Why consensus is not the same as being right: three opinion-dynamics models

DeGroot learning repeatedly averages trusted opinions

In a DeGroot model, each participant updates to a weighted average of existing opinions. The weights specify who influences whom and by how much. If the network and weights satisfy suitable conditions, the opinions can converge. The final value depends on the influence structure as well as the starting values; it need not equal their simple unweighted average.

Run the DeGroot-learning simulation with initial opinions zero, 0.4 and one. Under the ordinary displayed weights, the process approaches 0.45. DeGroot’s original paper presents the consensus framework. Toggle the first participant to stubborn and observe how a fixed opinion changes the path. The board’s opinion scale has no external truth marker, so convergence cannot be interpreted as improved accuracy.

Bounded confidence changes whom people will hear

Bounded-confidence models add a distance rule: an agent averages only with opinions sufficiently close to its own. A small confidence bound can keep distant groups from interacting. A larger bound may create bridges or a common cluster. The rule makes the interaction network depend on the current opinions, so changing the bound changes the mechanism itself.

In the bounded-confidence activity, eight agents update synchronously using an inclusive distance threshold and include their own opinion in the average. Hegselmann and Krause’s paper discusses this family of models. The toy agents have no identities, motivations or factual evidence. A cluster in the plot is an output of a selected update rule, not a measured account of why a particular community polarised.

Worked example: distinguish agreement from evidence

Use the fixed inputs before adding a real-world interpretation. These are constructed numerical demonstrations, not estimates of persuasion strength in a population.

1. Reset DeGroot learning and advance the ordinary-weight process. Record the initial opinions and the limiting value near 0.45. Note that the simple initial average is about 0.467, so influence weights matter.
2. Reset, make the zero-opinion agent stubborn and advance again. The other agents move toward zero through the connected influence network. Ask what evidence changed: in this model, none was introduced.
3. Open bounded confidence and compare a very small bound with a bound of one. The first can leave separated opinions; the second lets every agent hear everyone and moves to the mean. Neither setting includes a correct-answer variable, so neither alone establishes wiser beliefs.

The voter model copies instead of averaging

The voter-model game places binary opinions on an eight-person ring. At each step, a randomly selected person copies a randomly selected neighbor. There is no strength-of-argument calculation and no weighted compromise between zero and one. The stochastic sequence matters, so a different round can follow a different route even from the same starting split.

Finite systems of this type can eventually reach a common opinion under the model’s connected copying rules. That endpoint is not evidence that the surviving opinion was more accurate. To see why topology also matters, open the strength-of-weak-ties activity and add or remove a bridge between two groups. Reachability changes when the bridge changes, without establishing that every message crossing it is valuable.

Audit a group’s route to agreement

The useful application is a set of questions about a process you can observe, not a label attached to a group because its members agree.

• Identify the update rule: are participants averaging estimates, copying a source, exchanging reasons or responding to a new observation?
• Map influence as well as attendance. A person who speaks often, sets the agenda or cannot revise their position can have disproportionate impact.
• Check whether apparently separate sources share an origin. Independent evidence and multiple repetitions of one claim should not receive the same treatment.
• Look for missing connections and excluded perspectives, then ask whether adding a bridge improves access to evidence rather than simply accelerating repetition.

Make a real discussion easier to inspect

Before discussion, ask participants to write an estimate and the main observation supporting it. After discussion, record which observation changed and why. This creates a modest audit trail for learning without assuming that disagreement is always valuable or that a persistent minority must be right. The goal is to reveal how information affected a judgment.

The homophily network generator offers a related caution: denser within-group connection and sparser cross-group connection can arise from selected edge probabilities. In that toy model, the parameter changes expected connection counts as well as mixing. In a real network, shared opportunity, preference and institutional structure can overlap. A familiar visual pattern does not identify its cause.