# Novel Mathematical Framework Explores How Opinions and Decisions Coevolve

The dynamics of opinion formation and of collective decision-making are closely related. Exchanging opinions with peers affects what decisions people make, and the decisions they see other people make in turn affects their opinions. However, most of the existing studies on these two processes have not considered their interconnection.

Figure 1. A network, in which the circles at each node represent the status of that individual. The inner circle shows the action that the individual takes, which either corresponds to +1 (blue) or -1 (red). The outer circle shows the individualâ€™s opinion, with the color representing where in the range from -1 to +1 their opinion falls.
Ming Cao of the University of Groningen was interested in mathematically quantifying how the dynamics of opinions and decisions interact. “Can we build these two behaviors in the same model, and can we derive some analytical results that gives us new understanding?” Cao asked during a minisymposium presentation at the SIAM Conference on Control and Its Applications, which took place virtually this week concurrently with the SIAM Annual Meeting. He presented a mathematical model for the coevolution of opinions and decision-making in complex social networks, which he developed in collaboration with Lorenzo Zino (University of Groningen) and Mengbin Ye (Curtin University).

“Mathematical models have played a key role in understanding complex social dynamics, in particular interactive opinion-forming and decision-making processes,” Cao explained. Opinion formation models are typically based on linear and nonlinear systems theory, whereas decision-making models often draw on game theory—particularly evolutionary game theory—as that framework does well at representing individuals who act based on self-interest.

Drawing on the established paradigm of mathematical models for social dynamics, Cao and his collaborators developed their own model to connect opinion formation and decision-making. Their new model depicts a population of a certain number of individuals who are connected through an undirected weighted network. Each person in the network decides to take a certain action—in this model, there are only two possible actions to choose between—and have an opinion related to that action.

Figure 2. The flow of information in the co-evolutionary scheme, demonstrating the effects of both opinion and decision-making dynamics on the overall system.
The model represents the two possible actions with the numerical values +1 and -1; the related opinion can then be any number in the range from -1 to +1. Figure 1 depicts this network, with the color-coding of the inner and outer circles at each node on the network representing the action and opinion of each individual, respectively. At each time step of the model, a single individual revises their action and their opinion. The question then is how are those actions and opinions updating over time?

The decision-making calculations in the model determine which action each individual will take using a best response rule, which is based off of the payoff that the individual will receive for taking that action. The payoff function includes mathematical terms for an individual’s own opinion, their commitment to that opinion—i.e., how much of their action is influenced by their own conviction—and the evolutionary advantage of the action. The equation also gives a higher payoff when the individual coordinates with neighbors who are also taking same action, thereby accounting for the influence of other network members’ opinions on the decision-making process.

Simultaneous to the calculation of an individual’s new decision, the model uses an opinion dynamics framework to revise their opinion based on a combination of several factors. “A weighted average takes into account opinions exchanged, actions observed, and external influence,” Cao said. “This is how the actions are affecting opinions.” Observing how others make decisions can change an individual’s own opinions, as can external influences such as the government and media. The block diagram in Figure 2 demonstrates the flow of information in the model, showing how the processes of opinion dynamics and decision-making take effect.

Figure 3. Examples of simulations produced by the coevolution model for opinions and decision-making. 3a. Both the average action and the average opinion steady out at negative values, showing consistency between the two processes. 3b. Though the average action is -1, the average opinion is a positive value. This is an example of an unpopular social norm.
Cao and his collaborators used this model to simulate several interesting scenarios (see Figure 3). In the situation shown in Figure 3a, the average action in the network levels out at about -0.5 and the average opinion levels out at about -0.2, so there is a fair amount of consistency between the two processes — people’s opinions line up with their actions. But in the simulation shown in Figure 3b, even though everyone takes the -1 action, the average opinion is a positive value. “In other words, the actions and the opinions in the network are opposite from each other,” Cao said. This reflects the sociological phenomenon of the unpopular social norm, giving a possible mathematical explanation for the real-world phenomenon in which people collectively behave in a manner that is against their own interests.

Since this model is grounded in the theory of opinion dynamics and evolutionary games, Cao was able to establish and provide examples of analytical results using tools from linear systems and control theory. One example that he presented used a pure configuration, in which all of the individuals take the same action. Explicitly calculating how unpopular an action can become before the individuals in this configuration stop deciding to take it adds qualitative insight to the phenomenon of unpopular norms.

But how can a norm be changed? Cao considered a scenario in which the individuals in a network start from an established status quo at -1 for both their opinions and actions, but a stubborn individual “innovator” tries to steer the entire population to +1. Figure 4 shows three possible outcomes after introducing the innovator to the model. There might be an unpopular status quo, in which the average opinion of the population shifts to the positive side while the action remains at -1. The status quo may also stay popular, with the average opinion remaining close to the actions at -1. But third, there might be a paradigm shift: starting from the unpopular norm at -1, a phase transition occurs as people start to take the +1 action.

Figure 4. Three possible outcomes after introducing an innovator to the coevolutionary model: an unpopular status quo, a popular status quo, and a paradigm shift.

Overall, the coevolutionary model of opinions and actions on networks was able to reproduce real-life sociological phenomena, while the analytical tool also revealed how the model parameters affected the emergent behavior. These mathematical techniques also have a broader possibility of applications, such as in epidemiology or marketing. “Coevolution can be the key to understanding realistic phenomena, especially the emergence of unpopular norms and paradigm shifts,” Cao said.

 Jillian Kunze is the associate editor of SIAM News.