A mathematical model involving a decision maker and an expert is investigated. Through analyzing the model, we obtain several results on the expert’s information acquisition and disclosure strategy. When withholding information is costly to the expert, in equilibrium, an expert with a higher withholding cost acquires less information but discloses more acquired information. We also examine which expert is optimal to the decision maker among a group of experts with different costs of withholding information.

In many situations, a decision maker needs to make a decision but does not know which decision is the best one. For example, an investor may not know the future performance of companies in the stock market and is uncertain about which stock is the best. A patient does not have knowledge about the most cost-effective treatment for the health concern. Facing lack of information, the decision maker often looks for advice from investment consultants or doctors, who are experts and are better informed.

However, the interest of the expert is often different from that of the decision maker. In other words, the decision preferred by the expert is different from the one preferred by the decision maker. In our previous examples, an investment consultant may not be interested in picking the best stock. Instead, the consultant may want the investor to buy some particular stocks, perhaps because these stock companies will pay the consultant. Doctors often receive bonuses from expensive medical procedures, which might be unnecessary for patients with mild diseases. Due to this conflict of interests between experts and decision makers, experts often withhold information from decision makers.

We analyze a game between a decision maker (hereafter DM) and an expert (we use “she” to denote DM and “he” to denote the expert throughout this paper). DM needs to take a decision. The best decision of DM depends on an uncertain state of the world (hereafter the state). DM has no information about the state but the expert, through exerting costly effort to acquire information, can observe the state with some probability. The probability of observing the state is increasing in his effort. DM and the expert have different preferences. DM’s best decision is equal to the state but the expert prefers a decision strictly higher than the state. Upon observing the state, the expert can disclose the observation to DM or withhold the information. We analyze the perfect Bayesian Nash equilibrium of the game between DM and the expert. In equilibrium, the expert decides how much effort to spend in information acquisition and what range of acquired information to disclose to DM, and DM chooses which decision to be taken when the expert does not disclose information.

We consider a situation where withholding information is costly to the expert. This is because the expert may feel guilty if he withholds information from DM. The cost of withholding information reflects the loss in the expert’s utility caused by guilty feelings. An alternative explanation for the cost is that DM may discover the withholding behavior of the expert. As a result, DM will doubt the trustworthy of the expert and refuse seeking advice from the expert in the future. The cost of withholding information to the expert reflects the expert’s loss in his future revenue due to losing clients. Of course, different experts have different costs of withholding information. This is because different experts have different moral standards or have different concerns for future revenues.

We study how the expert’s equilibrium information acquisition and disclosure strategy changes as the expert’s withholding cost increases. We also study how the expected utility of DM changes. These questions are investigated under two different utility functions of the expert. Under both utility functions, the expert exerts less effort in acquiring information and discloses a greater proportion of collected information as withholding cost increases. If the expert prefers the decision to be as high as possible regardless of the state, the expert’s effort in acquiring information quickly decreases as withholding cost increases. As a result, the likelihood that the expert can obtain information quickly decreases. Although the expert discloses a greater proportion of acquired information to DM, DM gets less information and gets worse off. Alternatively, if the ideal decision of the expert is higher than that of DM by a constant distance, the expert’s effort in information acquisition slowly decreases as withholding cost increases. DM receives more information and gets better off.

This paper is related to a large literature on the strategic transmission of verifiable information between a decision maker and an expert. See Grossman [

It is assumed in Shin [

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The paper is organized as follows. Section

A decision maker needs to make a decision, modeled as a choice of a real number in the interval

Apparently, DM’s utility is higher when

However, DM has no information about

Initially, the expert does not have any information about

The expert’s preference is different from that of DM. In the benchmark case, we assume that when the decision taken by DM is

The expert’s utility only depends on the level of DM’s decision and is independent of the state. Therefore, the expert is not concerned about whether the decision is appropriate for the state or not. Instead, the expert only cares about the

When the expert observes

When the expert reports “

The equilibrium concept is perfect Bayesian Nash equilibrium (hereafter “the equilibrium”). It consists of three components: the expert’s strategy in how much effort to exert in acquiring information about the state and which states to be disclosed to DM after observing the state; DM’s strategy in what decision to be taken when the expert does not disclose information; and DM’s belief about the state when the expert does not disclose information. To be an equilibrium, these strategies and belief are such that, given DM’s strategy, the expert’s strategy maximizes his expected utility, net of the expected cost and given DM’s belief about the state, DM’s strategy maximizes her expected utility; DM’s belief is derived from the expert’s strategy using Bayes’ rule.

Denote the decision taken by DM by

Now we consider how much effort the expert will exert in information acquisition, given DM’s decision in the event of nondisclosure,

In order to maximize the expected utility, the expert will exert effort in acquiring information such that the probability

Through straightforward calculation, we have the following result regarding function

When

Intuitively, the incentive of the expert to acquire information is from the difference between the expert’s expected utility when information is successfully obtained and his expected utility when failing to obtain information. As

As the next step, we characterize DM’s posterior belief about the state when the expert does not disclose information. The belief is derived from the expert’s strategy in information acquisition and disclosure using Bayes’ rule.

Suppose that the expert exerts effort to acquire information, and the effort exerted costs the expert

Compared to DM’s prior belief about the state, DM’s posterior belief about the state when the expert does not disclose information features a higher probability density at

Under the posterior belief, DM’s optimal decision

Note that

To see how

It can be seen that if

Intuitively, when the probability that the expert obtains information is higher, it is more likely that the expert is withholding information when the expert does not disclose information. Given the value of the state is relatively low when the expert withholds information, DM’s expectation about the state is lower. Therefore, her optimal decision

When the cutoff value of expert’s disclosing strategy increases, there are two conflicting effects. On the one hand, with a higher cutoff value and more states under which the expert will withhold information, DM thinks that the probability that the expert withholds information is higher when the expert does not disclose information. Therefore, DM’s expectation of the state decreases and her optimal decision

A perfect Bayesian Nash equilibrium can be represented as a tuple

In this section, we address the following question: under a greater cost of withholding information, will the expert exert more effort in information acquisition and withhold less acquired information from DM in the equilibrium? In other words, when

We use numerical simulation to get the relationship between

How equilibrium variables change as

As

To see why the above relationship holds, suppose that

A higher

Intuitively, as

In this section, we consider the situation where DM faces a pool of experts, each with different cost of withholding information. We ask the following question: which expert should DM choose to acquire and report information, in order to maximize DM’s expected utility? As shown in the previous section, an expert with a greater cost of withholding information has a lower likelihood of successfully obtaining information and withholds less acquired information. Therefore, it seems ambiguous whether DM can receive more information from an expert with a greater cost of withholding information and receive a higher expected utility.

To tackle this problem, we write down DM’s expected utility when the equilibrium is

From an ex ante point of view, the expected utility of DM when the expert has withholding cost

Using numerical simulation, we can get the relationship between DM’s ex ante expected utility and the expert’s cost of withholding information. This is shown in Figure

How the expected utility of DM changes as

In this section, we consider an alternative utility function of the expert. In particular, we assume that when the decision is

When the state is

We characterize a perfect Bayesian Nash equilibrium between DM and the expert in a way similar to what we did before. Denote the decision taken by DM as

Given the expert’s optimal strategy in information disclosure, when the effort exerted by the expert in information acquisition costs

The probability

For DM, suppose that the expert obtains information with probability

Under this belief, the optimal decision of DM when the expert does not disclose information is

A perfect Bayesian Nash equilibrium is a tuple

Figure

How equilibrium variables change as

As the next step, we consider a situation where DM faces a pool of experts with different costs of withholding information. We only consider the case where the cost of withholding information is small; in particular,

For a fixed bias

How the expected utility of DM changes as

This paper studies a game between an expert and a decision maker. The expert first exerts costly effort to acquire decision-relevant information. Then the expert decides whether or not to disclose the acquired information to the decision maker. Finally, the decision maker takes a decision that affects the utilities of both the expert and the decision maker. The expert and the decision maker have different preferences about the decision. In addition, withholding information is costly to the expert. We find that an expert with a greater withholding cost exerts less effort to acquire information but discloses a greater proportion of acquired information to DM. When the expert prefers DM’s decision to be as high as possible, DM’s expected utility decreases when the expert’s withholding cost increases. When the decision preferred by the expert is higher than that preferred by DM by a constant distance, DM’s expected utility increases when the expert’s withholding cost increases.

Future work involves studying similar games with other utility functions of the expert. Another direction for future studies is to analyze a model where the expert can misreport the observed state but will incur a cost when doing so. In such a model, we can investigate the relationship between the expert’s cost of misreporting information and the expert’s incentive to acquire information. This model is more complicated than the one we examine in this paper and is left for future research.

The author declares that there is no conflict of interests regarding the publication of this paper.

This research is supported by the Fundamental Research Funds for the Central Universities (no. JBK140107).