Consider a communication in which the Sender \(S_i\) (the informed agents, \(i = 1, \ldots ,N\)) possesses private information about different issues that the Receiver \(R_j\) (the uninformed agents, \(j = 1, \ldots ,M\)) takes actions on. The informed agents are further divided into two groups. One is the honest informed agents \(S_h\), whose number is *h*, and who release the real information they receive, while the other is dishonest informed agents \(S_d\), the quantity of which is *d*, who release real or false information.

Uninformed agents only receive messages from their neighbors and do not share information with each other. After sorting uninformed agents by current payoffs in every period, some of the uninformed agents at the lowest ranked locations will reselect their information sources, thus cutting off the informed agent he trusts the least and searching for another informed agent who is not directly connected to him in the current communication, thereby excluding the informed agent at the lowest ranked location in this period.

### Decision-making and prediction of individuals

At any period *t* (\(t = 1, \ldots ,P\)) the real state of information is \(m_t\), which is a random sequence that follows a normal distribution *N*(0, 1). Information \({m_{i,t}}\) that the *i*th Sender receives at the same time is

$$\begin{aligned} {m_{i,t}} = {m_t} + b \cdot \gamma \end{aligned}$$

(1)

where \(b \cdot \gamma\) is the bias Sender *i* gets at period *t* from the real state, \(b \sim N(0,1),\,\gamma\) is the baseline of accuracy, which is a scalar parameter that we will use later to measure the accuracy of the information, such that the smaller that \(\gamma\) is, the higher the accuracy of the information the Sender receives is.

In our model, honest Senders would send the information as they receive it, \({m_{i,t}}\), at period *t*, while the dishonest Senders probably send noisy messages to Receivers. We define noise using the Sigmoid function, which is a strictly increasing function having an“S” shape. We use a sigmoid function to model the relationship between noise and expectation is because that sigmoid function is a strictly increasing function. This function could reflect that higher expected payoffs the dishonest agents would like to get, the more noise he will release. In addition, S shaped also inflects the marginal decrease of noise that an informed agent sends to his neighbors. As the increase on expectation of informed agent, he will send larger noise. However, a larger noise is hardly accepted by the unformed agents, and will lead more decrease on informed agents reputation. Therefore, we set a strictly increasing but marginal decrease function to describe the telling lies behavior of informed agents. Thus, for each dishonest Sender, the noise they will release is

$$\begin{aligned} {n_{i,t}} = {\left( { - 1} \right) ^{{x_{i,t}}}}\left( {\frac{1}{{1 + \exp \left( { - {E_{i,t}}} \right) }} - 1} \right) \end{aligned}$$

(2)

where \({x_{i,t}}\) is a pseudorandom integer equaling 0 or 1.

Thus, the information the dishonest Sender releases is

$$\begin{aligned} {m_{id,t}} = {m_{i,t}} + {n_{i,t}}. \end{aligned}$$

(3)

The payoffs depend on the difference between \({m_{i,t}}\), the real states of information and \({y_{j,t}}\), the action uninformed receiver *j* takes. When Receivers (uninformed agents) receive messages from the informed Senders whom they connects with at period *t*, they may accept or reject the messages depending on their strategies, which we will discuss in detail in the following sections. When uninformed receiver *j* chooses to trust the information received from informed sender *i* at period *t*, we define parameter \({y_{j,t}}\) as equal to the information he received, that is, \({m_{i,t}}\) (if the informed sender is honest) or \({m_{i,t}} + {n_{i,t}}\) (if the informed sender is dishonest). While if he does not trust the information received, parameter \({y_{j,t}}\) equals 0, which depends on mean of information real states of 1. So payoff of uninformed receiver *j* obtains at period *t* is

$$\begin{aligned} {\pi _{j,t}} = - {\left( {y_{j,t} - m_{t} } \right) ^2}. \end{aligned}$$

(4)

Because the actions of Receiver decide the payoffs of both sides, the payoff of informed senders depend on how many uninformed receivers choose to trust him in the period. Only uninformed receivers linked with him trust the information he released, he could obtain payoff from this communication. So, payoff of informed receiver *i* obtains at period *t* is

$$\begin{aligned} {\pi _{i,t}} = \sum \limits _{j=1}^{M}{ - {\left( {y_{j,t} - m_{t} } \right) ^2} }. \end{aligned}$$

(5)

### Strategy updates of individuals

In this section, we describe the informed and uninformed agents’ strategies when they make decisions in a typical period, which we will discuss below.

#### Strategies of informed agents

Information released by the honest and dishonest informed agents is given by Eqs. (1) and (3), respectively. The honest informed agents have only one strategy, to release the real information they receive. For the dishonest informed agents, they have a choice in every period to update their strategy and decide whether to release noisy information. In this paper, we follow the work of Borgers and Sarin (2000) and assume that dishonest informed agents update their strategies with a learning process. The learning process assumes that agents have certain expectations of his chosen strategy. If the payoff brought by the current strategy can satisfies his expectations, he will increase his probability of current strategy in the next period, otherwise he will change his strategy.

The learning process is a Markovian process. Whether an agent will be more likely to undertake an action at period \(t+1\) only depends on the payoff he receives at period *t*. At any period *t*, a dishonest informed agent *i* adopts a strategy \({s_{i,t}}\) with an expected payoff \({E_{i,t}}\), but at period *t*, he is actually paid \(\pi _{i,t}\). Thus, at period \(t+1\) the probability \({p_i}\left( {t + 1} \right)\) of the agent lying can be expressed as:

If the agent lies at period *t*,

$$\begin{aligned} {p_i}\left( {t + 1} \right) = \left\{ \begin{array}{ll} {p_i}\left( t \right) + \alpha \cdot \left( {1 - {p_i}\left( t \right) } \right) ,&{}\quad {\pi _{i,t}} \ge {E_{i,t}}\\ {p_i}\left( t \right) - \alpha \cdot {p_i}\left( t \right) ,&{}\quad {\pi _{i,t}} < {E_{i,t}} \end{array} \right. \end{aligned}$$

(6)

If the agent does not lie at period *t*,

$$\begin{aligned} {p_i}\left( {t + 1} \right) = \left\{ \begin{array}{ll} {p_i}\left( t \right) - \alpha \cdot {p_i}\left( t \right) ,&{}\quad {\pi _{i,t}} \ge {E_{i,t}}\\ {p_i}\left( t \right) + \alpha \cdot \left( {1 - {p_i}\left( t \right) } \right) ,&{}\quad {\pi _{i,t}} < {E_{i,t}} \end{array} \right. \end{aligned}$$

(7)

where the agent’s payoff at period \(t+1\) is \({E_{i,t + 1}} = \beta \cdot {\pi _{i,t}} + \left( {1 - \beta } \right) \cdot {E_{i,t}}\).

\({E_{i,t}}\) represents the expected payoff of an individual regarding his current strategy. With an exogenous aspiration level, if the actual payoff coming after strategy \({s_{i,t}}\) is above some threshold \({E_{i,t}}\), the *i*th agent would be satisfied with his current strategy, and therefore he tends to choose \({s_{i,t}}\) in the next period, and vice versa. \({E_{i,t+1}}\) reveals an agent’s endogenous expected payoff at period \(t+1\), which shows that the expected payoff in the next period will be between the expected payoff and actual return from this period. \(\alpha\) is the rate of selected strategy adjustment, and \(\beta\) is the rate of adjustment of \({E_{i,t+1}}\), both of which are fixed values in our model between 0 and 1.

#### Strategies of uninformed agents

Uninformed agents also adopt the learning process to update the credibility of the informed agents connected to them in the communication. Presume that \({b_{i,j,t}}\) is the confidence of the *j*th uninformed agent in the *i*th informed agent at the *t* information transmission period. Whether the uninformed agent trusts the informed agent depends on the credibility \({b_{i,j,t}}\) of all of the informed agents connected to him. At one period, one uninformed agent may receive multiple messages from adjacent informed agents in the process of information transmission. We introduce a mechanism of competition with the aim that an uninformed agent can choose a message released by an informed agent in whom he has the highest confidence and update the confidence parameter \({b_{i,j,t}}\) for every informed agent connected to him based on the information from these sources.

We suppose an uninformed agent can only choose a message released from one of the information sources in whom he has the highest confidence. In addition, each uninformed agent records all of the messages passed to him. When he receives the actual states \({m_{t + 1}}\) in period \(t+1\), the uninformed agents will examine whether the informed agents have ever lied and make an adjustment regarding \({b_{i,j,t + 1}}\) their confidence in them in the next period. The method of adjustment is as follows:

$$\begin{aligned} {b_{i,j,t + 1}} = \left\{ \begin{array}{ll} {b_{i,j,t}} + \left( {1 - {b_{i,j,t}}} \right) \cdot \theta ,&{}\quad {\mu _{i,j,t}} \le {{\bar{\mathrm{M}}}_{i,j,t}}\\ {b_{i,j,t}} - {b_{i,j,t}} \cdot \theta ,&{}\quad {\mu _{i,j,t}} > {{\bar{\mathrm{M}}}_{i,j,t}} \end{array} \right. \end{aligned}$$

(8)

where \({\mu _{i,j,t}}\) is the bias of information an uninformed agent *j* receives from the informed agent *i* connected to him \({m_{i,j,t}}\) and the real state \({m_t}\). \({\bar{\mathrm{M}}_{i,j,t}} = \frac{1}{M \cdot k}\sum \nolimits _i {\sum \limits _j {{\mu _{i,j,t}}} }\) is an average value of all of the information biases that all of the uninformed agents receive from their informed neighbors with the real state at period *t*. \(\theta\) is the rate of confidence adjustment of uninformed agents to informed agents, which is a fixed number between 0 and 1. When the information the *i*th informed agent provides to the *j*th uninformed agent is more precise than the average, \({\mu _{i,j,t}} \le {{\bar{\mathrm{M}}}_{i,j,t}}\), the *j*th uninformed agent will increase his confidence in the *i*th informed agent, and vice versa.

### Searching behavior

In the information network construction, we assume each uninformed agent has *k* informed neighbors. In this network, only informed agents can become an information source. If the number of informed Senders \(i \ge 2\), there is more than one source of information in the network.

Among informed agents, they do not convey messages to each other, they only release the messages to the uninformed agents who are linked to them. The uninformed agents can only receive messages from the informed agents who are linked to them, but they do not know whether the informed agents are honest. When they receive messages from informed agents in any period, they have the choice to believe one of the informed agent, which depends on the parameter called confidence in informed agents that is determined by the confidence one uninformed agent has in one informed agent based on the actions both have already taken. If an uninformed agent believes in one informed agent, the uninformed agent has the highest confidence, the uninformed agent will receive a messages from this informed agent and believe that it is true. Under the effect of this competitive atmosphere among informed agents, uninformed agents will preserve all of the messages released to them so they can update their confidence level in the informed agents who are adjacent to them. The informed agents also will be aware of how many uninformed agent accept his message. In each period, the payoff of one informed agents is the superposition of each payoff he gets from the uninformed agents he linked with. From the payoffs, an informed agent could learn whether the uninformed agents he linked with thinks he is honest or not. This is an endogenous and dynamic learning process that the informed adjust strategies through the increased or decreased payoffs.

At the end of each period, some of uninformed agents update their information sources by cutting off their connection with one of the informed agents linked to them in whom they have the lowest confidence. The uninformed agent will find a new information source to fill the vacancy. We define \(\Gamma\) as the percentage of relinks in each period. That is, after sorting uninformed agents by current payoffs in every period, \(\Gamma\) percent of uninformed agents at the lowest ranked locations will reselect their information sources. Suppose the *j*th uninformed agent is the agent who needs to take this action. The uninformed agent sorts the *k* informed agents who connect with him based on how well he trusts them (the parameter \({b_{i,j,t}}\) that we introduced above), and then he cuts off the informed agent \(k^{\prime }\) whom he trusts the least and searches for another informed agent who is not directly connected to him in the network, but excluding informed agent \(k^{\prime }\). By doing so, the uninformed agents can improve their searching behavior to receive more true messages as time passes. To simplify our modeling, there is no cost to identifying the level of confidence in information sources, and uninformed agents select a new source of information to reconnect with at random and reset their trust, thus giving an initial value of 0.5 to their confidence in the newly linked informed agent.

A typical information transmission process is shown in Fig. 1.