Introduction

Opinion dynamics is the process of studying the evolution of opinions through the social interaction between a group of agents. French and John (1956) first proposed the basic model of opinion dynamics. In last decades, a series of opinion dynamics models with different opinion evolution rules have been proposed, and they can be divided into two categories based on discrete and continuous opinion forms: (1) Discrete opinion models, this type of model is based on physics, such as Sznajd model (Sznajd-Weron and Sznajd 2000), voter model (Clifford and Sudbury 1973; Durrett et al. 2012), majority rule model (Galam 1986, 2002); (2) Continuous opinion models, this kind of model is established mathematically, such as FJ model (Friedkin and Johnsen 1990), DeGroot model (Degroot 1974), DW model (Deffuant et al. 2000; Weisbuch et al. 2002), HK model (Hegselmann and Krause 2002), continuous opinions and discrete actions (CODA) model (Martins 2008). With deeper research on opinion dynamics, some other classic models have been employed in this field, for example, the Ising model (Ising 1925; Glauber 1963; Bianconi 2002).

In opinion dynamics, the final state of opinion evolution tends to three stable states: consensus, polarization, and fragmentation. And the opinion dynamics models usually include three basic elements: opinion expression formats, opinion evolution rules, and opinion dynamics environments. Based on these features, the research on opinion dynamics can improve the understanding of some crucial phenomena in finance and business. There are studies to apply opinion dynamics to many aspects of this field, such as, marketing (Martins et al. 2009; Luo et al. 2014), finance (Kaizoji 2000; Johansen et al. 2000; Bornholdt 2001; Kaizoji et al. 2002), e-commerce (Wan et al. 2018; Zhao et al. 2018b), politics (Bernardes et al. 2001; Stauffer 2002a; Galam 2004), and group decision making (GDM) (Dong et al. 2020; Zha et al. 2019, 2020). Dong et al. (2018b) reviewed the opinion dynamics models within different environments. However, there is no literature review so far to analyze the application of opinion dynamics in finance and business. To this end, this paper provides a clear review of the application of opinion dynamics in finance and business. Moreover, from the insights gained from previous research, we analyze the challenges faced by future research to promote the research of this topic.

The rest of this paper is organized as follows. “Opinion dynamics models and social networks” section introduces the framework and some basic models of opinion dynamics. “Applications of opinion dynamics models in finance” section reviews the application of opinion dynamics in finance and business. Next, “Applications of opinion dynamics models in business” section presents the research challenges of this topic. Finally, the conclusions are drawn in “Summary, critical discussions and new directions” section.

Opinion dynamics models and social networks

Opinion dynamics is a process of individual opinion evolution, in which the interactive agents in the group constantly update their opinions on the same issue based on the evolution rules, and the opinions are stable at the final stage, forming a consensus, polarization, or fragmentation opinion distribution. As described before, the opinion dynamics are divided into two types: Continuous and discrete opinion models. We will introduce some basic models of these two types in “Continuous opinion models” and “Discrete opinion models” sections. A social network can be thought of as a connection between a group of agents who participate in and share various information for the purpose of friendship, marketing or business exchange. Some representative networks are introduced in “Social networks” section.

Continuous opinion models

  1. (1)

    DeGroot model

In opinion dynamics, the DeGroot model (DeGroot 1974) is considered as the classical model in general. DeGroot (1974) explicitly describes the process which leads to the consensus and specifies the weights that are to be used in Stone (1961). Let \(A = \{ a_{1} ,a_{2} , \ldots ,a_{m} \}\) be the set of agents, and \(o_{i}^{t} \in R\) be the opinion of agent \(a_{i}\) at round \(t\). Assume that \(w_{ij}\) is the weight agent \(a_{i}\) assigns to agent \(a_{j}\), where \(w_{ij} \ge 0\) and \(\sum\nolimits_{j = 1}^{m} {w_{ij} } = 1\). Then, the evolution rule of agent \(a_{i}\) will be:

$$o_{i}^{t + 1} = w_{i1} o_{1}^{t} + w_{i2} o_{2}^{t} + \cdots + w_{im} o_{m}^{t} \, t = 0,1,2 \ldots$$
(1)

which equals to:

$$O^{t + 1} = W \times O^{t} \, t = 0,1,2 \ldots$$
(2)

where \(W = (w_{ij} )_{m \times m}\) is constant and \(O^{t} = (o_{1}^{t} ,o_{2}^{t} , \cdots ,o_{m}^{t} )^{T} \in R^{m}\). DeGroot (1974) believed if there is a \(t\) such that every element in at least one column of the matrix \(W^{t}\) is positive, then a consensus can be reached, and the consensus opinion is a linear combination of the agents' initial opinions.

  1. (2)

    Bounded confidence model

The bounded confidence (BC) model is an extended study of the DeGroot model, and the weight in Eq. (1) changes with time or opinion. The BC model is based on the following idea: when the difference of opinions between two agents is lower than a given threshold, they will interact, otherwise they will not even bother to discuss. Let \(A = \{ a_{1} ,a_{2} , \ldots ,a_{m} \}\) be the same as before. Let \(o_{i}^{t}\) be the opinion of the agent \(a_{i}\) at round \(t\), which often takes the value from \([0,1]\). Let \(\varepsilon\) be the bounded confidence level, and the BC model will be homogeneous with the \(\varepsilon\) values be the same for all agents; otherwise, it will be heterogeneous. The BC model includes two essential models: the DW model (Deffuant et al. 2000; Weisbuch et al. 2002), and the HK model (Hegselmann and Krause 2002, 2005). In the following, we introduce these two models.

  1. (a)

    DW model

Two agents are randomly chosen from set \(A\), and they will determine whether to interact according to the bounded confidence. If \(|o_{i}^{t} - o_{j}^{t} | > \varepsilon\), the agents \(a_{i}\) and \(a_{j}\) will think that opinions are too far apart to interact; otherwise, the evolution rule will be:

$$\begin{gathered} o_{i}^{t + 1} = o_{i}^{t} + \mu (o_{j}^{t} - o_{i}^{t} ) \hfill \\ o_{j}^{t + 1} = o_{j}^{t} + \mu (o_{i}^{t} - o_{j}^{t} ) \hfill \\ \end{gathered}$$
(3)

where \(\mu \in [0,{0}{\text{.5}}]\) is the convergence parameter. Depending on the parameters \(\varepsilon\) and \(\mu\), a consensus, polarization, or fragmentation opinion distribution will be obtained in the DW model.

(b) HK model.

Let \(w_{ij}^{t}\) be the weight that agent \(a_{i}\) gives to \(a_{j}\) at round \(t\), which is described as:

$$w_{ij}^{t} = \left\{ {\begin{array}{*{20}l} {1/{|}S_{i}^{t} {|}} \hfill & {a_{j} \in S_{i}^{t} } \hfill \\ 0 \hfill & {a_{j} \notin S_{i}^{t} } \hfill \\ \end{array} } \right.$$
(4)

where \(S_{i}^{t} = \{ a_{j} ||o_{i}^{t} - o_{j}^{t} | \le \varepsilon \}\) is the confidence set of agent \(a_{i}\), and \({|} \cdot {|}\) denotes the absolute value of a real number and the number of elements for a finite set.

Then, the opinion evolution rule is as follows:

$$o_{i}^{t + 1} = \sum\limits_{{a_{j} \in S_{i}^{t} }} {w_{ij}^{t} o_{j}^{t} }$$
(5)

If there exists an ordering \(o_{i1} \le o_{i2} \le \cdots \le o_{im}\) such that two adjacent opinions are within the bounded confidence level \(\varepsilon\), then the opinion profile \(O = o_{1} ,o_{2} , \ldots ,o_{m}\) is called an ε-profile. Hegselmann and Krause (2002) argue that the opinion profile will be an ε-profile for all times if a consensus is reached for an initial profile. Moreover, two agents will remain separated forever if they split at some time.

Discrete opinion models

The discrete opinion models mostly use binary opinions for research at the beginning. As the research deepens, some extended models use multiple opinions to study more complex situations. Some basic models will be introduced below.

  1. (1)

    Voter model

The voter model (Clifford and Sudbury 1973) is a discrete opinion dynamics model with all agents been widely placed in a two-dimensional lattice. Let \(A = \{ a_{1} ,a_{2} , \ldots ,a_{m} \}\) be the set of agents as before, and \(o_{i}^{t}\) be a binary opinion of agent \(a_{i}\) at round \(t\), where \(o_{i}^{t} = 1\) or \(o_{i}^{t} = - 1\). The agent will randomly select an agent among the four neighbors and follow his/her opinion. Without loss of generality, agents \(a_{j}\) is assumed to be the selected neighbor. Then, the opinion \(o_{i}^{t + 1}\) of agent \(a_{i}\) will be \(o_{j}^{t}\), i.e., \(o_{i}^{t + 1} = o_{j}^{t}\). The voter model leads to two possible consensus states, and the probability of opinions to reach a consensus is determined by the initial distribution of opinions. Furthermore, a consensus is reached only for dimension \(d \le 2\) in an infinite system (Cox 1989).

  1. (2)

    Snajzd model

The Snajzd model is also a discrete opinion dynamics model for the one-dimensional case (Sznajd-Weron and Sznajd 2000), which is based on the characteristic of "United we Stand, Divided we Fall". And the opinion \(o_{i}^{t} = \pm 1\) is a binary opinion of agent \(a_{i}\) at round \(t\). Then, the opinions evolve according to the following rules:

  1. (a)

    In each round a pair of agents \(a_{i}\) and \(a_{i + 1}\) is selected to influence their nearest neighbors, i.e. the agents \(a_{i - 1}\) and \(a_{i + 2}\).

  2. (b)

    If \(o_{i}^{t} = o_{i + 1}^{t}\), then \(o_{i - 1}^{t + 1} = o_{i + 2}^{t + 1} = o_{i}^{t}\).

  3. (c)

    If \(o_{i}^{t} = - o_{i + 1}^{t}\), then \(o_{i - 1}^{t + 1} = o_{i + 1}^{t}\) and \(o_{i + 2}^{t + 1} = o_{i}^{t}\).

A different version of rule (c) proposed later is now more widely used and will be introduced in “Application of basic models” section. The difference from the voter or Ising model is the outflow of information. And, two types of stable states are always reached in this model: complete consensus or stalemate.

  1. (3)

    Ising model

Ising model is a well-known model in physics to explain the phase transition of ferromagnetic materials (Ising 1925). In opinion dynamics, it has been well studied over last decades to model the social interactions (Glauber 1963; Binder 1981; Harris 2001; Herrero 2002; Bianconi 2002). The energy \(E\) of interaction between two magnetic particles in Ising’s original work corresponds to the degree of conflict of opinions between two agents in opinion dynamics, as shown below

$$E = - Jo_{i} o_{j}$$
(6)

where \(o_{i}\) denotes the ith particle’s intrinsic spin and constant \(J\) is an energy coupling constant. And \(o_{i} { = + 1}\) or − 1 correspond to a up state and a down state respectively. In opinion dynamics, \(o_{i}\) indicates a binary opinion of agent \(a_{i}\) and \(J\) represents the interaction strengths among agents. When the spins of two interacting particles are parallel, the energy of the interaction is minimal, that is, the degree of conflict of opinions between two agents will be minimal when they have the same opinion. Furthermore, the interaction energy also exists between the magnetic particles and the external magnetic field \(H\), that is,

$$E = - Ho_{i}$$
(7)

where \(H\) represents the external information in opinion dynamics (e.g. media promotion). This energy will be minimized when the particle’s spin has the same value as the external field. The external field models the media promotion effect of opinions. If the opinions of agents are consistent with the media, the lower the energy and therefore the more harmonious. Then, the total energy of interaction can be described as

$$E_{t} = - J\sum\limits_{i,j} {o_{i} o_{j} } - H\sum\limits_{i} {o_{i} }$$
(8)

Social networks

Social network can be simply thought of as the structure of social relations. Social network can refer to not only the network between people, but also the network between organizations and organizations, even the network of relations between cities and cities, and the network of relations between countries and countries. The network approach greatly contributes to our understanding of complex system structure, trust propagation and influence, etc. In opinion dynamics, social network mainly refers to the network between agents, which can be defined by a graph with nodes representing agents and edges indicating relationships between agents. Three representative complex networks are introduced below:

Erdős–Rényi (ER) random network: The "random" of a random network is mainly reflected in the distribution of edges. A random network actually connects the given nodes randomly. The ER model means that in a given \(m\) nodes, it is specified that every two nodes are connected with a probability of \(p\) \((0 \le p \le 1)\). The clustering coefficient and average path length of the ER random network are very small, where the first property reflects the coincidence degree of friends between two neighbors, and the second one refers to the average of the shortest path length between any two nodes in a network.

Small-world (SW) network: In this network, most of the nodes are not connected to each other, but most of the nodes can be reached after a few connections. In daily life, sometimes you will find that some people who you think are "far away" from you are actually "close" to you. The SW network is a mathematical description of this phenomenon. Although the SW network has a small average path length, it also has a considerable higher clustering coefficient than the random network.

Scale-free (SF) network: The SF network has serious heterogeneity, and the connection status (degree) between the nodes has a serious uneven distribution: a few nodes in the network have extremely many connections, and most nodes have only a few connections. The nodes with extremely many connections play a leading role in the SF networks. Generally speaking, the scale-free nature is an inherent property that describes the serious uneven distribution of complex systems.

Applications of opinion dynamics models in finance

In this section, we introduce the applications of opinion dynamics from two aspects: Basic model application and extended application.

Application of basic models

The financial market is a very complex system, composed of competing and interacting economic agents with different strategies, motivations and investment horizons (Mitchell and Mulherin 1994; Feng and Seasholes 2005; Hou and Moskowitz 2005; Kaminsky and Schmukler 2007; Tetlock 2007; Diether et al. 2009; Easley et al. 2016). In the financial application, the binary opinion dynamics models are the most common. In this section, we present the application of three binary opinion dynamics models (i.e., the Ising model and the Snazjd model) in financial market. In these studies, the price changes proportional to the difference between demand and supply. An obvious requirement is that more demand than supply will push up the prices, and similarly for the opposite case.

  1. (1)

    Application of the Ising model

As described before, the Ising model is a well-known model in physics (Ising 1925), which also has attracted a lot of attention in the modeling of financial systems. Table 1 summarizes the applications of the Ising model in financial market. Here we introduce the basic applications of the Ising model or Ising-like model in financial market (Bornholdt 2001; Krawiecki et al. 2002; Sornette and Zhou 2006; Chowdhury and Stauffer 1999).

Table 1 The application of the Ising model in financial market
Full size table

The Ising model is a model with \(i = 1,2, \ldots ,m\) spins with orientations \(o_{i}^{t} = \pm 1\), corresponding to agent \(a_{i}\) with the actions at round \(t\) in financial market in Bornholdt (2001). And \(o_{i}^{t} = + 1\) is interpreted as a bullish trader \(a_{i}\) who places buy orders at round \(t\), while \(o_{i}^{t} = + 1\) is interpreted as a bearish trader \(a_{i}\) who places sell orders. Assume that \(H_{i}^{t}\) is the local field and \(\beta\) is a responsiveness parameter. And then the evolution rule is described as:

$$\left\{ {\begin{array}{*{20}l} {o_{i}^{t + 1} = + 1} \hfill & {with\,p = 1/[exp( - 2\beta H_{i}^{t} )]} \hfill \\ {o_{i}^{t + 1} = - 1} \hfill & {with\,{1} - p} \hfill \\ \end{array} } \right.$$
(9)

where \(p\) is the probability of opinion evolution. And the local field \(H_{i}^{t}\) is specified by

$$H_{i}^{t} = \sum\limits_{j = 1}^{m} {J_{ij} o_{j}^{t} } - \alpha C_{i}^{t} \frac{1}{m}\sum\limits_{j = 1}^{m} {o_{j}^{t} }$$
(10)

where \(J_{ij}\) represents the interaction strength (possibly 0) between traders \(a_{i}\) and \(a_{j}\); \(\alpha > 0\) is a global coupling parameter; \(C_{i}^{t}\) is a strategy of trader \(a_{i}\) at round \(t\).

Similar to Bornholdt (2001), \(o_{i}^{t} = \pm 1\) has the same meaning in Krawiecki et al. (2002). And the evolution rule is as follows:

$$\left\{ {\begin{array}{*{20}l} {o_{i}^{t + 1} = + 1} \hfill & {with\,p = 1/[{1 + }exp( - 2H_{i}^{t} )]} \hfill \\ {o_{i}^{t + 1} = - 1} \hfill & {with\,{1} - p} \hfill \\ \end{array} } \right.$$
(11)

where the local field \(H_{i}^{t}\) is below:

$$H_{i}^{t} = \frac{{1}}{m}\sum\limits_{j = 1}^{m} {J_{ij}^{t} o_{j}^{t} } - h_{i}^{t}$$
(12)

with interaction strength \(J_{ij}^{t}\) changing over time and external field \(h_{i}^{t}\) indicating the effect of environment.

Sornette and Zhou (2006) extended the Ising model and introduced the following evolution rules:

$$o_{i}^{t} = {\text{sign}} \left[ {\sum\limits_{j = 1}^{m} {J_{ij}^{t} E^{t} [o_{j}^{t} ]} + \sigma_{i} G(t) + \varepsilon_{i}^{t} } \right]$$
(13)

Equation (13) embodies three contributions:

  1. (a)

    Mutual influences \(\sum\nolimits_{j = 1}^{m} {J_{ij}^{t} E^{t} [o_{j}^{t} ]}\): \(J_{ij}^{t}\) quantifies the interaction strength of the expected decision of trader \(a_{j}\) on trader \(a_{i}\); \(E^{t} [o_{j}^{t} ]\) is the expected decision of trader \(a_{j}\) estimated by trader \(a_{i}\).

  2. (b)

    External news \(\sigma_{i} G(t)\): \(\sigma_{i}\) is the relative sensitivity of trader \(a_{i}\) to the external news; \(G(t)\) is defined as the influence of the external news on the decision of trader \(a_{i}\).

  3. (c)

    Idiosyncratic judgements \(\varepsilon_{i}^{t}\): \(\varepsilon_{i}^{t}\) indicates the trait of the decision of trader \(a_{i}\) for the explanation of her personal information.

Chowdhury and Stauffer (1999) proposed an Ising-like stock market model, which includes three states of a trader: \(+ |o_{i} |\) (buy), \(- |o_{i} |\) (sell), and 0 (not to trade). And the evolution rule is below:

  1. (a)

    A trader picks up the \(+ |o_{i} |\) with probability \(b\), \(- |o_{i} |\) with probability \(b\) and the 0 with probability \(1 - 2b\).

  2. (b)

    Then the trader \(a_{i}\) changes into the state picked up with probability \(e^{{ - \Delta E_{i} /(k_{B} T)}}\), where \(T\) represents the fictitious temperature, and \(\Delta E\) is the change of disagreement connected with this transition. Let \(h_{i}\) be the individual bias of the trader \(a_{i}\). And the disagreement functions of noise trader and fundamentalist trader are defined as Eqs. (14) and (15), respectively.

    $$E_{i} = - o_{i} H_{i}$$
    (14)
    $$E_{i} = - o_{i} (H_{i} + b_{i} )$$
    (15)

    with the local field \(H_{i}\):

    $$H_{i} = \sum\limits_{j = 1,j \ne i}^{m} {J_{ij} o_{j} }$$
    (16)

    where \(J_{ij}\) is the interaction strength between traders \(a_{i}\) and \(a_{j}\)

.

  1. (2)

    Application of the Sznajd model

So far, the possible financial application of the Sznajd model has not received much attention. Sznajd-Weron and Weron (2002) studied the price formation in a financial market based on the one-dimensional Sznajd model. Let \(A = \{ a_{1} ,a_{2} , \ldots ,a_{m} \}\) be the set of agents. In the Sznajd model, \(o_{i}^{t}\) is a binary opinion of agent \(a_{i}\) at round \(t\), here \(o_{i}^{t}\) is defined as the attitude of market participant \(a_{i}\). And \(o_{i}^{t} = + 1\) has the same meaning as the application of the Ising model in “Applications of opinion dynamics models in finance” section (1). Then, the opinion evolution represents the dynamic state of the trader, and the dynamic rule is as shown below:

  1. (a)

    In each round, a pair of traders \(a_{i}\) and \(a_{i + 1}\) is chosen to affect their neighbors, i.e. the traders \(a_{i - 1}\) and \(a_{i + 2}\).

  2. (b)

    If \(o_{i}^{t} o_{i + 1}^{t} = 1\), that is, traders \(a_{i}\) and \(a_{i + 1}\) have the same state, then traders \(a_{i - 1}\) and \(a_{i + 2}\) follow the state of the traders \(a_{i}\) and \(a_{i + 1}\), that is \(o_{i - 1}^{t + 1} = o_{i + 2}^{t + 1} = o_{i}^{t} = o_{i + 1}^{t}\). The reason for this dynamic rule is that many market participants are trend followers who place orders according to the opinions of local gurus.

  3. (c)

    If \(o_{i}^{t} o_{i + 1}^{t} = - 1\), then traders \(a_{i - 1}\) and \(a_{i + 2}\) randomly choose to buy or sell, namely, \(o_{i - 1}^{t + 1} ,o_{i + 2}^{t + 1} = - 1{\text{ or } + \text{ 1}}\) at random. This dynamic rule incorporates the fact that the absence of a local guru, i.e., traders \(a_{i}\) and \(a_{i + 1}\) are in different state, will cause the trend followers in market to act randomly.

  4. (d)

    In financial market, trend followers are not the only remaining market participants (Bak et al. 1997). Some rational traders also exist, and they know the system better and have strategies. For simplicity, Sznajd-Weron and Weron (2002) introduced one rational trader in the proposed model, who known exactly the difference between demand and supply in the current market. He/she places buy orders when there is more demand than supply, and vice versa. Sznajd-Weron and Weron (2002) argued that \(p_{t} = {1 \mathord{\left/ {\vphantom {1 m}} \right. \kern-\nulldelimiterspace} m}\sum\nolimits_{i = 1}^{m} {o_{i}^{t} }\) is the price at round \(t\). Then, this rule is that the rational trader \(a_{k}\) will buy (i.e., \(o_{k}^{t} = + 1\)) at round \(t\) with probability \(|p_{t} |\) when \(p_{t} < 0\), and sell (i.e., \(o_{k}^{t} = - 1\)) with probability \(|p_{t} |\) when \(p_{t} > 0\).

Sabatelli and Richmond (2004) proposed a model of trading orders based on a Sznajd-like interaction, and this study showed how the proposed model is compatible with some of the main statistical characteristics observed in asset volumes in financial markets.

  1. (2)

    Application of the voter model

Krause and Bornholdt (2012) used a two-dimensional voter model with a tunable social temperature to study the opinion evolution process among traders. In this model, agent \(a_{i}\) adapts the opinion \(o_{i}^{t} { = } \pm {1}\) (buy or sell) based on his/her nearest neighbors. Let \(u_{i}\) be the number of agreeing neighbors of agent \(a_{i}\). Then, the opinion evolution is described with the flip probabilities \(p_{{u \to 4{ - }u}}\), where \(p_{{u \to 4{ - }u}} { + }p_{{{4 - }u \to u}} { = 1}\). With suppressed voluntary isolation this model argues that \(p_{{{2} \to {2}}} { = 1/2}\), \(p_{{{0} \to 4}} { = 1}\), and \(p_{{{4} \to {0}}} { = 0}\). Based on the inverse temperature \(\beta { = 1/}T\), the remaining probability of join local minorities \(p_{{{3} \to {1}}}\) is set as follows:

$$p_{3 \to 1} = \frac{1}{1 + \exp (4\beta )}$$
(17)

The social temperature \(T\) can be regarded as the market temperature, because it affects the uncertainty of all agents' investment strategies. For low temperatures the persuasiveness of local groups of agents will be increased (\(p_{1 \to 3} > 3p_{{{3} \to {1}}}\)). Meanwhile, the persuasiveness of local majorities will be suppressed for higher temperatures.

There are other applications of voter and voter-like models in financial markets. For example, Zubillaga et al. (2019) studied the financial market with the majority-vote model, and includes three states, i.e., buy, sell or remain inactive; Vilela et al. (2019) also applied the majority-vote model to research the financial market; Wang et al. (2019) studied the complex and composite entropy fluctuation behaviors through a voter interacting system.

The basic models follow the statistical physics-based models and are traditionally designed to capture several regulatory real-life phenomena. These opinion dynamics studies use different opinion evolution rules to model the microscopic dynamics among agents, meanwhile, study the trends, bubbles and crashes of the financial market from the macroscopic level. And the basic binary opinion dynamics models provides tools and methods to study the phase transition, metastable phase and structured lattice critical point.

Extension application

The application of the opinion dynamics models in financial research have been studied from many aspects. Here we mainly introduce the following two extension applications.

  1. (1)

    Kinetic models of opinion formation

The basic opinion dynamics models study the financial market by the methods and tools from classical statistical mechanics, where the complex behaviors arise from relatively simple interaction rules. The mathematical framework that characterizes financial markets is another research line (Maldarella and Pareschi 2012; Cordoni and Di Persio 2014). The kinetic models of opinion formation consider opinion dynamics analysis from the point of view of mathematical framework involving both opinion exchange between agents and information diffusion (Toscani 2006). Compared to the basic opinion dynamics models with the discrete opinion and time, the kinetic models of opinion formation provide tools to analyze microscopic dynamics of each trading agent in financial market using the continuous opinion and time. Furthermore, in contrast to the basic opinion dynamic models that usually only study behavior empirically through computer simulation, the kinetic models based on partial differential equations allow us to obtain analytical general information about the model and its asymptotic behavior. Maldarella and Pareschi (2012) studied the kinetic models for socio-economic dynamics of speculative markets characterized by two different market strategies, chartists and fundamentalists, similar to Lux and Marchesi model (Lux and Marchesi 1999, 2000). This study uses the kinetic system coupling a description for the price formation mechanism and the propensity of investment of chartists.

  1. (2)

    Data-driven opinion dynamics models

Data-driven opinion dynamics models are study the opinion evolution from a more computational point of view. Das et al. (2014) observed three distinct types of opinion evolution processes derived from stubbornness, compromise, and biased conformity in the carefully crafted experiments and proposed a biased-voter model based on these observations. De et al. (2014) proposed a linear influence opinion dynamics model, where the edge influence strengths are estimated, not assumed, from an observed series of opinions of agents using the projected gradient descent algorithm. De et al. (2016) proposed a probabilistic modeling framework of opinion dynamics with efficient model simulation and parameter estimation from historical fine grained event data, where opinions over time are represented by means of marked jump diffusion stochastic differential equations. To extend the contribution in De et al. (2016), t, Kulkarni et al. (2017) applied a network-guided recurrent neural network architecture to capture a generic form of nonlinear dependencies between the social network and the past events to model the nonlinear opinion dynamics in social networks. Di Persio and Honchar (2016) argue that even if only training on plain time series data, neural networks can predict the movements of financial time series. Although data-driven opinion dynamics models have not been studied in the financial market environment, all the studies described above provide a good research foundation. Using machine learning to study the microscopic dynamics of the trading agents and the collective behaviors will be of great help in financial forecasting.

Applications of opinion dynamics models in business

In this section, we introduce the applications of opinion dynamics in marketing, e-commerce, politics and decision making.

Marketing

The use of opinion dynamics models in marketing focuses on modeling two influential factors: advertising and word-of-mouth recommendations from friends. Specifically, we introduce the use of some basic opinion dynamics models in marketing below.

  1. (1)

    Application of the CODA model

Martins et al. (2009) and Luo et al. (2014) studied the dynamics of customers opinions affected by advertising and word of mouth using the CODA model. When using the CODA model in marketing, two choices are often considered, i.e., adoption and non-adoption of a new product. Let \(o_{i}^{t}\) (continuous variable) be the inner opinion of agent \(a_{i}\) associated with a product or service at round \(t\). Let \(s_{i}\) be the binary choices in regards to agent \(a_{i}\). If agent \(a_{i}\) chooses to adopt the product or service, then \(s_{i} = + 1\); otherwise, \(s_{i} = 0\). And the relation between the action \(s_{i}\) and the opinion \(o_{i}\) of agent \(a_{i}\) can be:

$$s_{i} = \left\{ {\begin{array}{*{20}c} 1 & {o_{i} \ge 0.5} \\ 0 & {o_{i} < 0.5} \\ \end{array} } \right.$$
(18)

Luo et al. (2014) studied both influential factors, and then, the opinion evolution of agent \(a_{i}\) can be described as:

$$o_{i}^{t + 1} = o_{i}^{t} + F_{i}^{t} + AD_{i}^{t}$$
(19)

where \(F_{i}^{t}\) indicates the influence that agent \(a_{i}\) affected by his/her friends, and is related to the actions of his/her friends; \(AD_{i}^{t}\) represents the influence of the advertising on agent \(a_{i}\).

  1. (2)

    Application of the Sznajd model

Schulze (2003) and Sznajd-Weron and Weron (2003) studied that how strong an advertising must be to support one of the two products to conquer the market depending on the Sznajd model in a two-dimensional setup. Assuming that \(o_{i}^{t}\) is the binary opinion of agent \(a_{i}\) of the Sznajd model, it corresponds to the situation where the customer \(a_{i}\) supports product \(x_{A}\) or \(x_{B}\). Then, the opinion evolution rule is described as follows (Sznajd-Weron and Weron 2003):

  1. (a)

    Randomly select an agent (that is, a customer *), and then form a panel (dark blue site) with his/her three neighbors. This panel can affect the eight nearest neighbors (light blue site), as shown in Fig. 1.

    Fig. 1
    figure 1

    Rules affecting neighbors

    Full size image
  2. (b)

    If the four agents in a panel support the same product, all eight neighbors will be affected to support the same product.

  3. (c)

    If one of the agents in a panel supports the product opposite to the other three agents, the neighbors will support the product supported by the majority with probability 3/4 and will be responsive to the advertising with probability 1/4.

  4. (d)

    In the case of two agents in the panel supporting product A and two agents supporting product B, any neighbor will not be persuaded by the panel, but will respond to advertising.

  1. (3)

    Application of the BC model

Salehi and Taghiyareh (2014) studied the opinion prediction of agents based on the BC model to identify suitable products and customers and determine the correct marketing strategy. Suppose that \(o_{ik}^{t}\) is the opinion of agent \(a_{i}\) about product \(x_{k}\), where \(o_{ik}^{t} \in [ - 1,1]\), and opinions of agents can be divided into three categories, namely, positive, negative and don't care. Let \(d\) be the bounded confidence. Similar to Pazzani (1999), two agents \(a_{i}\) and \(a_{j}\) are selected to interact. If \(|o_{ik}^{t} - o_{jk}^{t} | < d\), the opinion evolution will be:

$$\begin{gathered} o_{ik}^{t + 1} = o_{ik}^{t} + \mu (o_{jk}^{t} - o_{ik}^{t} ) \hfill \\ o_{jk}^{t + 1} = o_{jk}^{t} + \mu (o_{ik}^{t} - o_{jk}^{t} ) \hfill \\ \end{gathered}$$
(20)

where \(\mu\) is a parameter about the degree of trust \(T_{ij}\) between the selected two agents, i.e., \(\mu { = }T_{ij}\). And \(T_{ij}\) is a dynamic value according to the opinions of agents.

Applying this method, a prediction of opinions and some clusters for each product will be obtained. Salehi and Taghiyareh (2014) argued that these results can be used to create management reports and support managers in choosing the right products to produce, and choosing the right customers or communities to promote different products.

Moreover, there are other studies using the opinion dynamics models research marketing. Taghiyareh's team studied this issue from different aspects (Nazemian and Taghiyareh 2012; Salehi and Taghiyareh 2014; Vadoodparast et al. 2014; Vadoodparast and Taghiyareh 2015; Salehi and Taghiyareh 2019). Günther et al. (2011) proposed an agent-based simulation method to support managers in marketing activities, and illustrated the method by taking the new product diffusion of a novel biomass fuel as an example. Chasparis and Shamma (2012) studied the issue of obtaining the optimal marketing policies for the diffusion of innovations based on the FJ model. Maghami and Sukthankar (2012, 2013) studied the influence of advertising in marketing using independent cascade model. Bimpikis et al. (2016) examines a game-theoretic model of competition between firms with the optimal targeted advertising strategies based on the binary voter model. Varma et al. (2017) analyzed the competition between two firms to capture a larger market share based on the DeGroot model. Table 2 summarizes the applications of opinion dynamics in marketing.

Table 2 The applications of opinion dynamics in marketing
Full size table

E-commerce

In this section, the BC model is a useful tool to study the opinion evolution of consumer in the e-commerce environment. Online consumer reviews and opinion leaders are two important factors that affect consumers' opinions on products. Wan et al. (2018) and Zhao et al. (2018b) used the BC model to study the fluence of the two factors in the e-commerce environment, respectively. Here we mainly introduce their use of the BC model in the e-commerce environment.

Wan et al. (2018) argued that a consumer will retain his/her opinion, but change it slightly based on the average of all other trustworthy reviews. Let \(\mu\) be the convergence parameter which indicates consumers’ level of trust in reviewers. Then, the evolution rule on an e-commerce platform is as follows:

$$o_{i}^{t + 1} = (1 - \mu )o_{i}^{t} + \mu (average\,of\,{\text{opinions}}\,{\text{in}}\,{\text{trustworthy}}\,{\text{reviews}})$$
(21)

where the trustworthy reviews are selected by:

$$|o_{i}^{t} - o_{r}^{t} | \le \varepsilon$$
(22)

where \(\varepsilon\) is the bounded confidence.

Zhao et al. (2018b) studied the influence power of opinion leaders in e-commerce networks, and divided the consumers into two groups: opinion leaders and followers. Zhao et al. (2018b) considered a social network composed of \(m\) agents, including \(m_{1}\) followers, \(m_{2}\) leaders with a positive target opinion, \(m_{2}\) leaders with a negative target opinion, and \(m_{1} + m_{2} + m_{3} = m\). Then, the updating rule of followers is below:

$$\begin{aligned} o_{i}^{{t + 1}} = & (1 - \alpha _{i} - \beta _{i} )\frac{1}{{m_{i}^{{F,t}} }}\sum\limits_{{j = 1}}^{{m_{1} }} {A_{{ij}}^{t} o_{i}^{t} } + \alpha _{i} \frac{1}{{m_{i}^{{P,t}} }}\sum\limits_{{j = m_{1} + 1}}^{{m_{1} + m_{2} }} {A_{{ij}}^{t} o_{j}^{t} } \\ &\quad + \beta _{i} \frac{1}{{m_{i}^{{N,t}} }}\sum\limits_{{j = m_{1} + m_{2} + 1}}^{m} {A_{{ij}}^{t} o_{j}^{t} } \quad i = 1, \ldots ,m_{1} \\ \end{aligned}$$
(23)

The updating rule of leaders with a positive target opinion is below:

$$o_{i}^{t + 1} = (1 - w_{i} )\frac{1}{{m_{i}^{P,t} }}\sum\limits_{{j = m_{1} + 1}}^{{m_{2} }} {A_{ij}^{t} o_{j}^{t} } + w_{i} d\quad i = m_{1} + 1, \ldots ,m_{1} + m_{2}$$
(24)

The updating rule of leaders with a negative target opinion is below:

$$o_{i}^{t + 1} = (1 - z_{i} )\frac{1}{{m_{i}^{N,t} }}\sum\limits_{{j = m_{1} + m_{2} + 1}}^{m} {A_{ij}^{t} o_{j}^{t} } + z_{i} g\quad i = m_{1} + m_{2} + 1, \ldots ,m$$
(25)

where

$$A_{ij}^{t} = \left\{ {\begin{array}{*{20}l} 1 \hfill & {||o_{i}^{t} - o_{j}^{t} || \le \varepsilon_{i} } \hfill \\ 0 \hfill & {otherwise} \hfill \\ \end{array} } \right.$$
(26)

and \(\varepsilon_{i}\) represents the bounded confidence of consumer \(a_{i}\); \(m_{i}^{F,t} = \sum\nolimits_{j = 1}^{{m_{1} }} {A_{ij}^{t} }\) is the number of neighbors of follower \(a_{i}\); \(m_{i}^{P,t} = \sum\nolimits_{{j = 1 + m_{1} }}^{{m_{2} }} {A_{ij}^{t} }\) and \(m_{i}^{N,t} = \sum\nolimits_{{j = 1 + m_{1} + m_{2} }}^{m} {A_{ij}^{t} }\) are the total number of opinion leaders of agent \(a_{i}\) who is from the positive and negative leader subgroups, respectively; \(\alpha_{i}\), \(\beta_{i}\), and \(1 - \alpha_{i} - \beta_{i}\) are the trust degrees; \(d \in [0,1]\) is the positive target opinion value; \(g \in [ - 1,0]\) is the negative target opinion value; and \(w_{i}\) and \(z_{i}\) are the influence weights. It is worth mentioning that \(A_{ij}^{t}\) does not directly evolve over time, but depends on the value of \(o_{i}^{t}\).

Applications of opinion dynamics models in GDM

GDM is an important research content of business, for example, the selection of suppliers is a matter of GDM. In the process of supplier selection, multiple members from the company constitute a decision-making committee. The decision-making committee needs to work together to select a suitable supplier. In GDM, consensus reaching is an important research direction (Li et al. 2019; Xu et al. 2020; Zhang et al. 2020). Dong et al. (2018a) summarized the use of opinion dynamics models in this field, and also in Ureña et al. (2019a). Here we mainly summarize the research results of the past two years to supplement these two literatures, as shown in Table 3.

Table 3 The application of opinion dynamics models in GDM
Full size table

In a rational and democratic society, "voting" is one of the two basic methods of GDM, usually used to make political decisions. For example, the decision-making of the UN Security Council is a matter of group decision-making. When the Security Council makes decisions, the five permanent members form a group, and they need to negotiate and discuss together to find a solution to a problem. The opinion dynamics models have been widely used in "political" decisions. Most of the models are the discrete opinion dynamics models, such as Sznajd model (González et al. 2004; Sznajd-Weron 2005), voter model (Yildiz et al. 2013; Pérez et al. 2015); majority rule model (Galam 1999, 2004, 2007; Galam and Jacobs 2007). It is worth to notice that the minority opinion spreading is important for the voting result, which explains why an initially minority opinion can become a majority in the long run (Galam 2002; Stauffer 2002b; Tessone et al. 2004; Kułakowski and Nawojczyk 2008). A summary of the applications of the opinion dynamics models in politics is shown in Table 4. In this section, we mainly introduce the use of the voter and majority rule models in politics because of the application details of other discrete models has been introduced in the previous sections.

Table 4 The application of opinion dynamics models in politics
Full size table
  1. (1)

    Application of the voter model

Yildiz et al. (2013) studied the voter model in a social network with Stubborn voters, and the basic use of the voter model in politics is described in the following. Let \(G(A,E)\) be a directed graph to represent the social network, where \(A = \{ a_{1} ,a_{2} , \ldots ,a_{m} \}\) is the set of voters and \(E\) is the set of edges representing the relationships among the voters. \((a_{i} ,a_{j} ) \in E\) is an edge from voter \(a_{i}\) to voter \(a_{j}\), and the set \(N_{i} = \{ a_{j} |(a_{i} ,a_{j} ) \in E\}\) is defined as the neighbor set of voter \(a_{i} \in A\). Assume that \(o_{i}^{t} \in \{ 0,1\}\) is the voting status of agent \(a_{i}\) at round \(t\) representing the selection of agent \(a_{i}\) among two candidates in the election. Then, the evolution rule is as follows:

  1. (a)

    One of the neighbors of the agent \(a_{i}\) is randomly and uniformly selected, i.e., agent \(a_{j}\).

  2. (b)

    Then, the voting status of agent \(a_{i}\) at round \(t + 1\) will be:

    $$o_{i}^{t + 1} = o_{j}^{t}$$
    (27)
  1. (2)

    Application of the majority rule model

Galam (1999) studied the majority rule model in hierarchical structures, and the basic use of the majority rule model in this paper is described as follows. Let \(A = \{ a_{1} ,a_{2} , \ldots ,a_{m} \}\) and \(o_{i}^{t} \in \{ 0,1\}\) be the same as before. Then, the evolution rule is below:

  1. (a)

    Randomly select \(r\) voters to form a cell.

  2. (b)

    In each cell, the voting status of agent \(a_{k}\) at round \(t + 1\) will be:

    $$o_{k}^{t + 1} = \left\{ {\begin{array}{*{20}c} {\begin{array}{*{20}c} 1 \\ {o_{k}^{t} } \\ 0 \\ \end{array} } & {\begin{array}{*{20}l} {if{{ \, \sum\nolimits_{i = 1}^{r} {o_{i}^{t} } } \mathord{\left/ {\vphantom {{ \, \sum\nolimits_{i = 1}^{r} {o_{i}^{t} } } r}} \right. \kern-\nulldelimiterspace} r} > 0.5} \hfill \\ {if{{ \, \sum\nolimits_{i = 1}^{r} {o_{i}^{t} } } \mathord{\left/ {\vphantom {{ \, \sum\nolimits_{i = 1}^{r} {o_{i}^{t} } } r}} \right. \kern-\nulldelimiterspace} r} = 0.5} \hfill \\ {otherwise} \hfill \\ \end{array} } \\ \end{array} } \right.$$
    (28)

Summary, critical discussions and new directions

Opinion dynamics is a useful tool to model the diffusion and opinion evolution among a group of interactive agents. In opinion dynamics models, there are three key elements in general: opinion expression formats, evolution rules and opinion evolution environments. Due to different opinion expression formats, the opinion dynamics models with different evolution rules can be divided into continuous and discrete categories. Opinion dynamics mainly studies the mutual influence of opinions among agents by exchanging information and the evolution of opinion to form a consensus, fragmentation or polarization phenomenon. Based on this feature, the opinion dynamics models have been applied to the fields of finance and business with different opinion evolution environments, such as marketing, finance, e-commerce, politics, and GDM. Although opinion dynamics models have been studied in many aspects in finance and business, which can help us understand the rules and important factors of opinion evolution in different scenarios, there are still some limitations that need to be paid attention to:

  1. 1

    Most opinion dynamics researches mainly follows the methods and tools of the statistical physics-based models, and the data sets used usually employ random data in the simulation (Dong et al. 2016, 2017, 2018b; Ding et al 2017, 2019). Few studies (Das et al. 2014; De et al. 2014, 2016; Kulkarni et al. 2017) have used real data to drive opinion dynamics processes.

  2. 2

    Although the opinion dynamics model has been studied in many aspects in the finance and business, it is still focused on the characteristics of the opinion dynamics model. The combination with the characteristics of its application field is still insufficient.

  3. 3

    In opinion dynamics, there are mainly two kinds of opinion expression forms, discrete and continuous, which are mainly expressed in the form of numbers (Martins et al. 2009; Maghami and Sukthankar 2012, 2013; Luo et al. 2014; Liang et al. 2016). In the actual opinion expression, the linguistic term is very common, for example, word-of-mouth, and advertising.

  4. 4

    Network is a powerful method for modeling and studying various complex phenomena, and collective phenomena emerges from the interactions between dynamical processes in multiplex networks (Nicosia et al. 2017). However, in the applications, the social network generally is assumed to be static during the interaction (Kaizoji 2000; Sznajd-Weron and Weron 2003; Sznajd-Weron 2005; Varma et al. 2018).

Thus, future research on this topic can follow the directions below:

  1. 1

    Using real data to study opinion dynamics and developing a real data driven opinion dynamics model will be an interesting topic. Meanwhile, studying data-driven opinion dynamics models in finance and business will be interesting.

  2. 2

    It would be interesting to consider opinion dynamics from the point of view of mathematical finance works more oriented to the “continuous time” treatment of technical topics.

  3. 3

    There will be cost/resource constraints associated with opinion changes in product selection, it will be an interesting research direction to use the opinion dynamics models to study marketing strategy with minimum cost.

  4. 4

    The Ising model is a basic model to simulate the stock market (Kaizoji 2000; Johansen et al. 2000; Bornholdt 2001; Kaizoji et al. 2002; Eckrot et al 2016; Takaishi 2017), and the strategies of trader's buy and sale are diverse. Therefore, it would be interesting to use the Ising model to study financial market from the perspective of individual personalization.

  5. 5

    In marketing, word-of-mouth and advertising are presented in the form of linguistic term. It is necessary to study the opinion dynamics of agents with different forms of opinion expression, especially the interaction using linguistic term. And this will make research on the effects of word-of-mouth and advertising more realistic in marketing.

  6. 6

    Complex networks [e.g. ER random network (Erdős and Rényi 1960), SW network (Newman and Watts 1999), and SF network (Barabasi and Albert 1999)] have been extensively studied in opinion dynamics. However, research on the impact of the dynamic network in the existing literature is still insufficient. The Hopfield neural network model can express homogenous attraction and heterogeneous repulsion (Li and Tang 2013). Therefore, using this model to study group dynamics such as global differentiation and local convergence in financial markets is a promising research direction.

  7. 7

    The application of opinion evolution in GDM and e-commerce is beginning to be recognized (Dong et al. 2017, 2018a, 2018b, 2020; Wan et al. 2018; Zhao et al. 2016, 2018a, b; Liang et al. 2020), and it is still necessary to further develop the theoretical basis for in-depth interdisciplinary integration research.

Conclusions

This paper reviews the application of the opinion dynamics models in finance and business. Firstly, we introduce some basic opinion dynamics models, including the DeGroot model, the BC model, the Snajzd model, and the voter model. Then, we review the use of opinion dynamics in different aspects of finance and business, including marketing, finance, e-commerce, politics, and group decision making. In the end, we note some limitations of the existing studies that need to be paid attention to and suggest several new directions for future research.