The Value of Context:
Human versus Black Box Evaluators

Our paper is situated at the intersection of the literatures on learning (Section 1.1.1) and strategic information disclosure (Section 1.1.2), where our analysis is primarily differentiated from the previous frameworks by our assumption that the agent has model uncertainty (see Section 1.1.1). Our paper is also inspired by a recent empirical literature that compares human and AI evaluation, which we review in Section 1.1.3.

Agents are each described by a binary covariate vector $\mathbb{x}_{n}=(x_{1},x_{2},\dots,x_{n})$ and a type $y\in\left[-\overline{y},\overline{y}\right]$ (where $0\leq\overline{y}<\infty$), which are structurally related by the function

We refer to $f$ henceforth as the type function. The distribution over covariate vectors is uniform in the population.⁸⁸8All of our results extend for arbitrary finite-valued covariates.

We refer to the covariates indexed to $\mathcal{S}=\{1,\dots,s\}$ as standard covariates and the covariates indexed to $\mathcal{N}=\{s+1,\dots,n\}$ as nonstandard covariates.

An evaluation of the agent, $\hat{y}\in\left[-\overline{y},\overline{y}\right]$, is described in the following section. The agent has a Lipschitz continuous utility function $u:\left[-\overline{y},\overline{y}\right]^{2}\rightarrow\mathbb{R}$, which maps the evaluation $\hat{y}$ and the agent’s true type $y$ into a payoff.

There are two evaluators: a black box evaluator, henceforth Black Box (it), and a human evaluator, henceforth Human (she). Both form evaluations as an expectation of the agent’s type $y$ given observed covariates, so we will introduce notation for these conditional expectations. For any covariate vector $\mathbb{x}_{n}=(x_{1},\dots,x_{n})$ and subset of nonstandard covariates $A\subseteq\mathcal{N}$, let

be the set of all covariate vectors that agree with $\mathbb{x}_{n}$ on the covariates with indices in $\mathcal{S}\cup A$. Further define

to be the conditional expectation of the agent’s type given their standard covariates and their nonstandard covariates with indices in $A$. We use

to denote the agent’s payoff given this evaluation.

Both the human and black box evaluation take the form (2.2), but the observed sets of nonstandard covariates $A$ are different across the evaluators. Black Box observes the nonstandard covariates in the set $B=\{s+1,\dots,s+b_{n}\}$ where $b_{n}=\lfloor\alpha_{b}\cdot n\rfloor$.¹⁰¹⁰10One can instead assume that these nonstandard covariates are selected uniformly at random. This will not affect the results of this paper. Importantly, this set is held fixed across agents. So an individual with covariate vector $\mathbb{x}_{n}$ receives the evaluation $\hat{y}_{\mathbb{x}_{n}}^{f}(B)$ and payoff $U_{\mathbb{x}_{n}}^{f}(B)$ when evaluated by the Black Box.¹¹¹¹11It is not important for our results that $B$ is common across individuals; what we require is that any randomness in $B$ is independent of the agent’s covariates and type. For example, if the set $B$ were drawn uniformly at random for each agent, our results would hold.

Human differs from Black Box in two ways. First, Human has a capacity of $h_{n}=\lfloor\alpha_{h}\cdot n\rfloor$ nonstandard covariates per agent, where $\alpha_{h}<\alpha_{b}$ (i.e., Human cannot process as many inputs as Black Box). Second, Human does not pre-specify which nonstandard covariates to observe, but rather learns these through conversation, and thus potentially observes different nonstandard covariates for each agent. For example, a doctor (evaluator) may pose different questions to different patients (agents) depending on their answers to previous questions. Or a job candidate (agent) might choose to disclose to an interviewer (evaluator) certain nonstandard covariates that are favorable to him.

Rather than modeling the complex process of a conversation, we study the quantity

which is the agent’s payoff when the posterior expectation about his type is based on those $\alpha_{h}\cdot n$ or fewer covariates that are best for him.

We can interpret this quantity as an upper bound for the agent’s payoffs under certain assumptions. First, if the evaluator selects which covariates to observe, then (2.3) is an upper bound on the agent’s possible payoffs across all possible evaluator selection rules. Second, if covariates are disclosed by the agent, but the evaluator updates to the disclosed covariates as if they had been chosen exogenously, then again (2.3) represents an upper bound on the agent’s possible payoffs.¹²¹²12Jin et al. (2021) and Farina et al. (2023) report that the beliefs of experimental subjects falls somewhere in between this naive benchmark and equilibrium beliefs, since subjects do not completely account for the strategic nature of disclosure.

If however the covariates are disclosed by the agent in a disclosure game, and the evaluator accounts for the strategic nature of this disclosure, then whether (2.3) represents an upper bound will depend on what we assume that the agent knows at the time of disclosure.¹³¹³13This is roughly because the agent can potentially “sneak in” information about the other covariates via the covariates that are revealed. We show in Section 4.2 that if the agent knows his entire covariate vector, then (2.3) need not upper bound every agent’s payoffs. Nevertheless, we present a different quantity that does upper bound the maximum payoff that any agent can obtain in this disclosure game, and show that our main results extend when we replace (2.3) with this quantity. To streamline the exposition we focus on the prior two interpretations (in which the human evaluator either selects the covariates herself or updates to the agent’s disclosures naively), and discuss disclosure games in Section 4.2.

A key input towards understanding the comparison between Human and Black Box is quantifying the extent to which individualized context improves the agent’s payoffs.

In general, the value of context depends on the type function $f$ as well as on the agent’s own covariate vector $\mathbb{x}_{n}$.¹⁴¹⁴14The value of context given a specific function $f$ is spiritually related to the communication complexity of $f$ (Kushilevitz and Nisan, 1996).

In what follows, we give the agent uncertainty about $f$ and characterize the agent’s expected value of context and expected payoffs, integrating over the agent’s belief about $f$.¹⁵¹⁵15If we interpret the covariates in our model as signals about the type, then the function relating covariates to type corresponds to the signal structure.

We do this for two reasons. First, in many applications it is not realistic to suppose that the agent knows $f$. For example, a patient who anticipates that a diagnosis will be based on an image scan of his kidney may recognize that there are properties of the image that are indicative of whether he has the condition or not, but likely does not know what the relevant properties are, or how they determine the diagnosis.¹⁶¹⁶16In the case of a job interview, the function $f$ may reflect particular subjective preferences of the firm, which are initially unknown to the agent.

Second, the case with uncertainty about $f$ turns out to yield a more elegant analysis than the one in which $f$ is known. That is, although the value of context for specific functions $f$ depends on details of that function and on the agent’s own covariate vector, there is a large class of prior beliefs (described in the following section) for which it is possible to draw strong detail-free conclusions about the expected value of context.

We impose two assumptions on the agent’s prior belief about $f$. Together, these assumptions deliver a setting in which many different structural relationships between the covariates and the type are possible (including both ones where the value of context is high and low), but ex-ante those relationships are not known.

The first assumption says that while the agent may know how standard covariates impact the type, he has no ex-ante knowledge about the roles of the nonstandard covariates.

The set $\{(f(x_{1},\dots,x_{s},x_{s+1},\dots,x_{n}):(x_{s+1},\dots,x_{n})\in\{0,1\}^{n-s}\}$ ranges over all covariate vectors that share the standard covariate values $(x_{1},\dots,x_{s})$. Assumption 1 says that the joint distribution of these types is ex-ante invariant to permutations of the labels and values of the nonstandard covariates. An agent whose prior satisfies Assumption 1 is thus agnostic about how the nonstandard covariates impact the type.

While our assumption of no prior knowledge about the role of nonstandard covariates is strong, it is consistent with our interpretation of the nonstandard covariates as those covariates for which there is little historical data about correlations. For example, if it were known that a higher GPA positively correlates with on-the-job performance, but not how a large number of social media followers predicts performance, then we would think of GPA as a standard covariate and the number of social media followers as a nonstandard covariate.

Assumption 1 does not restrict how the agent’s prior varies with $n$, the number of covariates. In a model in which $x_{1},x_{2},\dots$ were drawn i.i.d. from a type-dependent distribution $F_{y}$, the total quantity of information about $y$ would increase in the number of covariates, and the evaluator’s uncertainty about $y$ would vanish as $n$ grew large. This does not seem descriptive of real applications: credit scoring algorithms and healthcare algorithms use millions of covariates, but there remains substantial residual uncertainty about the agent’s type. We take the opposite extreme in which the predictability of the agent’s type is a primitive of the setting, which is held constant for all $n$. In our model, $n$ is not a measure of the total quantity of information, but instead moderates the richness of the informational environment and the potential complexity of the mapping $f$. Loosely speaking, as $n$ grows large, the agent has a more extensive set of words to describe a fixed unknown $y$.¹⁷¹⁷17As $n$ grows large, the smallest possible informational size of each covariate (in the sense of McLean and Postlewaite (2002)) vanishes. But we do not require each covariate to be equally informationally relevant in the realized function. So, for example, $f(x_{1},\dots,x_{n})=x_{1}$ can be in the support of the agent’s beliefs for $n$ arbitrarily large (see Example 7).

The statement of Assumption 2 formally depends on the ordering of types within the vector $(Y^{n^{\prime}}_{1},\dots,Y^{n^{\prime}}_{2^{n^{\prime}}})$, since this determines which types appear in the truncated sequence $(Y^{n^{\prime}}_{1},\dots,Y^{n^{\prime}}_{2^{n}})$. But if we further impose Assumption 1 (and we will always impose these assumptions jointly), then the ordering of types is irrelevant: That is, when Assumption 2 holds for one such ordering, it will hold for all orderings.

It is important to note that in our model, Assumptions 1 and 2 are placed ex-ante on the agent’s prior, and not ex-post on the realized function $f$. For example, the function $f(x_{1},\dots,x_{n})=x_{1}$, which says that the only covariate that matters is $x_{1}$, is strongly asymmetric ($x_{1}$ is differentiated from the other covariates) and also features a single “large” covariate (the realization of $x_{1}$ completely determines $y$). Our assumptions do not rule out the possibility of this function. Rather, they require that if this function is considered possible, then certain other functions are as well.¹⁸¹⁸18Assumption 1 requires that for every permutation $\pi:\{0,1\}^{n}\rightarrow\{0,1\}^{n}$, the function $g_{\pi}$ satisfying $g_{\pi}(x_{1},\dots,x_{n})=f(\pi(x_{1},\dots,x_{n}))$ is also in the support of the agent’s beliefs.

Simple examples of priors satisfying these two assumptions are given below.

Priors that violate these assumptions include the following.

We view these two assumptions as useful conceptual benchmarks, but neither is necessary for our subsequent results. In Section 5, we explore how far our main results generalize under different relaxations of Assumptions 1 and 2.

We now define the expected value of context from the point of view of an agent who knows his covariate vector $\mathbb{x}_{n}$ but does not know the function $f$ (and hence also does not know his type $y=f(\mathbb{x}_{n})$). As we show in Section 4.1, the assumption that the agent knows $\mathbb{x}_{n}$ is immaterial for the results.

We similarly compare evaluators based on the expected payoff that the agent receives.

These definitions correspond to a thought experiment in which (for example) a patient has a choice between being seen by a doctor or assessed by an algorithm. If the patient chooses the algorithm, his standard covariates and $\alpha_{b}\cdot n$ arbitrarily chosen nonstandard covariates will be sent to the algorithm. If the patient chooses the doctor, he will engage in a conversation with the doctor, where his standard covariates and $\alpha_{h}\cdot n$ selected nonstandard covariates will be revealed. Which should the patient choose?

The first part of Definition 2.3 compares the agent’s expected payoff under black box evaluation with the best-case expected payoff under human evaluation, namely when the human evaluator observes those (up to) $\alpha_{h}\cdot n$ covariates that maximize the agent’s payoffs. If the agent’s expected payoff is nevertheless higher under black box evaluation even after biasing the agent towards the human in this way, we say that the agent prefers to be evaluated by the black box. The second part of the definition compares the agent’s expected payoff under black box evaluation with the worst-case expected payoff under human evaluation, namely when the human evaluator observes those (up to) $\alpha_{h}\cdot n$ covariates that minimize the agent’s payoffs. If the agent’s expected payoff is lower under black box evaluation even after biasing the agent against the human in this way, then we say that the agent prefers to be evaluated by the human.²⁰²⁰20In Section 4.2 we further discuss the extent to which these interpretations are valid when the evaluator also updates her beliefs to the selection of covariates.

These are clearly very conservative criteria for what it means to prefer the human or the black box. In practice, we would expect the set of revealed covariates $H$ to be intermediate to the two cases considered in Definition 2.3, i.e., that $H$ neither maximizes nor minimizes the agent’s payoffs.²¹²¹21Angelova et al. (2022) provide evidence that some judges condition on irrelevant defendant covariates when predicting misconduct rates. But if we can conclude either that the agent prefers the black box evaluator or the human evaluator according to Definition 2.3, then the same conclusion would hold for these more realistic models of $H$.

Suppose there are two covariates, $x_{1}$ and $x_{2}$, both nonstandard. For each covariate vector $\mathbb{x}\in\{0,1\}^{2}$, define the random variable $Y_{\mathbb{x}}=f(\mathbb{x})$, where the randomness is in the realization of $f$.

The agent has utility function $u(\hat{y},y)=\hat{y}$ and covariate vector $(1,1)$. Suppose Human observes up to one nonstandard covariate; then, there are three possibilities for what the evaluator observes. If Human observes $x_{1}=1$, her evaluation is

If Human observes $x_{2}=1$, her evaluation is

And if Human observes no nonstandard covariates, then her evaluation remains the unconditional average

So the expected value of context for this agent is

Suppose $n$ grows large with up to $h_{n}=\lfloor\frac{n}{2}\rfloor$ covariates observed. There are two opposing forces affecting the value of context. First, when $n$ is larger there are more distinct sets of covariates that can be revealed to Human, and hence the max in (3.1) is taken over a larger number of posterior expectations. This increases the value of context. On the other hand, each $Z_{k}$ is a sample average, and the number of elements in this sample average also grows in $n$.²²²²22For example, observing $X_{1}=1$ with $n=2$ gives the evaluator a posterior expectation of $(Y_{10}+Y_{11})/2$, while the same observation gives the evaluator a posterior expectation of $(Y_{100}+Y_{101}+Y_{110}+Y_{111})/2$ if $n=3$. By the law of large numbers, each $Z_{k}$ thus concentrates on its expectation (which is common across $k$) as $n$ grows large, so the difference between any $Z_{k}$ and $Z_{k^{\prime}}$ grows small. What we have to determine is whether the growth rate in the number of subsets of nonstandard covariates (of size $\leq h_{n}$) is sufficiently large such that the maximum difference in evaluations across these sets is nevertheless asymptotically bounded away from zero. The answer turns out to be no.

Our main result says that for every agent, the expected value of context (as defined in Definition 2.2) vanishes as $n$ grows large.

Thus although the value of context may be substantial for certain type functions (such as in Example 5), it does not matter on average across these functions when the agent’s prior satisfies Assumptions 1 and 2. This also implies that for sufficiently large $n$, the provision of context does not “typically” matter; that is, the probability that the agent gains substantially from targeted information acquisition is small.

The core of the proof of Theorem 3.1 is an argument that the extent to which context can change the evaluator’s posterior expectation vanishes in the number of covariates. We outline that argument here. For each $n$, there are $K_{n}=\sum_{j=0}^{\lfloor\alpha_{h}n\rfloor}\binom{n-s}{j}$ sets of $\alpha_{h}n$ (or fewer) nonstandard covariates that can be disclosed. We can enumerate and index these sets to $k=1,\dots,K_{n}$. Each set $k$ induces a posterior expectation $Z_{k}$ which is a sample average of random variables $Y_{x}\equiv f(x)$. The expected value of context (for this utility function) is

where $Z_{\varnothing}$ is Human’s posterior expectation given observation of standard covariates only. Normalizing $E[Z_{\varnothing}]=0$, it remains to study properties of the first-order statistic $\max_{1\leq k\leq K_{n}}Z_{k}$.

There are two challenges to analyzing this quantity. First, the correlation structure of $Z_{1},\dots,Z_{K_{n}}$ can be complex: The variables $Z_{k}$ are neither independent (because the same posterior expectation $Y_{x}$ can appear as an element in different sample averages $Z_{k},Z_{k^{\prime}}$) nor identically distributed (because the sample averages are of different sizes depending on how many nonstandard covariates are revealed). The second challenge is that the length of the sequence $(Z_{1},\dots,Z_{K_{n}})$ grows exponentially in $n$. Thus even though each term within the maximum eventually converges to a normally distributed random variable (with shrinking variance), the errors of each term may in principle accumulate in a way that the maximum grows large.

Our approach is to first construct new i.i.d. variables $Z^{iid}_{k}$, with the property that

Applying a result from Chernozhukov et al. (2013), we show that $\max_{1\leq k\leq K_{n}}Z_{k}^{iid}$ (properly normalized) converges to $\max_{1\leq k\leq K_{n}}Z_{k}^{Normal}$ in distribution, where (due to properties of our problem) $Z_{k}^{Normal}\sim_{iid}\mathcal{N}\left(0,\frac{1}{2^{n(1-\alpha_{h})-s}}\right)$. Having reduced the analysis to studying the expected maximum of i.i.d.  Gaussian variables, classic bounds apply to show that this quantity is no more than

This display quantifies the importance of each of the two forces discussed in Section 3.1. First, as $n$ grows larger, the number of posterior expectations $K_{n}=\sum_{j=0}^{\lfloor\alpha_{h}n\rfloor}\binom{n-s}{j}\leq 2^{n-s}$ grows exponentially in $n$, increasing the expected value of context. But second, as $n$ grows larger, each $Z_{k}$ concentrates on its expectation, where its variance, $\frac{1}{2^{n(1-\alpha_{h})-s}}$, decreases exponentially in $n$. What the bound in display (3.3) tells us is that the exponential growth in the number of variables is eventually dominated by the exponential reduction in the variance of each variable, yielding the result.

This proof sketch also clarifies the role of Assumption 1. As we show in Section 5.4, the statement of the theorem extends so long as the evaluator’s posterior expectation $Z_{k}$ concentrates on its expectation sufficiently quickly as $n$ grows large. Roughly speaking, this means that the informativeness of any specific set of covariates is decreasing in the total number of covariates. Thus the precise symmetry imposed by Assumption 1 is not critical for Theorem 3.1 to hold.

On the other hand, the conclusion of Theorem 3.1 can fail if the agent has substantial prior knowledge about how $y$ is related to the covariates.

In Section 5 we explore several relaxations of Assumptions 1 and 2. Our first relaxation of Assumption 1 supposes that there is a “low-dimensional” set of covariates that predictive of the agent’s type, while the remaining covariates are irrelevant. The second relaxation supposes that there is a subset of nonstandard covariates whose effects are known. We also consider a relaxation of Assumption 2 where the evaluator’s ability to predict $Y$ is increasing in the total number of covariates that define the type. We formalize these extensions of our main model and examine the extent to which Theorem 3.1 extends.

We now turn to the question of when the agent should prefer the human evaluator and when the agent should prefer the black box evaluator.

Roughly speaking, the numerator describes the sensitivity of the function $\phi$ to the evaluation, and the denominator describes the curvature of the function $\phi$. The assumption thus says that the curvature of the function must be sufficiently large relative to its slope. While there is no formal relationship, the LHS is evocative of the coefficient of absolute risk aversion of the function $\phi$.²⁴²⁴24Recall that the coefficient of absolute risk aversion of the function $\phi$ is $-\frac{\phi^{\prime}(\hat{y})}{\phi^{\prime\prime}(\hat{y}))}$.

The comparisons in this theorem apply for reasonably small $N$. For example, let $C=100$, in which case the restriction in Assumption 3 is quite weak. Figure 1 fixes $\alpha_{h}=0.1$ and plots $N$ for different values of $\alpha_{b}$. If (say), the human evaluator observes 10% of covariates while Black Box observes 90%, then the comparisons in Theorem 3.2 hold for all $n\geq 14$.

The case of convex $\phi$ (Part (a)) corresponds to a preference for more accurate evaluations.²⁵²⁵25Consider any two sets of covariates $A\subset A^{\prime}$ and let $\hat{y}_{A}$, $\hat{y}_{A^{\prime}}$ be the corresponding posterior expectations. The distribution of $\hat{y}_{A^{\prime}}$ (i.e., the posterior expectation that conditions on more information) is a mean-preserving spread of the distribution of $\hat{y}_{A}$. When $\phi$ is convex, the former leads to a higher expected utility. Such an agent “prefers more accurate evaluations” in the sense that giving the evaluator better information (in the standard Blackwell sense) leads to an improvement in the agent’s expected utility. Such an agent prefers for the evaluation to be based on more information (advantaging Black Box), but also prefers for the evaluation to be based on more relevant covariates (advantaging Human). We show that what eventually dominates is how many covariates the evaluators observe, not how they are selected; for an agent who prefers accuracy, this favors the Black Box.

Part (b) of Theorem 3.2 says that if instead $\phi$ is concave, then the agent should eventually prefer the human evaluator. We conclude this section with example decision problems that induce utility functions satisfying the conditions of either part of the theorem.

So far we have studied the the expected value of context for a single agent. If we instead ask whether the firm should use human or algorithmic evaluation—for example, whether a hospital should automate diagnoses or rely on doctor evaluations—other statistics may also be relevant. For example, it may matter whether the value of context is large for any agent in the population (e.g., because a lawsuit regarding algorithmic error may be brought on the basis of harm to any specific individual (Jha, 2020)). We thus study the expected maximum value of context, as defined below.

The following corollary says that this quantity also vanishes as $n$ grows large.

So far we’ve remained agnostic as to whether the agent or evaluator chooses which nonstandard covariates are revealed, assuming that in either case the evaluator updates as if the covariates were revealed exogenously. We now consider a more traditional disclosure game, in which the agent chooses which nonstandard covariates are revealed, and the human evaluator updates her beliefs about the agent’s type in part based on which covariates are chosen.

For any fixed function $f$, call the following an $f$-context disclosure game: There are two players, the agent and the evaluator. The function $f$ is common knowledge.²⁶²⁶26We do not interpret this assumption literally. At the other extreme where $f$ is unknown to the agent, there is no informational content in which covariates the agent chooses to reveal, as they are all symmetric from the agent’s point of view. The set of possible disclosures $\mathcal{D}$ is the set of all pairs $(H,(x_{i})_{i\in H})$ consisting of a set of nonstandard covariates $H\subseteq\mathcal{N}$ and values for those covariates. A disclosure $d=(H,(x^{\prime}_{i})_{i\in H})$ is feasible for an agent with covariate vector $(x_{1},\dots,x_{n})$ if the disclosed covariate values are truthful, i.e., $x_{i}=x_{i}^{\prime}$ for every $i\in H$.

The agent chooses a disclosure strategy, which is a map

from covariate vectors to feasible disclosures. The agent then privately observes his covariate vector $\mathbb{x}_{n}$ and discloses $\sigma(\mathbb{x}_{n})$. The evaluator observes this disclosure and chooses an action $\hat{y}$. That is, the evaluator’s strategy is a function $\sigma_{E}:\mathcal{D}\rightarrow[-\overline{y},\overline{y}]$. The evaluator’s payoff is $-(\hat{y}-y)^{2}$ and the agent’s payoff is some function $u(\hat{y})$.

In this section we focus on pure strategy Perfect Bayesian Nash equilibria (PBE) of this game, henceforth simply equilibria. (A similar result holds for mixed strategy equilibria, which is demonstrated in the appendix.)

We show that the best payoff that an agent can receive in any pure strategy $f$-context equilibrium is bounded above by the maximum value of context across agents.

Thus, applying Proposition 4.1 and Corollary 1, our previous results extend.

Under Assumption 1, it cannot be known ex-ante that some covariates are irrelevant for predicting the type. The assumption thus rules out settings such as the following.

If irrelevant covariates cannot be disclosed to the evaluator, then we return to our main model with a smaller $n$ and our previous results extend directly. The more novel case is the one in which it is known that $n-r_{n}$ covariates are irrelevant, but those covariates can still be disclosed to the evaluator (for example, because it is not commonly understood that they are irrelevant).²⁷²⁷27To see the difference, consider the case in which the agent simply wants the evaluator to hold a higher posterior expectation. The irrelevant covariates create noise, and for some realizations of $f$ it may be that disclosing an irrelevant covariate leads to a higher evaluation.

To model this, we suppose there is a sequence of sets of relevant covariates $(R_{1},R_{2},\dots)$ such that each $R_{n}$ includes the standard covariates in $\mathcal{S}$ and is of size $s+r_{n}$, where $r_{n}=\lfloor\alpha_{r}\cdot n\rfloor$ is the (known) number of relevant nonstandard covariates. We will moreover without loss index these sets so that the relevant covariates are $x_{s+1},x_{s+2},\dots,x_{s+r_{n}}$. The irrelevance of the remaining covariates is reflected in the following assumption, which says that, holding fixed the values of the relevant covariates, the values of the irrelevant covariates do not change the type.

We then modify Assumptions 1 and 2 to apply only to the relevant covariates.

Our main model is otherwise unchanged—in particular, we allow the agent to disclose any of the $n-s$ nonstandard covariates, including those which are irrelevant. We show that our previous results extend so long as $\alpha_{h}<\alpha_{r}$.

The case where $\alpha_{r}<\alpha_{h}$ (violating the assumption of the result) corresponds to a setting in which the number of relevant covariates is so small that the agent can disclose all of them. For example, if a job candidate is convinced that only 10 nonstandard covariates are actually relevant for predicting his on-the-job ability, and all of these nonstandard covariates can be shared during a job interview, then our main results do not extend and we should think of the value of context as being potentially large. On the other hand, if the set of relevant covariates are low-dimensional relative to the total number of covariates, but are still too numerous to be fully revealed, then our main results do extend.

This result suggests that whether human or black box evaluation is more appropriate should be determined in part based on whether the available signal is concentrated in a small number of covariates (favoring the human evaluator) or spread out across a large number of covariates (favoring the black box evaluator). The same application may transition between these regimes over time. For example, in a medical setting where black box diagnosis is highly accurate based on non-interpretable features of an image scan, it may not be possible to communicate sufficient information via any small number of covariates. But if the predictive features of the image are subsequently better understood and defined, then it may be that a small set of (newly defined) features does eventually capture all of the signal content, and can be fully disclosed in a conversation.

Another possibility is that the agent knows how certain nonstandard covariates are correlated with the type.

Specifically suppose there is a set $K\subseteq\{1,\dots,n\}$ of covariate indices whose effects are known. The set $K$ includes the standard covariates, but possibly also includes some nonstandard covariates. Without loss, we can index these $x_{1},\dots,x_{K}$. We weaken Assumption 1 to the following:

This assumption imposes exchangeability only over the nonstandard covariates whose effects are not ex-ante known. Clearly if $K$ is a strict superset of $\mathcal{S}$, then the expected value of context need not vanish under Assumptions 2 and 7. A simple example is the following.

But if we modify the definition in (2.2) to

with $K$ replacing $\mathcal{S}$, and again let $U^{f}_{\mathbb{x}_{n}}(A)=u\left(\hat{y}^{f}_{\mathbb{x}_{n}}(A),y\right)$, then the modified expected value of context

evaluates the value of context beyond those covariates with known effects. The same proof shows that this expected value of context vanishes to zero as $n$ grows large. That is, beyond the value of context that is already clear to the agent based on private knowledge about his nonstandard covariates, the agent does not expect substantial additional gain from the remaining covariates.

So far we have assumed that the predictability of $Y$ is constant in the number of covariates. This is not essential for our results. Suppose instead that

where $\varepsilon_{n}$ is a mean-zero random variable that is independent of the covariates $(x_{1},\dots,x_{n})$. This describes a setting in which the covariates are not sufficient to reveal the agent’s type, and there is a residual unknown.

Our previous results extend directly when the distribution of $\varepsilon_{n}$ is the same for all $n$. Another natural case is one in which $\text{Var}(\varepsilon_{n})$ decreases monotonically in $n$, with $\lim_{n\rightarrow\infty}\text{Var}(\varepsilon_{n})=0$; that is, in environments with a larger number of covariates, the agent’s type is more predictable. In Appendix B.5 we show that Theorem 3.1 directly extends. We also show that the comparisons in parts (a) and (b) of Theorem 3.2 hold for sufficiently large $n$, under the following assumption:

Loosely speaking, our comparisons in Theorem 3.2 continue to hold so long as the variance of $\varepsilon_{n}$ does not increase too fast in $n$.

In this final section, we provide an abstract condition on the evaluator’s learning environment, under which Theorem 3.1 extends.

For each $n$, let $\mathcal{D}_{n}$ denote the set of all disclosures respecting the human evaluator’s capacity constraint, i.e., all pairs $(H,(x_{i})_{i\in H})$ consisting of a set $H\subseteq\{s+1,\dots,n\}$ with $\lfloor\alpha_{h}\cdot n\rfloor$ or fewer nonstandard covariates, and values $(x_{i})_{i\in H}$ for those covariates. Further define $\mathcal{D}=\cup_{n\geq 1}\mathcal{D}_{n}$ to be the set of all disclosures. Similarly, for each $n$ let $\mathcal{F}_{n}$ be the set of all type functions $f:\{0,1\}^{n}\rightarrow[-\overline{y},\overline{y}]$, and define $\mathcal{F}=\cup_{n\geq 1}\mathcal{F}_{n}$. An evaluation rule is any family $\rho=(\rho_{f})_{f\in\mathcal{F}}$ where each $\rho_{f}:\mathcal{D}\rightarrow[-\overline{y},\overline{y}]$ maps disclosures into evaluations for the given function $f$. Finally, fixing any update rule $\rho$, number of covariates $n$, and disclosure $d\in\mathcal{D}_{n}$, let

be the random evaluation when $f$ is drawn from $\mathcal{F}_{n}$ according to the agent’s prior.

We impose two assumptions below on the evaluation rule. The first says that the expected evaluation $Z_{d}^{n}$ is equal to the prior expected type $\mu\equiv\mathbb{E}[Y]$; the second says that the distribution of the evaluation concentrates on $\mu$ sufficiently fast as the number of hidden covariates $n$ grows large. Intuitively, the assumption requires that as the number of residual unknowns—i.e., the covariates which are predictive of the type, but are not revealed to the evaluator—grows large, the informativeness of any fixed disclosure becomes small.²⁸²⁸28In the limit with an uninformative disclosure, the distribution of the evaluation is degenerate at the prior expectation $\mu$ for any Bayesian updating rule.

These assumptions do not in general represent a weakening of our main model. Previously we studied the evaluation rule $\rho$ mapping each disclosure into the conditional expectation of the agent’s type, and imposed Assumption 1 on the agent’s prior about $f$. In this model, the evaluation $Z_{d}^{n}$ for any disclosure $d=(H,(x_{i})_{i\in H})$ could be represented as a sample average consisting of $2^{n-s-|H|}$ elements. Assumption 9 is clearly satisfied (because the update rule is Bayesian), but one can select a sequence of disclosures $(d_{n})$ such that $Var(Z_{d_{n}}^{n})=\frac{1}{2^{n(1-\alpha_{h})-s}}$ (see the proof of Theorem 3.1 for details). Thus the speed of convergence demanded in Assumption 10 is not met when $\alpha_{h}$ is sufficiently large.

Nevertheless, Assumption 10 identifies the qualitative property of our main setting that gave us Theorem 3.1: residual uncertainty must have the power to overwhelm any information revealed through disclosure. Under these assumptions, our main result extends.

This result also clarifies that neither the precise symmetry imposed by Assumption 1, nor the assumption of Bayesian updating in our main model, are crucial for our main result.