Discrete Dynamics in Nature and Society

Volume 2016 (2016), Article ID 6507104, 6 pages

http://dx.doi.org/10.1155/2016/6507104

## A Mathematical Model of Communication with Reputational Concerns

^{1}School of Public Finance and Taxation, Southwestern University of Finance and Economics, 555 Liutai Avenue, Wenjiang, Chengdu, Sichuan 611130, China^{2}School of Securities and Futures, Southwestern University of Finance and Economics, 555 Liutai Avenue, Wenjiang, Chengdu, Sichuan 611130, China^{3}School of Finance, Southwestern University of Finance and Economics, 555 Liutai Avenue, Wenjiang, Chengdu, Sichuan 611130, China

Received 29 January 2016; Accepted 9 March 2016

Academic Editor: Christos K. Volos

Copyright © 2016 Ce Huang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We investigate a mathematical model where an expert advises a decision maker for two periods. The decision maker is initially unsure about whether the expert is biased or not. After consulting the expert on the decision problem of period one, the decision maker updates belief about the expert’s bias and consults the expert on the problem of period two. We find that more information is delivered in the model’s first period than in the one-period situation of communication.

#### 1. Introduction

This paper studies a mathematical model involving a decision maker and an expert with two periods. In each period, the decision maker needs to make a decision but does not know which one is the best. There is an expert who knows the optimal decisions but may be biased in favor of decisions that are different from the ideal ones of the decision maker. Initially unsure about whether the expert is biased or not, the decision maker consults the expert for the ideal decision in period one. After receiving the expert’s advice, the decision maker makes the first-period decision and updates belief about the expert’s bias. Then the decision maker consults the expert for the optimal decision in period two. We study whether the expert delivers more information on the optimal decision in period one to the decision maker, compared to the case where the decision maker consults the expert only in period one.

Many situations in real life are captured by the model described above. For instance, a patient usually has less information about his illness than a doctor does. So the former consults the latter. However, the latter may prefer the former to buy expensive medicine or to take medical procedures, even when the patient’s disease is mild and these medicine and procedures are unnecessary. The patient is often uncertain about whether the doctor has such preferences. But he can form some belief about the doctor’s preferences based on the doctor’s former prescriptions. The doctor expects that former prescriptions will affect the patient’s belief of the doctor’s preferences and influence future communications between the patient and the doctor. Will the doctor give more accurate prescriptions in period one, compared to the case where the doctor advises the patient only in period one?

We find that the expert communicates more information to the decision maker about the optimal decision in period one. This is because if the decision maker believes that the expert is more likely to be unbiased at the end of period one, the expert’s payoff will be higher in period two. In period one, the expert engages less in misreporting information, in order to increase the probability that the decision maker believes that the expert is unbiased at the end of period one. As a result, more information is communicated between the expert and the decision maker in period one.

There are several papers that study similar issues as the current paper does. In these papers, the decision maker is uncertain about the expert’s bias and the expert is concerned about establishing a reputation for being unbiased. There is another strand of literature studying the case where experts observe signals about the state with different accuracies and each expert prefers to be perceived as having accurate information. In Ottaviani and Sørensen [1], it is shown that experts with reputational concerns for having accurate information typically do not wish to tell the truth. In Bourjade and Jullien [2], the expert cannot misreport the information but can conceal the information. In the current paper, the expert is not concerned with establishing a reputation for having accurate information since it is assumed that all experts have perfect information about the state. In Sobel [3], the expert communicates with the decision maker repeatedly. The expert may be biased in favoring a particular decision. In Benabou and Laroque [4], a model similar to the one in Sobel [3] is studied. Li [5] studies the information transmission between the expert and the decision maker through an intermediary. All papers mentioned above assume that the unbiased expert truthfully reports information. In this paper, the unbiased expert may lie in order to enhance his reputation.

The paper that is closest to the current paper is that of Morris [6], who also studies a two-period information transmission model between a decision maker and an expert. It is found that the unbiased expert may send a report different from the observed signal in order to enhance the expert’s reputation for being unbiased. Our study is different from that of Morris [6] in the following ways. First, we assume that the expert can perfectly observe the state. Second, there is a continuum of possible states and decisions in the current paper, whereas, in Morris [6], only two states and decisions are possible.

The remaining part of the paper proceeds as follows. Section 2 builds a two-period model of strategic information transmission between a decision maker and an expert. In addition, it characterizes an equilibrium in period two of the model and an equilibrium in period one. Section 3 compares the ex ante expected payoff of the decision maker in period one of the model to that of the decision maker when the decision maker consults the expert in only one period. Section 4 concludes and discusses a direction for future work.

#### 2. Model

There are two periods. A decision maker (hereafter DM) needs to take a decision in each period. The decision is to choose a real number in the interval . In period , after DM takes decision , DM receives some payoff, which depends on the underlying state in period denoted as . Assume that the payoff of DM in period is

Apparently, the payoff is higher when the decision is closer to . Therefore, the decision that gives DM the highest payoff in period (hereafter the optimal decision of DM in period ) is equal to . However, DM does not know except that it is uniformly distributed over and is distributed independently of .

There is an expert who observes at the beginning of period (throughout the paper, we use “he” to denote the expert and “she” to denote DM). However, the ideal decision of the expert may be different from that of DM. We assume that, in period , the payoff received by the expert when DM takes decision is

The expert can be of two types. For one type of the expert, and the expert’s ideal decision in period is the same as that of DM. We call this type of the expert “unbiased.” For the other type, and the expert’s ideal decision is higher than that of DM by . We call this type of the expert “biased.” The expert knows his own type, but DM does not. Initially, DM believes that the expert is of either type with equal probabilities. In addition, the expert’s type is distributed independently of the state in each period.

In each period, DM communicates with the expert about the state of that period before making a decision. The communication process is as follows. After observing , the expert sends a report to DM. We assume that the expert can send any report in , which can be different from the state. Also, the expert does not incur any cost in sending a report. After receiving the report from the expert, DM updates her belief about the distribution of the state in that period and her belief about the expert’s type. Then DM takes a decision . The timeline of the game between the expert and DM is shown in Figure 1.