\).

Lemma 1 is the dilution or constant-marginal-cost property of [31]: the marginal cost of increasing the success probability of an experiment is constant. This property has a sharp implication: expected-cost budgets are too weak to constrain rare posteriors under uniform posterior separability; the next result formalizes this degeneracy.

Observation 3 shows why standard rational-inattention capacity constraints must be adapted in our institutional setting. Since our planner aggregates beliefs after signals are observed, expected-cost budgets would deem a posterior admissible whenever it can be produced by some sufficiently rare signal realization. In other words, to pass any expected-cost budget, an individual can simply “dilute” their experiment. This makes admissibility depend on how rarely the posterior is reached, rather than on how informative the posterior is. Thus, for a posterior-level capacity constraint to be nontrivial, it must control the cost per unit probability of producing a posterior. This unit-cost notion is defined below.

4.2.2 Unit cost of producing a posterior↩︎

The next definition introduces the unit cost of producing a posterior belief.

The normalization in ?? divides expected cost of a signal structure by the probability of the signal realization that induces posterior \(q\). This converts an ex-ante cost into a posterior-level cost. The conversion is needed because expected cost depends on how often a posterior is reached, while admissibility of a reported posterior should depend on the informational cost required to produce that posterior when its signal realization occurs. The unit cost \(\kappa(q|q_0)\) captures this cost per realized occurrence of \(q\). The next result sharpens this intuition.

Observation 4 gives the unit cost in Definition 5 an operational interpretation. Let \(c^\star(\varepsilon)\) be the minimum expected cost of a signal structure in which some signal realization inducing \(q\) occurs with probability at least \(\varepsilon\). Thus, \(c^\star(\varepsilon)=\varepsilon \kappa(q|q_0)+o(\varepsilon)\), for \(\varepsilon\downarrow0\). That is, \(\kappa(q|q_0)\) is the marginal expected cost of making posterior \(q\) occur with small probability. Then, Appendix 7 shows that the unit cost \(\kappa\) is the unique nondegenerate cost index for evaluating posterior reports under constant marginal costs. We now identify this unit cost.

This result is the main identification step of our epistemic foundation because it reveals that, for uniformly posterior-separable cost functions, the unit cost of producing a candidate posterior \(q\) is equal to the difficulty measure \(D_G(q\lVert q_0)\) of distinguishing \(q\) from the prior \(q_0\).

4.3 Rational inattention, posterior certification, and representation↩︎

Interpret each individual \(i\in I\) as a distinct information source. For each \(i\), fix a full-support prior (or reference belief) \(p_i\in\Delta(\Omega)\) and a cost function \(C_i\) with generator \(G_i\), where \(\kappa_i\) is the unit cost in ?? induced by \(C_i\). Every source \(i\) is rationally inattentive, in the sense that it can generate posteriors only within its fixed unit-cost budget, denoted \(\eta_i\geq0\), i.e., the set of admissible posteriors for \(i\) consists of all posteriors \(q\in\Delta(\Omega)\) such that \(\kappa_i(q|p_i)\leq \eta_i.\)¹¹ We interpret this capacity constraint as a posterior-level analogue of the standard rational-inattention capacity constraint, adapted to our institutional environment.

The planner is aware that sources are rationally inattentive in the sense above, and following [5], she is not assumed to have superior (or privileged) knowledge about the state of the world.¹² As a result, she cannot verify posterior reports directly. She therefore relies on a mechanism in which sources provide evidence for candidate posteriors. As in [69], this evidence is only partially verifiable: a source can substantiate a posterior only if this posterior does not exceed its capacity constraint, and it cannot manufacture the verification of a posterior outside this constraint. Thus, any mechanism that certifies a posterior as a common evidentiary basis must be able to obtain verification from every source. This necessity requirement is what we call joint admissibility: a collective posterior must satisfy all sources’ unit-cost budgets simultaneously.

Appendix 8 provides the technical details that microfound joint admissibility using an institutional mechanism with partially verifiable evidence and shows that the implementable certification region must consist of those posteriors that all sources can verify.

This equivalence is the main representation result of our foundation. Building on Lemma 2, uniform posterior separability turns each source’s posterior-level budget into a dual Bregman ball, and joint admissibility is therefore equivalent to inclusion in the intersection of those balls. Section 4.4 provides a discussion of Theorem 1 and relates it to Observation 1, then Section 4.5 completes our foundation by providing a parallel result for primal balls.

We also know from [21] that the Shannon cost function is uniquely identified within the uniformly posterior-separable family by an axiom called “invariance under compression.”¹³ This means that including that axiom in Theorem 1 would pin down the dual entropy ball \(\hat{B}^{\text{\normalfont\scriptsize KL}}_{\eta_i}(p_i)\) in Theorem 1.(2). We should note that a similar axiomatization of other Bregman divergences remains an open problem in information theory.

4.4 Discussion↩︎

We can now relate Theorem 1 and Observation 1. DGHT provide a normative justification for collective beliefs to lie in an intersection. We complement their agenda by providing an epistemic justification: each rationally-inattentive source generates a posterior-feasibility set, and the planner’s certification region contains the posteriors consistent with all of these feasibility constraints. Theorem 1 identifies this region with the intersection of dual Bregman balls, i.e., the \(\hat{B}^{G_i}_{\eta_i}(p_i)\)’s in Theorem 1.(2) play the role of the \(P_i\)’s in Observation 1.

4.4.1 A partial-identification analogy↩︎

Our planner’s problem can be interpreted as an institutional version of the partial-identification problem studied by [70]. In [70], the object of interest is a probability distribution that is not point-identified by the available evidence alone. An analyst therefore characterizes the identification region—the set of distributions that remain admissible after the evidence and the maintained restrictions are imposed. Our planner faces the same kind of set-valued problem, but at the level of posterior beliefs. The maintained restrictions are the source-specific unit-cost constraints generated by rational inattention. The identification region is therefore an analogue of the certification region formalized in Appendix 8: it describes which candidate posterior every source can substantiate.

The key feature that relates our environment to [70] is that the identification region in his chapters above also takes an intersection form: different pieces of evidence or maintained restrictions generate sets of admissible probability distributions, and the identified set consists of the distributions that satisfy all of them at once.¹⁴ The analogy is useful because both frameworks produce an intersection of admissible probability distributions. In our institutional environment, the restrictions are sources’ capacity constraints generated by rational inattention and interpreted via the institutional mechanism in Appendix 8.

4.5 Primal Bregman balls↩︎

Section 4.1.1 imposes uniform posterior separability, under which the local cost of signal realization \(y\) is \(D_G(q_y\lVert q_0)\). This order of arguments delivers dual Bregman balls. The opposite order has a different interpretation. Rather than pricing the difficulty of moving from the reference belief \(q_0\) to the realized posterior \(q_y\), it prices the regret from using the contingency-specific model \(q_y\) when the reference belief \(q_0\) is the benchmark predictive model. We show that proper scoring rules provide a natural foundation for this interpretation.

A scoring rule assigns a numerical reward to a reported distribution after the state is realized. It is proper if the expected reward is maximized by truthful reporting [71]. Let \(\iota_{\omega}\in\mathbb{R}^m\) denote the Dirac vector on state \(\omega\in\Omega\). Given a convex differentiable real-valued function \(G\), the score associated with \(G\) is \(s_G(r,\omega):=G(r)+\langle\nabla G(r),\iota_{\omega}-r\rangle\). For any data-generating distribution \(q\), the expected score from report \(r\) is \(G(r)+\langle\nabla G(r),q-r\rangle\), which is maximized at \(r=q\) by convexity of \(G\). Notably, the expected regret from reporting \(r\) instead of \(q\) is the Bregman divergence: \[\label{eq:score95regret} \mathbb{E}_q\big[s_G(q,\omega)-s_G(r,\omega)\big] = D_G(q\lVert r).\tag{6}\] This scoring-rule regret interpretation of Bregman divergences is standard in the probabilistic forecasting and elicitation literature [60], [72]–[74]. For example, when \(G(q)=\sum_{\omega} q(\omega)\log q(\omega)\), the score is the log score and 6 becomes the log-loss regret \(D_{\mathrm{KL}}(q\lVert r)\); when \(G(q)=\sum_{\omega}q(\omega)^2\), the score is the so-called Brier score [75] and 6 becomes the quadratic-loss regret.

After observing signal \(y\), a source reports the predictive model \(q_y(\sigma,q_0)\). In this reporting environment, the reference belief \(q_0\) is a benchmark model and the realized posterior \(q_y(\sigma,q_0)\) is the model proposed after the signal realization. A proper-scoring-rule audit evaluates the regret from validating this contingency-specific report relative to the benchmark. Thus, the local validation cost of report \(q_y(\sigma,q_0)\) is \(D_G(q_0\lVert q_y(\sigma,q_0)).\) Averaging this report-level regret across signal realizations yields a uniformly posterior-separable cost, as in [21], but with the reverse order of arguments. The Blackwell experiment, Bayes rule, and averaging over posteriors are unchanged; the Bregman term now measures the cost of certifying posterior reports against a benchmark predictive model. This is formalized below.

4.5.1 Blackwell informativeness↩︎

The reverse formulation is also disciplined by Blackwell informativeness. If an experiment is garbled, then the posterior induced by each garbled signal is an average of the posteriors induced by the original experiment. Thus, when the report-level regret \(q\mapsto D_G(q_0\lVert q)\) is convex (e.g., entropy or quadratic losses), garbling weakly lowers expected validation cost.

Observation 5 gives the reverse cost an information-theoretic interpretation. More informative experiments generate posterior reports that are weakly more costly to validate under scoring-rule regret. Thus, reverse uniform posterior separability represents a scarce model-validation capacity: a source can report posterior models only up to the amount of scoring-rule regret it can certify relative to its benchmark model.

4.5.2 Representation↩︎

Define \(\tilde{\kappa}(\cdot|\cdot)\) as in ?? by replacing \(C\) by \(\tilde{C}\). For each source \(i\in I\), let \(\tilde{\kappa}_i\) be the unit cost generated by \(\tilde{C}_i\). The unit-cost argument is unchanged from Lemma 2, except for the order of the Bregman divergence; the proof is given in Appendix 10. Joint admissibility is also defined just as in Definition 6 with \(\tilde{\kappa}_i\) in place of \(\kappa_i\), so we omit its statement for brevity.

Thus, replacing \(C\) by \(\tilde{C}\) in Theorem 1 changes only the order of the Bregman divergence. The admissible belief sets therefore switch from dual balls \(\hat{B}^{G_i}_{\eta_i}(p_i)\) to primal balls \(B^{G_i}_{\eta_i}(p_i)\).

5 Application: Financial Markets↩︎

This application studies some observable market implications of our epistemic foundation. Financial markets are relevant even though they do not contain a planner who explicitly chooses collective beliefs. In complete markets, equilibrium prices aggregate heterogeneous probabilistic beliefs into a state-price kernel. In the incomplete-preference economy of [41], called “Bewley economy,” a shared belief in the intersection of individual belief sets can support full-insurance equilibria; when the intersection is a singleton, it is the unique normalized Arrow price vector. Thus, the intersection rule has a market interpretation: it identifies which belief is capitalized into prices. We use this observation to show the pricing implications of the Bregman geometry: entropy balls make the Bewley economy mimic log-utility and exponential-utility risk economies, while the broader separable Bregman family delivers power-pooled prices with an isoelastic risk-sharing interpretation.

5.1 Setup and preliminaries↩︎

Consider a two-date Arrow-Debreu exchange economy. At date \(1\), one of the states \(s\in S\) realizes. There is one consumption good in each state and a complete set of Arrow securities traded at date \(0\). Let \(I\) be a finite set of market participants. We impose the assumptions in [41]: for each individual \(i\in I\), let \(\omega^i\in\mathbb{R}^S_{++}\) be the date-\(1\) endowment, let \(u_i:\mathbb{R}_+\to\mathbb{R}\) be differentiable, strictly increasing, concave, such that \(u_i'(z)>0\) for all \(z\), and let \(P_i\subseteq\Delta(S)\) be a nonempty closed convex set of beliefs. Individual \(i\)’s unambiguous preference over consumption bundles has the following representation: \[x^i\succ_i y^i \quad\Longleftrightarrow\quad \sum_{s\in S}\pi(s)u_i(x^i_s) > \sum_{s\in S}\pi(s)u_i(y^i_s) \quad\forall\pi\in P_i,\] where this strict preference is imposed in [41]. Assume no aggregate uncertainty: \(\sum_{i\in I}\omega^i_s=\bar W\) \(\forall s\in S\), for \(\bar W>0\). For \(x^i\in\mathbb{R}^S_{++}\), define the marginal-belief set \[P_i(x^i) := \left\{ q\in\Delta(S): q(s)= \frac{\pi(s)u_i'(x_s^i)}{\sum_{t\in S}\pi(t)u_i'(x_t^i)} \text{ for some }\pi\in P_i \right\}.\] A feasible allocation \((x^i)_{i\in I}\) is a full-insurance allocation if each \(x^i\) is constant across states, say \(x_s^i=a^i\) for every \(s\). Notice that if \(x^i\) is constant across states, then \(P_i(x^i)=P_i\).

Full insurance isolates the pricing role of beliefs. With no aggregate uncertainty, variation in normalized Arrow prices is not driven by aggregate consumption risk, but instead, it is driven by the shared belief selected from the intersection of belief sets. Thus, Lemma 3 clarifies the economic role of the singleton intersection: in a full-insurance Bewley equilibrium, a singleton intersection of belief sets pins down the unique normalized Arrow price.

For comparison, Lemma 4 presents two classical complete-preference benchmarks that aggregate heterogeneous priors in different ways. Throughout, every individual \(i\) has a full-support reference belief \(p_i\in P_i\), which later will represent the center of their Bregman ball.

Lemma 4 formalizes how two standard complete-preference risk economies price beliefs in the absence of aggregate uncertainty. In the log-utility benchmark, normalized Arrow prices average probability levels across individuals. In the exponential-utility benchmark, normalized Arrow prices average log probabilities across individuals and therefore take a geometric-pooling form. These benchmarks differ from the Bewley economy above, where individuals have incomplete preferences represented by belief sets, and prices are disciplined by shared beliefs—those beliefs that remain in their intersection. The next result uses Lemmas 3 and 4 to contrast these two ways of generating normalized Arrow prices.

5.2 Market implications of Bregman balls↩︎

We now specialize the full-insurance pricing result to entropy balls, where the only difference between the two cases is the order of arguments in the underlying KL divergence.

This result gives the Bregman geometry a pricing interpretation in financial markets. The order of arguments selects which shared belief is capitalized into equilibrium prices in full-insurance Bewley economies. Specifically, primal entropy yields the prices of a log-utility risk economy, whereas dual entropy yields the prices of an exponential-utility risk economy. Proposition 5 extends the same logic to power pooling and isoelastic risk sharing.

5.3 Illustration: financial losses and information technologies↩︎

Section 3 indicated that primal entropy aggregates probability levels, whereas dual entropy aggregates log probabilities. Proposition 4 shows that this distinction has an asset-pricing implication: using the wrong entropy order is equivalent to using the prices of a different complete-preference economy. We now illustrate how this mistake generates financial losses.

Consider two states, \(S=\{A,B\}\), and two individuals, \(I=\{1,2\}\), with reference beliefs \(p_1=(0.96,0.04)\) and \(p_2=(0.36,0.64)\). Individual \(1\) views state \(B\) as very unlikely, while individual \(2\) views it as fairly likely. Suppose the information technology is the dual-entropy technology, so individual belief sets are dual KL balls, \(P_i=\hat{B}^{\text{\normalfont\scriptsize KL}}_{\eta_i}(p_i)\). Suppose the radii satisfy \(\eta_1=D_{\mathrm{KL}}(q^{\mathrm H}\lVert p_1)=0.202\) and \(\eta_2=D_{\mathrm{KL}}(q^{\mathrm H}\lVert p_2)=0.379\), where \(q^{\mathrm H}=(0.786,0.214)\). This belief is the equal-weight geometric pool of \(p_1\) and \(p_2\): its odds ratio is the geometric average of the two individual odds ratios, since \(q^{\mathrm H}(A)/q^{\mathrm H}(B)=\sqrt{(p_1(A)/p_1(B))(p_2(A)/p_2(B))}=\sqrt{13.5}\). Moreover, it follows that the intersection \(P_1\cap P_2=\{q^{\mathrm H}\}\) is a singleton. By Proposition 4, every full-insurance Bewley equilibrium has the unique normalized Arrow price vector \(q^{\mathrm H}\); in particular, the state-\(B\) Arrow security has price \(q^{\mathrm H}(B)=0.214\).

Suppose a financial authority evaluates whether buying one unit of state-\(B\) security at date-\(0\) price \(\rho\) is socially valuable. If the authority ignores the dual-entropy technology and instead uses a linear pooling, it acts as if the relevant benchmark were the log-utility economy in Lemma 4.(i), rather than the exponential-utility economy in Lemma 4.(ii). The equal-weight linear pooling is \(q^{\mathrm L}=(0.66,0.34)\), so the mistaken authority values the state-\(B\) security at \(q^{\mathrm L}(B)=0.34\). Figure 1 illustrates the financial loss caused by applying the incorrect aggregation rule. The “wrong-rule approval region” \((0.214 < \rho < 0.34)\) denotes prices where the linear-pooling authority incorrectly evaluates the security as underpriced and authorizes a purchase. Within this region, the downward-sloping line (\(0.34-\rho\)) plots the authority’s perceived gain per unit, while the upward-sloping line (\(\rho-0.214\)) plots the actual financial loss evaluated under the correct dual-entropy technology. The intersection at \(\rho=0.277\) marks the price where the perceived gain equals the actual loss. To the left of this point (\(0.214 < \rho < 0.277\)), the perceived gain exceeds the actual loss, leading the authority to view the trade as favorable despite it reducing overall value. To the right (\(0.277 < \rho < 0.34\)), the actual loss surpasses the perceived gain. The shaded region traces the actual per unit loss across prices at which the authority approves the trade. In summary, this example has illustrated how failing to account for information technologies when aggregating beliefs can lead a planner to endorse trades that generate aggregate financial losses.

5.4 Extension: power pooling and risk aversion↩︎

The entropy comparison isolates the effect of changing the order of relative entropy. The same pricing logic extends to the broader separable Bregman family from Section 3.4. When the singleton intersection is a power pool, the resulting state-price kernel has a standard risk-sharing interpretation: it is the Arrow price vector generated by an isoelastic risk (CRRA) economy. Thus, the power parameter \(r\) has a direct economic meaning in financial markets.

6 Conclusion↩︎

This paper develops an epistemic foundation for the intersection rule by modeling the formation and aggregation of imprecise beliefs through the lens of rational inattention, partial verifiability, and information theory. We establish that when individuals face standard information-acquisition costs and capacity constraints, the set of all collectively admissible posteriors forms an intersection of Bregman balls. This geometric structure is economically relevant: its inherent asymmetry unifies linear, geometric, power, and multiplicative pooling rules under a single framework. These distinct epistemic rules carry observable market implications; when applied to financial markets, the underlying information-cost primitive acts as the main driver dictating how conflicting beliefs are capitalized into equilibrium prices.

7 Appendix: Uniqueness of Unit Cost↩︎

This appendix explains why the unit-cost restriction is the only nondegenerate way to impose a cost constraint on posterior reports. The issue is due to the timing of the planner’s problem. Observation 3 shows that imposing an expected-cost budget on posterior reports is vacuous: whenever a posterior can be produced, the experiment can be diluted so that the same posterior is reached with arbitrarily small probability and arbitrarily small expected cost.

The constant-marginal-cost property in Lemma 1 identifies the appropriate normalization. If posterior \(q\) is required to occur with probability \(\varepsilon\), then a report budget \(\eta\) corresponds to the ex-ante budget \(\varepsilon\eta\). Thus, the relevant object is the least expected cost of producing \(q\) per unit probability of reaching \(q\). This appendix shows that this normalization is necessary.

Throughout this appendix, fix \(q_0\in\Delta^\circ(\Omega)\) and a target posterior \(q\in\Delta(\Omega)\) for which the unit cost in Definition 5 is finite. For each \(\varepsilon\in(0,1]\), recall from Observation 4 that \[c_q^\star(\varepsilon):= \inf\Big\{ C(\sigma,q_0;Y): \exists y^\star\in Y \text{ with } q_{y^\star}(\sigma,q_0)=q \text{ and } \pi_{y^\star}(\sigma,q_0)\geq\varepsilon \Big\}\] is the least expected cost of making \(q\) occur with probability at least \(\varepsilon\). Define the critical report cost \(\Gamma(q|q_0):=\lim_{\varepsilon\downarrow0} \frac{c_q^\star(\varepsilon)}{\varepsilon},\) whenever the limit exists. All proofs are in Appendix 10.

A proposed report-cost index \(\rho(q|q_0)\) is cost-calibrated if it uses the same units as the primitive expected cost in the following sense: for every budget \(\eta\geq0\), if \(\rho(q|q_0)<\eta\), then \(c_q^\star(\varepsilon)\leq\varepsilon\eta\) for all sufficiently small \(\varepsilon>0\), while if \(\eta<\rho(q|q_0)\), then \(c_q^\star(\varepsilon)>\varepsilon\eta\) for all sufficiently small \(\varepsilon>0\). In words, budgets above the report cost are sufficient to substantiate the report at small probability, and budgets below the report cost are not sufficient.

Cost calibration is the relevant economic concept because it captures the idea that two posterior reports should be regarded as equally informative whenever they require the same marginal informational investment to produce, regardless of how frequently they arise in expectation. The constant-marginal-cost property says that requiring a report with probability \(\varepsilon\) scales the relevant ex-ante budget by exactly \(\varepsilon\). Dividing by \(\varepsilon\) in Definition 5 is therefore the unique way to express the cost of the report itself in the same units as the primitive cost. Any proposed restriction with a different threshold either admits reports below the cost needed to substantiate them or rejects reports above the cost needed to substantiate them. Proposition 6 rules out both possibilities and leaves exactly the restriction \(\kappa(q|q_0)\leq\eta\).

8 Appendix: Microfoundation of Joint Admissibility↩︎

We provide a microfoundation of joint admissibility (Definition 6) by applying the partial-verifiability mechanism-design framework of [69] to posterior reports. The environment is a posterior-reporting institution in which information sources have private abilities to substantiate posterior claims. A posterior can serve as a common evidentiary basis precisely when every source can provide the verifying message associated with that posterior. Joint admissibility is the resulting implementable certification region.

The climate-policy example in DGHT’s introduction illustrates the economic content of this certification problem. They describe a European climate-policy decision in which different public authorities (e.g., French and British governments) may use different ranges for the probability of severe global warming. In our terminology, each authority is an information source whose models, data, and forecasting capacity determine which posterior claims it can substantiate. A posterior probability for severe warming can serve as a common evidentiary basis for policy evaluation only when every source can verify it. Joint admissibility therefore captures the institutional requirement that a climate-policy assessment be supported by all evidentiary constraints before it is used to rank uncertain policies at the European level.

The appendix proceeds as follows. Appendix 8.1 recalls the relevant Deneckere-Severinov primitives. Appendix 8.2 embeds posterior reports into those primitives. Appendix 8.3 proves that joint admissibility is the set of posterior claims that can be certified by an incentive-compatible mechanism with partial verifiability. All proofs are in Appendix 10.

8.1 Partial verifiability↩︎

[69] study mechanism design when agents have type-dependent abilities to provide evidence. Their model has a public-decision set \(\mathcal{X}\), a verifying-message space \(\mathcal{C}\), and agents \(i=1,\dots,L\). In the basic model, a type is \(t_i=(\vartheta_i,\mathcal{M}_i)\), where \(\vartheta_i\) is a preference parameter and \(\mathcal{M}_i\subseteq\mathcal{C}\) is the set of verifying messages available to agent \(i\). A verifying message is a feasible piece of evidence: a type can be asked to provide only messages available to that type. We build on their formulation with combinations of messages [69]. Here, a type is \(t_i=(\vartheta_i,\mathcal{E}_i)\), where \(\mathcal{E}_i(t_i)\) is the set of feasible message combinations. A feasible mechanism is a dynamic game form in which, at every information set where \(i\) can send verifying messages, type \(t_i\) is restricted to combinations in \(\mathcal{E}_i(t_i)\); along any path, the union of verifying messages sent by \(t_i\) must also belong to \(\mathcal{E}_i(t_i)\). Thus, evidence requests are supported on feasible message combinations.

The implementation results in [69] identify the incentive consequences of this evidence structure. Under their Assumption 1, their Theorem 3 shows that their revelation mechanism is without loss. Under the worst-outcome version used in their Corollary 4, the incentive constraints are exactly those in which a type imitates another type whose feasible message combinations it can reproduce. This implication is central in our posterior-reporting setting: a source that can verify a posterior may withhold the evidence, while a source that cannot verify the posterior cannot manufacture the verifying message. We find this property to be natural in institutional settings such as health policy and climate policy, where manufacturing messages may cause significant irreversible damages.

8.2 Embedding posterior reports into partial verifiability↩︎

Fix a profile \(\mathscr{E}:=\{(p_i,C_i,G_i,\eta_i)\}_{i\in I}\). For each \(i\), let \(\kappa_i(\cdot|p_i)\) denote the unit cost generated by \(C_i\) (Definition 5). From Section 4.3, \(i\) is rationally inattentive with posterior-verifiability set \(Q_i(\mathscr{E}) := \{q\in\Delta(\Omega):\kappa_i(q|p_i)\leq \eta_i\}.\) This is the set of posterior claims that source \(i\) can substantiate within its unit-cost capacity, which captures [69]’s ([69, p. 499]) insight that some messages may be “infeasible or too costly to send.”

Fix a posterior claim \(q\in\Delta(\Omega)\). We build a Deneckere-Severinov environment for certifying this posterior. The public-decision set is \(\mathcal{X}_q:=\{x_q,x_{\varnothing}\}\cup\{x_i^\bot:i\in I\},\) where \(x_q\) means that posterior \(q\) is certified as the common evidentiary basis, \(x_{\varnothing}\) means that no posterior is certified, and \(x_i^\bot\) is the punishment decision for source \(i\). The verifying-message space is \(\mathcal{C}_q:=\{m_i(q):i\in I\},\) where \(m_i(q)\) is \(i\)’s evidence for posterior \(q\).

Source \(i\) has two possible verifiability types for posterior \(q\). Type \(V_i^q\) can verify \(q\), while type \(N_i^q\) cannot. Their feasible message-combination sets are \(\mathcal{E}_i^q(V_i^q)=\big\{\{m_i(q)\},\emptyset\big\},\) and \(\mathcal{E}_i^q(N_i^q)=\{\emptyset\}.\) Thus, \(V_i^q\) can imitate \(N_i^q\) by withholding evidence, while \(N_i^q\) cannot imitate \(V_i^q\) because it cannot send \(m_i(q)\). For the source profile \(\mathscr{E}\), define the induced verifiability type by \(t_i^q(\mathscr{E})=V_i^q\) if \(q\in Q_i(\mathscr{E})\), and \(t_i^q(\mathscr{E})=N_i^q\) otherwise. Hence, \[\label{eq:joint95type95equiv} q\in\bigcap_{i\in I}Q_i(\mathscr{E}) \quad\Longleftrightarrow\quad t_i^q(\mathscr{E})=V_i^q \qquad \forall i\in I .\tag{7}\]

Let \(T_i^q:=\{V_i^q,N_i^q\}\), \(T^q:=\prod_{i\in I}T_i^q\), and let \(F^q\) be a full-support prior on \(T^q\); write \(F_i^q(t_{-i}|t_i)\) for its conditional probability. Let \(v_i(x;t)\) denote source \(i\)’s payoff from public decision \(x\) when the type profile is \(t\). A mechanism certifies \(q\) only at histories where every \(i\) has sent a message combination containing \(m_i(q)\). Since \(x_q\) represents certification of \(q\) as common evidence, certification therefore requires the full collection of source-specific verifying messages \(\{m_i(q)\}_{i\in I}\). This concept formalizes the requirement that a certified collective posterior be supported by every source’s posterior evidence.

8.3 Joint admissibility as implementable certification↩︎

The next result is the main result of this microfoundation, showing that certification requires joint admissibility, i.e., every source must be able to verify a candidate posterior.

Notice that this necessity result uses only feasibility of posterior-specific evidence without requiring any additional incentive restrictions. We will now complement this result with the converse under two standard conditions. (1) The Deneckere-Severinov worst-outcome condition: \(v_i(x_i^\bot;t)=-\infty\) \(\forall t\), while \(v_i(x;t)>-\infty\) \(\forall x\in\mathcal{X}_q \setminus\{x_i^\bot\}\) and \(\forall t\).¹⁵ This penalty is natural in high-stakes institutional settings such as the climate-policy example where endorsing a fraudulent climate model may ruin an authority’s reputation. (2) The alignment condition: a source that can verify \(q\) weakly prefers, in expectation, certification to withholding verification when all other sources can verify \(q\). Formally, \[\label{eq:alignment} \sum_{t_{-i}: t_j=V_j^q\forall j\neq i} \Big[v_i(x_q;V_i^q,t_{-i})-v_i(x_{\varnothing};V_i^q,t_{-i})\Big] F_i^q(t_{-i}|V_i^q) \geq 0 \qquad \forall i\in I,\tag{8}\] i.e., this is the Deneckere–Severinov incentive constraint specialized to the certification rule. In the climate-policy example, alignment is natural because it means that if the authorities can all substantiate the same posterior probability that severe warming will occur, then each authority weakly prefers that this posterior be officially certified and used for policy evaluation. Now, define the certification social choice function \(f_q^\star:T^q\to\mathcal{X}_q\) by \(f_q^\star(t)=x_q\) if \(t_i=V_i^q\) \(\forall i\in I\), and \(f_q^\star(t)=x_{\varnothing}\) otherwise.

We can now show the necessity and sufficiency of joint admissibility for certification.

It is also straightforward to extend from a single posterior to a set of posteriors.

The institutional content of joint admissibility is therefore the following: each source’s information technology determines which posterior-verifying messages it can provide, partial verifiability determines which certification rules are incentive compatible, and the planner’s admissible collective beliefs are exactly the posteriors that every source can verify.

9 Appendix: Regularity Conditions↩︎

We impose some regularity conditions on Bregman balls that are standard in information geometry, where, for example, all balls are typically assumed to have the same radius [61], [76]. Some of these conditions can be relaxed, e.g., most of our results extend to infinite-dimensional spaces, but we omit such extensions to simplify proofs. We separate the conditions used in Section 3 and Section 4.

9.1 Regularity: Section 3↩︎

Throughout the aggregation results in Sections 3.2–3.5, \(S\) is finite and define the set of full-support beliefs as \(\Delta^{\circ} := \{q\in\Delta:q(s)>0 \forall s\in S\}.\) In all our aggregation results, we maintain the assumption that the intersection of belief sets is nonempty, and all reference beliefs satisfy \(p_i\in\Delta^{\circ}\) for every \(i\in I\). Whenever a result assumes that an intersection of belief sets is a singleton, the singleton element is assumed to lie in \(\Delta^{\circ}\).¹⁶ This guarantees that the derivatives used in the singleton-intersection arguments are well-defined.

Notice that the entropy generator \(G(q)=\sum_{s\in S}q(s)\log q(s)\) is not differentiable on the boundary of the simplex. Thus, whenever a generator \(G\) is defined only on an open convex set \(\mathcal{D}\) containing \(\Delta^{\circ}\), the associated Bregman divergence is understood on \(\Delta\) through its lower-semicontinuous extension whenever this extension exists. For the entropy generator, we use the standard conventions \(0\log 0=0\) and \(a\log(a/0)=+\infty\) for \(a>0\).

For the primal Bregman-ball results in Section 3.2, the maintained conditions are the following. The generator \(G\) is twice continuously differentiable and convex on an open convex domain \(\mathcal{D}\subseteq\mathbb{R}^S\) containing \(\Delta^{\circ}\).¹⁷ For every \(i\in I\), define the extended-valued function \[F_i(q):= \begin{cases} D_G(p_i\Vert q), & q\in\Delta\cap\mathcal{D},\\ +\infty, & q\in\Delta\setminus\mathcal{D}. \end{cases}\] We assume that \(F_i\) is convex and lower semicontinuous on \(\Delta\). We also assume that, for every \(q\in\Delta\cap\mathcal{D}\), the Hessian \(\nabla^2G(q)\) is positive definite on the tangent space \(T_\Delta:=\big\{z\in\mathbb{R}^S:\sum_{s\in S}z(s)=0\big\}.\) These conditions imply that the primal balls \(B^G_{\eta_i}(p_i)=\{q\in\Delta:F_i(q)\leq \eta_i\}\) are closed and convex subsets of \(\Delta\), which are the two required properties of belief sets in unambiguous preferences (see, Definition 1).¹⁸ For example, they include primal entropy balls because, for \(p_i\in\Delta^{\circ}\), \(D_{\mathrm{KL}}(p_i\Vert q)\) is convex and lower semicontinuous on \(\Delta\), finite on \(\Delta^{\circ}\), and the entropy Hessian is positive definite on \(T_\Delta\).

For the dual Bregman-ball results in Sections 3.3–3.4, no constraint qualification is imposed. We use only the following implication: if \(q^*\in\Delta^{\circ}\) and \(\bigcap_{i\in I}\big\{q\in\Delta:f_i(q)\leq0\big\}=\{q^*\},\) where each \(f_i\) is convex on \(\Delta\) and differentiable in a neighborhood of \(q^*\), then, with \(A(q^*)=\{i\in I:f_i(q^*)=0\}\), there exist multipliers \((\lambda_i)_{i\in A(q^*)}\), with \(\lambda_i\geq0\) and not all zero, and a scalar \(\zeta\in\mathbb{R}\), such that \(\sum_{i\in A(q^*)}\lambda_i\nabla f_i(q^*)=\zeta\mathbf{1}\), where \(\mathbf{1}\) denotes the vector of all ones. This implication is proved in Appendix 10 as Lemma 7. Applied to \(f_i(q)=D_G(q\Vert p_i)-\eta_i\), it gives the active-normal relation used in the proofs of Proposition 2 and Proposition 3.

9.2 Regularity: Section 4↩︎

For Section 4, all priors have full support. The generator \(G\) in Definition 4 is convex and differentiable at every full-support prior under consideration, and \(D_G(q\lVert q_0)\) is well-defined for every posterior \(q\in\Delta(\Omega)\) used in Lemma 2 and Theorem 1. This convention includes the entropy generator \(G(q)=\sum_{\omega\in\Omega}q(\omega)\log q(\omega)\), because \(D_{\mathrm{KL}}(q\lVert q_0)\) is finite for every \(q\in\Delta(\Omega)\) whenever \(q_0\in\Delta^{\circ}(\Omega)\). For the reverse-order results in Section 4.5, the generator \(G\) is continuously differentiable on a neighborhood of every posterior at which \(D_G(q_0\lVert q)\) is evaluated; in particular, the reverse entropy case is applied only to \(q\in\Delta^{\circ}(\Omega)\).

10 Appendix: Proofs↩︎

10.1 Proof of Proposition 1 and Corollary 1↩︎

10.2 Proof of Propositions 2–3, Observation 2 and Corollary 3↩︎

10.3 Proof of Lemma 1 and Observations 3–4↩︎

The proof of Lemma 1 is straightforward, but we include it for completeness.

10.4 Proof of Lemma 2 and Theorem 1↩︎

10.5 Proof of Observation 5 and Theorem 2↩︎

10.6 Proofs from Section 5↩︎

10.7 Proofs from Appendices 7–8↩︎

References↩︎

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1. Department of Economics, MIT, florianm@mit.edu.↩︎

2. I sincerely thank Drew Fudenberg and Stephen Morris for their guidance and support during this project.↩︎

3. DGHT’s Section II.B propose two ways to deal with empty intersections: (1) omit individuals whose belief sets are incompatible with others by giving them zero utility weight; (2) consider a weaker aggregation result where the intersection is replaced with the convex hull of the union of belief sets.↩︎

4. For example, [57] and [58] require existence of shared beliefs to rationalize Pareto improvements in exchange economies under uncertainty.↩︎

5. The squared error is (up to scaling) the unique Bregman divergence that is symmetric [60].↩︎

6. In statistics, [17] discuss dual and primal entropy balls.↩︎

7. If \(\Phi:\Delta^{|I|}\rightarrow\Delta\) is a belief-aggregation rule, the unanimity axiom requires \(\Phi(p,\dots,p)=p\). Now, suppose \(p_i=p\) \(\forall i\in I\) and \(|I|>1\). The multiplicative rule yields \(q^\times(s)=p(s)^{|I|}/\sum_{t\in S}p(t)^{|I|}\), which differs from the common \(p\), except when \(p\) is uniform. In contrast, all other rules above satisfy unanimity.↩︎

8. Full-support-prior assumption is standard in information theory [31].↩︎

9. For a survey on the standard capacity constraint, see [66].↩︎

10. Our model is not the first to adapt aspects of rational inattention; it is actually common to do so in applications. For example, [28] adapt the framework to portfolio choice by letting investors choose what asset-payoff information to acquire before investing, and they compare entropy-based, linear-precision, and decreasing-returns learning technologies. [29] adapt it to financial contagion by letting investors allocate limited information-processing resources across two stock markets, imposing a linear constraint of the form \(K_{i1}+K_{i2}\leq K\), and distinguishing anticipated from unanticipated crises depending on whether capacity can be adjusted. [30] adapt it to mutual funds by letting every manager \(j\) allocate signal precisions across \(L\) risk factors under \(\sum^L_{\ell=1}K_{\ell j}\leq K\), with signals that may concern linear combinations of asset payoffs or risk factors.↩︎

11. This form of capacity restriction imposed on Bregman divergences appears also in other fields; e.g., [67] impose an ex-post privacy capacity by requiring the KL divergence between a posterior and prior to be bounded above; see also [68] for a version in Bayesian persuasion.↩︎

12. As [5] remark: “In many realistic situations, the planner does not observe the objective belief and faces the same difficulty as individuals do in discriminating different beliefs based on available data.”↩︎

13. This axiom states that behavior is invariant to changes in the decision problem that leave the probabilistic structure of payoffs unchanged—meaning the nature of the state space itself is behaviorally irrelevant.↩︎

14. [70] studies the partial identification of probability distributions—a problem that arises when an analyst does not have enough information or evidence to point-identify a distribution of interest. Chapter 1.1 introduces the basic logic: when an additional maintained restriction is imposed, the original evidence-based region is intersected with the set allowed by that restriction. Chapter 1.4 then studies finitely many sampling processes, each of which implies its own set of admissible distributions, and the identification region is their intersection. Chapter 2.4 shows that a closely related finite-intersection structure arises when the maintained restriction is imposed across finitely many observable conditions.↩︎

15. The infinite punishment is a convenient way to use [69]. The results hold with finite punishments by using their general incentive constraints and requiring, in addition, that a non-verifier who reports \(V_i^q\) and cannot produce \(m_i(q)\) weakly prefers truthful non-verification. In the certification mechanism used here, this amounts to \(\sum_{t_{-i}: t_j=V_j^q \forall j\neq i} \bigl[v_i(x_\emptyset;N_i^q,t_{-i})-v_i(x_i^\perp;N_i^q,t_{-i})\bigr]F_i^q(t_{-i}|N_i^q)\ge 0,\) which holds whenever \(x_i^\perp\) is a sufficiently severe finite punishment for source \(i\).↩︎

16. Focusing on interior solutions is standard in the aggregation literature [77].↩︎

17. These conditions on \(G\) are standard in economics [24].↩︎

18. These assumptions are necessary because [61] show that while dual balls are always convex, primal balls need not be, so we must restrict attention to primal balls that are convex.↩︎