The Information Environment

We formalize this information structure as follows.

State of the World. Let the state of the world be $$ \upomega \in \Omega \equiv \left\{0,1\right\} $$. The state of the world encodes the decision-relevant information for the DM (e.g., $$ \upomega =1 $$ if the chemical in question is harmful to humans and $$ \upomega =0 $$ if not). It is unknown to the DM at the outset, which is why she consults an expert.

Let $$ {p}_1 $$ represent the common knowledge probability that the state is 1, so it is equal to 0 with probability $$ 1-{p}_1 $$. To reduce the cases to consider, we also assume that $$ \upomega =1 $$ is the ex ante more likely state, so $$ {p}_1\ge 1\divslash 2 $$.Footnote ⁶

Expert Types. The expert has a type $$ \uptheta \in \Theta \equiv \left\{g,b\right\} $$, which indicates whether he is good (able to digest the relevant scientific literature) or bad (not able to digest the scientific literature). For linguistic variety, we often call good experts “competent” and bad experts “incompetent”. Let $$ {p}_g\in \left(0,1\right) $$ be the probability than an expert is good, and so $$ 1-{p}_g $$ represents the probability of a bad expert. This probability is common knowledge, and the expert knows his type.

Question Difficulty. The difficulty of the question is captured by another random variable, $$ \updelta \in \Delta \equiv \left\{e,h\right\} $$. That is, the question may be easy (the scientific literature is informative about whether the chemical is harmful), or hard, (the best research does not indicate whether the chemical is harmful). Let $$ {p}_e $$ be the common knowledge probability that $$ \updelta =e $$, so $$ \updelta =h $$ with probability $$ 1-{p}_e $$. The difficulty of the question is not directly revealed to either actor at the outset.

Expert Signal. The expert’s type and the question difficulty determine what he learns about the state of the world. In our main analysis, we assume that the expert receives a completely informative signal if and only if he is good and the question is easy. If not, he learns nothing about the state. (In section 3 of the Supplemental Information, we analyze a more general information structure, which only assumes that a signal is more likely to be informative when the expert is good and the question is easy). Formally, let the signal be

(1)$$ s=\left\{\begin{array}{cc}{s}_1& \upomega =1,\uptheta =g\ \mathrm{and}\ \updelta =e\\ {}{s}_0& \upomega =0,\uptheta =g\ \mathrm{and}\ \updelta =e\\ {}{s}_{\varnothing }& \kern-7em \mathrm{otherwise}.\end{array}\right.\kern1.00em $$

In what follows, we will often refer to an expert who observes $$ {s}_0 $$ or $$ {s}_1 $$ as informed and an expert who observes $$ {s}_{\varnothing } $$ as uninformed. Importantly, this distinction is not the same as good and bad. If an expert is informed he must be good because bad experts always observe $$ {s}_{\varnothing } $$. However, an uninformed expert may be good (if $$ \updelta =h $$) or bad.

An important implication of our signal structure is that the good expert infers $$ \updelta $$ from his signal because he observes $$ {s}_0 $$ or $$ {s}_1 $$ when $$ \updelta =e $$ and $$ {s}_{\varnothing } $$ if $$ \updelta =h $$. More concretely, a competent expert can review the relevant literature and always determine whether the harmfulness of a chemical is known. On the other hand, bad experts, who always observe $$ {s}_{\varnothing } $$, learn nothing about the question difficulty.

Sequence of Play and Payoffs

The game proceeds in the following sequence: first, Nature picks the random variables (ω, θ, δ), according to independent binary draws with the probabilities $$ {p}_1 $$, $$ {p}_g $$, and $$ {p}_e $$, specified above. Second, the expert observes his competence and signal and chooses a message from a infinite message space $$ \mathbf{\mathcal{M}} $$. The information sets of the expert are summarized in Figure 1. There are four: first, the expert may be bad; second, the expert may be good and the question hard; third, the expert may be good, the question easy, and the state 0; and finally, the expert may be good, the question easy, and the state 1.

FIGURE 1.

Nature’s Play and Experts’ Information Sets

Next, the DM observes m and takes an action $$ a\in \left[0,1\right] $$, the policy choice. Her information set in this stage consists only of the expert report; that is, $$ {\mathbf{\mathcal{I}}}_{DM1}=(m) $$.

In the running example, we can interpret a as the stringency of regulations applied to the chemical in question (restrictions on emissions, taxes, resources to spend on enforcement). To map cleanly to the formalization, $$ a=0 $$ corresponds to the optimal policy if the chemical is not harmful. Policy $$ a=1 $$ corresponds to the optimal level of regulation if the chemical is harmful.

Formally, let $$ v\left(a,\upomega \right) $$ be the value of choosing policy a in state $$ \upomega $$. We assume the policy value is given by $$ v\left(\mathrm{a,}\upomega \right)\equiv 1-{\left(a-\upomega \right)}^2 $$. If taking a decisive action of $$ a=0 $$ or $$ a=1 $$, the value of the policy is equal to 1 for making the correct choice ($$ a=\upomega $$) and 0 for making the wrong choice ($$ a=1-\upomega $$). Taking an interior action ($$ 0<a<1 $$) gives an intermediate policy value, where the v function implies an increasing marginal cost the further the action is from the true state. This function is common knowledge, but since the DM may be uncertain about $$ \upomega $$, he may be uncertain about how the policy will turn out and hence the ideal policy. Let $$ {\uppi}_1=\mathrm{\mathbb{P}}\left(\upomega =1|{\mathbf{\mathcal{I}}}_{DM1}\right) $$ denote the DM’s belief that $$ \upomega =1 $$ when he sets the policy. Then, the expected value of taking action a is

(2)$$ {\displaystyle \begin{array}{ccc}1-\left[{\uppi}_1{\left(1-a\right)}^2+\left(1-{\uppi}_1\right){a}^2\right],& & \end{array}} $$

which is maximized at $$ a={\uppi}_1 $$.

The quadratic loss formulation conveniently captures the realistic notion that when the expert does not learn the state, the decision maker makes a better policy choice (on average) when learning this rather than being misled into thinking the state is zero or one. That is, in our formalization it is best to pick an intermediate level of regulation when it is unclear whether the chemical is harmful (e.g., modest emissions restrictions, labeling requirements). Formally, if the question is unsolvable, the optimal action is $$ a={p}_1 $$, giving an average payoff of $$ 1-{p}_1\left(1-{p}_1\right) $$, which is strictly higher than the average value of the policy for any other action.

After the policy choice is made, the DM may receive some additional information (validation), and she then forms an inference about the expert competence $$ {\uppi}_g\equiv \mathbb{P}\left(\theta =g|{\mathbf{\mathcal{I}}}_{DM2}\right) $$. Different validation regimes (described below) affect what information the DM has at this stage, $$ {\mathbf{\mathcal{I}}}_{DM2} $$.

The decision maker only cares about the quality of the policy:

(3)$$ {\displaystyle \begin{array}{ccc}{u}_{DM}=v\left(\mathrm{a,}\upomega \right).& & \end{array}} $$

So, as derived above, the DM will always pick a policy equal to her posterior belief that the state is 1, $$ {\uppi}_1 $$.

The expert cares about the belief that he is competent ($$ {\uppi}_g $$) and potentially also about the quality of the policy choice. We parameterize his degree of policy concerns by $$ \upgamma \ge 0 $$ and write his payoff

(4)$$ {\displaystyle \begin{array}{ccc}{u}_E={\uppi}_{\mathrm{g}}+\upgamma v\left(a,\upomega \right).& & \end{array}} $$

We first consider the case where $$ \upgamma =0 $$; that is, the expert only cares about his reputation. We then analyze the case where $$ \upgamma >0 $$, focusing attention on the case where policy concerns are small ($$ \upgamma \to 0 $$) in the main text.

Validation

Finally, we formalize how different kinds of ex post validation affect what the DM knows ($$ {\mathbf{\mathcal{I}}}_{DM2} $$) when forming a belief about the expert competence. For our regulation example, there are several ways the DM might later learn things about the state or difficulty of the question.

First, there may be studies in progress which will later clearly indicate whether the chemical is harmful. Alternatively, if the question is whether the chemical has medium or long-term health consequences, this may not become clear until after the initial regulation decisions are made. To pick a prominent contemporary example, the degree to which we should regulate “vaping” of nicotine and cannabis depends on the long-term health consequences of this relatively new technology—perhaps relative to smoking—which are not currently known.

Second, the DM could consult other experts (who may themselves have strong or weak career concerns). These other experts could be asked about the state of the world or perhaps the quality of the evidence on dimensions the DM may not be able to assess (Anecdotal evidence or scientific? Randomized control trials or observational studies? Have they been replicated? Human or animal subjects?).

In other contexts, validation about the state of the world could naturally arise if it is about an event that will later be realized (the winner of an election, the direction of a stock’s movement) but where the DM must make a choice before this realization. Alternatively, in many policy or business settings, the decision maker may be able to validate the expert’s message directly, whether through experimental A/B testing or observational program evaluation. Closer to difficulty validation is the familiar notion of “peer review,” whereby other experts evaluate the feasibility of an expert’s design without attempting the question themselves. Another possibility is when the decision maker has expertise (substantive or methodological) in the general domain of the question but does not have the time to investigate herself, so she may be able to validate whether the question was answerable only after seeing what the expert comes up with.

Alternatively, subsequent events may reveal auxiliary information about whether the state should have been knowable, such as an extremely close election swayed by factors which should not have been ex ante predictable (i.e., this reveals that forecasting the winner of the election was a difficult question).

Formally, we consider several variations on $$ {\mathbf{\mathcal{I}}}_{DM2} $$. In all cases, the structure of $$ {\mathbf{\mathcal{I}}}_{DM2} $$ is common knowledge.

In the no validation case, $$ {\mathbf{\mathcal{I}}}_{DM2}=(m) $$. This is meant to reflect scenarios where it is difficult or impossible to know the counterfactual outcome had the decision maker acted differently. The state validation case, $$ {\mathbf{\mathcal{I}}}_{DM2}=\left(m,\upomega \right) $$, reflects a scenario in which the DM can check the expert’s advice against the true state of the world. In the difficulty validation case, $$ {\mathbf{\mathcal{I}}}_{DM2}=\left(m,\updelta \right) $$, meaning that the DM learns whether the question was hard (i.e., whether the answer could have been learned by a good expert). In the full validation case, $$ {\mathbf{\mathcal{I}}}_{DM2}=\left(\mathrm{m,}\upomega, \updelta \right) $$, the DM learns both the state and the difficulty of the question.

To be clear, many of the motivating examples blend validation about the state and difficulty. We consider the cases of “pure” difficulty or state validation to highlight what aspects of checking expert claims are most important for getting them to admit uncertainty.

Markov Sequential Equilibrium (MSE)

The Markov restriction in repeated games requires that if two histories of play ($$ {h}_1 $$ and $$ {h}_2 $$) result in a strategically equivalent scenario starting at both histories (meaning that the players have the same expected utilities over the actions taken starting at both $$ {h}_1 $$ and $$ {h}_2 $$), then the players must use the same strategies starting in both histories. The analogous restriction in our setting has implications for strategies and beliefs.

In terms of strategies, we require that if two types of expert face a strategically equivalent scenario, they must play the same strategy. Most important to our model, in some cases a competent but uninformed expert and an incompetent expert have the exact same expected utility (for any DM strategy). Perfect Bayesian equilibrium would allow these two types to play different strategies, despite the fact that they face identical incentives; the Markov restriction requires them to play the same strategy. Our restriction to beliefs essentially assumes that, even when observing an off-path message, the DM still believes that the expert plays some Markov strategy.

Formally, let $$ {\upsigma}_{\uptheta, s}(m) $$ be the probability of sending message m as a function of the sender type, $$ {\uppi}_1(m) $$ be the posterior belief that the state is 1 given message m, $$ {\uppi}_g\left(m;{\mathbf{\mathcal{I}}}_{DM2}\right) $$ be the posterior belief about the expert’s competence, and $$ a(m) $$ be the policy action given message m. Let $$ {U}_E $$ and $$ {U}_{DM} $$ be the expected utilities for each player.

Perfect Bayesian equilibrium requires that both players maximize their utility given the other’s strategy and their beliefs and that these beliefs are formed by Bayes’ rule when possible. To these we add two requirements:

The key implication of Markov consistency is to rule out off-path inferences about payoff-irrelevant information, because off-path beliefs that condition on payoff-irrelevant information cannot be reached by a sequence of Markov strategies. Our restriction is related to that implied by D1 and the intuitive criterion (Cho and Kreps Reference Cho and Kreps1987). However, where these refinements require players to make inferences about off-path play in the presence of strict differences of incentives between types, our restriction rules out inference about types in the absence of strict differences of incentives.

We use this solution concept for two reasons. The first is theoretical. People have endless “private information” that they could, in principle, condition their behavior on. In a more complex but realistic version of our model, the expert may have private information about not only what he learns about the chemical in question but also about his personal policy preferences, views on what kinds of scientific evidence are credible, and what he had for breakfast in the morning. When observing an off-path message, PBE would allow for updating (or not) on any of these dimensions regardless of their relevance to the interaction at hand. The Markov strategies restriction provides a principled and precise justification for which kinds of private information experts can condition their strategies onFootnote ⁷ (Beliefs about the effects of the chemical in question? Usually. What evidence is credible? Sometimes. Breakfast? RarelyFootnote ⁸). The Markov beliefs restriction is then a logical implication of what observers can update on when observing off-path messages.

The second reason is more practical: our main result about the importance of difficulty validation for getting experts to admit uncertainty is nearly immediate when using this restriction. As demonstrated in Appendix A, similar results hold when analyzing PBE, but since the set of PBE is much larger, the comparisons are not as clean.

Properties of Equilibria

Since we allow for a generic message space, there will always be many equilibria to the model even with the Markov restrictions. To organize the discussion, we will focus on how much information can be conveyed (about $$ \upomega $$ and $$ \uptheta $$). On one extreme, we have babbling equilibria, in which all types employ the same strategy, and the DM learns nothing.

On the other extreme, there is never equilibrium with full separation of types. To see why, suppose there is a message that is only sent by the good but uninformed types $$ {m}_{g,\varnothing } $$ (“I don’t know because the optimal policy isn’t clear”) and a different message only sent by the bad uninformed types $$ {m}_{b,\varnothing } $$ (“I don’t know because I am incompetent”). If so, the policy choice upon observing these messages would be the same. However, the reputation payoff for sending $$ {m}_{g,\varnothing } $$ is strictly higher, so the bad types have an incentive to deviate.

Still, such full separation is not required for the expert to communicate what he knows about the state. That is, there may still be equilibria where the uninformed types say “I don’t know” (if not why), and the informed types report the state of the world. We call these honest equilibria. In the main text, we focus on the simplest version of an honest equilibrium, where there is a unique message $$ {m}_x $$ sent by each type observing $$ {s}_x $$ with probability 1, $$ x\in \left\{0,,,1,,,\varnothing \right\} $$; see Appendix B for a more general definition. This is a particularly important class of equilibria in our model because it conveys the most information about the state:

This result formalizes our intuition that it is valuable for the DM to learn when the expert is uninformed, and it follows directly from the convexity of the loss function for bad policies. For example, if the expert on environmental policy sometimes says that a chemical is harmful (or is not harmful) when the truth is that he isn’t sure, the regulator will never be entirely sure what the optimal policy is. Her best response to this garbled message is to not regulate as aggressively as she would if knowing the chemical is harmful for sure, even when the expert fully knows that this is the case.

As we will see, honest equilibria tend to fail when one of the uninformed types would prefer to “guess,” mimicking the message of the informed types. So, the other equilibria we consider in the main text are ones where the informed types still send their respective messages $$ {m}_0 $$ and $$ {m}_1 $$, but the uninformed types at least sometimes send one of these messages. We refer to sending one of these messages as “guessing,” and sending $$ {m}_{\varnothing } $$ as “admitting uncertainty.” See Appendix B for a definition of what it means to admit uncertainty with more general messaging strategies and section 2 of the Supplemental Information for an extensive discussion of why focusing on this class of messaging strategies sacrifices no meaningful generality for our results.

Combined with proposition 1, these definitions highlight why admission of uncertainty is important for good decision-making. In any equilibrium with guessing, the fact that the uninformed types send messages associated with informed types leads to worse policies than in an honest equilibrium. This is for the two reasons highlighted in our opening paragraph. First, when the expert actually is informed, his advice will be partially discounted by the fact that the DM knows some uninformed types claim to know which policy is best. That is, decision makers may ignore good advice. Second, when the expert is uninformed, he will induce the DM to take more decisive action than the expert’s knowledge warrants; that is, decision makers may take bad advice.

Equilibrium

We continue to study the most straightforward class of messaging strategies where the informed types send one message each, to which we give the natural labels $$ {m}_0 $$ and $$ {m}_1 $$, and the uninformed types either send one of these messages or a third message labeled $$ {m}_{\varnothing } $$ (“I Don’t Know”).Footnote ¹⁰

In an honest equilibrium, messages $$ {m}_0 $$ and $$ {m}_1 $$ are only sent by informed types. So, when observing these messages, the DM picks actions $$ a=0 $$ and $$ a=1 $$, respectively. At the evaluation stage, when observing $$ \left({m}_0,e\right) $$ or $$ \left({m}_1,e\right) $$, the DM knows the expert is competent with certainty.

Upon observing $$ {m}_{\varnothing } $$, the DM picks policy $$ {p}_1 $$. The competence assessment when the expert says “I don’t know” depends on the result of the validation. When the problem is easy, the DM knows that the expert is incompetent because this fact means that a competent expert would have learned whether the chemical is harmful and sent an informative message. So, $$ {\uppi}_g\left({m}_{\varnothing },e\right)=0 $$. Upon observing $$ \left({m}_{\varnothing },h\right) $$, the DM learns nothing about the expert competence because no expert gets an informative signal when the scientific literature is uninformative.

Combining these observations, the payoff to the good but uninformed type for sending $$ {m}_{\varnothing } $$ (who knows the validation will reveal $$ \updelta =h $$) is

$$ {\displaystyle \begin{array}{ccc}{p}_g+\upgamma \left[1-{p}_1\left(1-{p}_1\right)\right].& & \end{array}} $$

The bad uninformed type does not know if the validation will reveal the problem is hard and so receives a lower expected competence evaluation and hence payoff for sending $$ {m}_{\varnothing } $$:

$$ {\displaystyle \begin{array}{ccc}\left(1-{p}_e\right){p}_g+\upgamma \left[1-{p}_1\left(1-{p}_1\right)\right].& & \end{array}} $$

Since no types are payoff equivalent, the Markov consistency requirement places no restrictions on off-path competence evaluations, and we can set these to zero.Footnote ¹¹ In the case with only difficulty validation, these are the information sets $$ \left({m}_0,,,h\right) $$ and $$ \left({m}_1,,,h\right) $$—getting caught guessing when validation reveals that the scientific literature is not informative. Importantly, the expert is not caught because the DM realizes his claim is wrong but because she comes to learn that the question was not answerable. As formalized below, such an off-path belief is also “reasonable” in the sense that the bad uninformed types face a strictly stronger incentive to guess than the good uninformed types do.

If the DM believes that the expert is bad with probability one upon observing $$ \left({m}_0,,,h\right) $$ or $$ \left({m}_1,,,h\right) $$, then a good but uninformed type knows he will get a reputation payoff of zero if sending either of these messages. Further, if claiming he knows whether the chemical is harmful, the policy is worse as well, so he has no incentive to deviate. A bad type guessing that the chemical is harmful ($$ {m}_1 $$) gets the expected payoff:

$$ {\displaystyle \begin{array}{ccc}{p}_e+\gamma {p}_{1,}& & \end{array}} $$

which is strictly higher than the payoff for sending $$ {m}_0 $$. So, the constraint for an honest equilibrium is

$$ {\displaystyle \begin{array}{ccc}\left(1-{p}_e\right){p}_g+\upgamma \left[1-{p}_1\left(1-{p}_1\right)\right]\ge {p}_e+\upgamma {p}_1& & \\ {}\upgamma \ge \frac{p_e\left(1+{p}_g\right)-{p}_g}{{\left(1-{p}_1\right)}^2}.& & \end{array}} $$

Not surprisingly, when policy concerns are very strong, uninformed experts will admit uncertainty because it leads to better policy choices. In fact, this holds for any validation regime; see Section 1 of the Supplemental Information for a comparison of “how strong” policy concerns must be to induce honesty for each case. However, this analysis also highlights that when policy concerns are small relative to reputation concerns, there is never an honest equilibrium with no validation or just state validation. In contrast, with just difficulty validation,Footnote ¹² as $$ \upgamma \to 0 $$ there is an honest equilibrium when

(5)$$ {\displaystyle \begin{array}{ccc}{p}_e\le \frac{p_g}{1+{p}_g}.& & \end{array}} $$

This inequality implies that an honest equilibrium is easy to sustain when $$ {p}_e $$ is low and $$ {p}_g $$ is high. The former holds for two reasons: a prior belief that the literature is likely not informative about the chemical means that uninformed experts are likely to be competent, and the uninformed expert is more likely to be “caught” claiming to have an answer to an impossible problem. A situation with more competent experts (high $$ {p}_g $$) makes honesty easier because it means the uninformed types are frequently competent, making admitting uncertainty look less bad.

What if Equation 5 does not hold? Of course, there is always a babbling equilibrium, and there is also an always guessing equilibrium where the good and bad uninformed types always send $$ {m}_0 $$ or $$ {m}_1 $$. More interesting for our purposes, there can also be an MSE where all of the good types report honestly and the bad types play a mixed strategy over $$ \left({m}_0,,,{m}_1,,,{m}_{\varnothing}\right) $$.Footnote ¹³ There is a more subtle incentive compatibility constraint that must be met for this equilibrium to hold: if the bad types are indifferent between sending $$ {m}_0 $$ and $$ {m}_1 $$, it can be the case that the informed types prefer to deviate to sending the other informed messages (i.e., the $$ {s}_1 $$ type prefers to send $$ {m}_0 $$) when policy concerns get very small. See the proof of proposition S.5 in section 1 of the Supplemental Information for details. In short, if the probability of a solvable problem is not too low or the probability of the state being 1 is not too high, then this constraint is not violated and there is an MSE where all of the good types send their honest message.Footnote ¹⁴

Proposition 4. As $$ \upgamma \to 0 $$ with difficulty validation, there is an honest MSE if and only if $$ {p}_e\le \frac{p_g}{1+{p}_g} $$. If not, and $$ {p}_e\ge 2{p}_1-1 $$, then there is an MSE where the good types send their honest message and the bad types use the following strategy:

(6)$$ {\displaystyle \begin{array}{ccc}{\upsigma}_b^{\ast}\left({m}_{\varnothing}\right)=\left\{\begin{array}{cc}\frac{1-{p}_e\left(1+{p}_g\right)}{1-{p}_g}& {p}_e\in \left(\frac{p_g}{1+{p}_g},\frac{1}{1+{p}_g}\right)\\ {}0& {p}_e\ge \frac{1}{1+{p}_g},\end{array}\right.,& & \\ {}{\upsigma}_b^{\ast}\left({m}_0\right)=\left(1-{p}_1\right)\left[1-{\upsigma}_b^{\ast}\left({m}_{\varnothing}\right)\right],\hfill & & \\ {}{\upsigma}_b^{\ast}\left({m}_1\right)={p}_1\left[1-{\upsigma}_b^{\ast}\left({m}_{\varnothing}\right)\right].\hfill & & \end{array}} $$

In sum, other than in a restrictive corner of the parameter space,Footnote ¹⁵ small policy concerns and difficulty validation are sufficient to induce at least good and uninformed experts to admit uncertainty—and potentially full honesty among all experts.

When Do We Observe Admission of Uncertainty, and from Whom?

Figure 4 illustrates the equilibrium described by proposition 4, which helps provide some more concrete insights into when we should expect to see admission of uncertainty and good decisions. All claims that follow are comparisons within this equilibrium, though again this is the only MSE where the good types report honestly (see footnote 13).

FIGURE 4.

Illustration of Expert Strategies in the Equilibrium Identified by Proposition 4, as a Function of $$ {p}_e $$ and $$ {p}_g $$, with $$ {p}_1=1\divslash 2 $$.

In the bottom right corner (most experts are competent, most problems are hard), there is an honest equilibrium. Substantively, this corner of the parameter space plausibly corresponds to environments where the best experts are tackling questions on the research frontier, be it at conferences or in the pages of academic journals. In general, the admission of uncertainty is quite frequent in these settings: even good experts often don’t know the answer but feel comfortable admitting this, and as a result bad experts can pool with them and admit uncertainty too. So, while decision makers may not always get useful advice here—recall, most questions have no good answer—they will at least have no problems getting experts to honestly report what they know.

In the top right corner, most experts are competent and most problems are easy. We can think of this part of the parameter space as “business as usual” in bureaucracies, companies, or other organizations where qualified experts are addressing relatively mundane problems that they generally know how to solve. In this case, there is little admission of uncertainty, both because experts typically get informative signals and because admitting uncertainty in this setting is too risky for bad experts to ever do it. While experts have the most information in this region, this comes at a cost from the perspective of the decision maker, as bad types always guess, which dilutes the value of the informative messages.

Comparing across these two cases also hints at the question of when good or bad experts are more likely to admit uncertainty. Our model contains a straightforward assumption about which experts are likely to be uncertain about the state of the world: good experts sometimes and bad experts always. And, in an honest equilibrium—when $$ {p}_e<{p}_g\divslash (1+{p}_g) $$—this is what they will report, so bad experts admit uncertainty more often.

However, on the other extreme, when $$ {p}_e>1\divslash (1+{p}_g) $$, it is only the good experts who will ever admit uncertainty. Put another way, if an outside observer (who the expert does not care to impress) were to see an expert admit uncertainty in this part of the parameter space (and in this equilibrium), she would know for sure that anyone who says “I don’t know” is in fact competent.

More generally, good experts admit uncertainty whenever the problem is hard, with probability $$ 1-{p}_e $$. For parameters where good experts sometimes admit uncertainty (the triangle in the left of Figure 4), bad experts send $$ {m}_{\varnothing } $$ with probability $$ \frac{1-{p}_e\left(1+{p}_g\right)}{1-{p}_g} $$, which is less than $$ 1-{p}_e $$ if and only if $$ {p}_e>1\divslash 2 $$. So, in domains of difficult problems, bad experts are more likely to admit uncertainty, and in domains with easier questions good experts are more likely to admit uncertainty.

Comparative Statics

We now ask how changing the probability parameters of the model affects the communication of uncertainty by uninformed experts in the MSE identified by proposition 4.

In general, one might expect that adding more competent experts will lead to less admission of uncertainty and better decisions, and that making problems easier will also lead to less admission of uncertainty and better decisions. Both of these always hold within an honest equilibrium, but they can break down elsewhere. Here we highlight cases where these intuitive results do not hold, and we then summarize with a complete description of the comparative statics.

More Competent Experts. Adding more competent experts always leads to better decisions (on average), but there are two scenarios where adding more competent experts (increasing $$ {p}_g $$) can lead to more admission of uncertainty.

First, in the region where the bad types always guess, the good types are the only experts who send $$ {m}_{\varnothing } $$, so increasing $$ {p}_g $$ leads to more admission of uncertainty.

More subtly, when the bad types sometimes send $$ {m}_{\varnothing } $$ and $$ {p}_e<1\divslash 2 $$ (the bottom left triangle), adding more competent experts leads the bad types to admit uncertainty more often. And since this is the part of the parameter space where good types usually admit uncertainty as well, adding more competent types leads to more admission of uncertainty overall:

More Easy Questions. When the probability of an easy problem increases, this always decreases the admission of uncertainty because good types are more likely to be informed and bad types are more apt to guess. The potentially counterintuitive result here is that more easy problems can sometimes lead to worse decisions. This is possible because when problems are more likely to be easy, the bad types are more tempted to guess. If the bad types never actually guess (bottom right corner) or always guess (top right corner), this does not matter. However, when making problems easier actually makes the bad types guess more, the messages $$ {m}_0 $$ and $$ {m}_1 $$ become less informative. As shown in the following proposition, this can sometimes lead to worse decisions: