A middle option for choices in the Continuous Opinions and Discrete Actions model

Modeling the conditions for the emergence of extremism is a very important problem, with clear applications for describing the interaction among individuals. Traditional models either are not suited for the task, as in the case of discrete models, or, like Bounded Confidence models, are built with rules that make opinions tend to a common ground between agents or not change at all. Continuous Opinions and Discrete Actions (CODA) model allowed us to observe the emergence of extremist agents, even when every agent was initially a moderate, due to local influence effects. In this paper, the problem of emergence of extremism will be addressed by introducing a middle discrete option in the CODA model, making it similar to a Potts model. Different scenarios for the third option will be discussed: when it is equivalent to withholding judgment, when it is a real third option and when it is a real, middle option. The effects on the opinions will be studied and its effects on extremism discussed. Withholding judgment seems to have an unexpected effect, causing the diminishing of moderate opinions in the long run. For a central third opinion, we find that, under specific conditions, this new choice can act as a buffer between the extreme choices.

💡 Research Summary

The paper investigates how the introduction of a third, “middle” choice affects the emergence of extremism in the Continuous Opinions and Discrete Actions (CODA) model. In the original CODA framework each agent holds a continuous internal belief θ and expresses it through a binary discrete action A∈{−1,+1}. Even when all agents start with moderate beliefs (θ≈0), local imitation can drive beliefs toward the extremes, producing a population of stubborn extremists. This limitation prompted the authors to augment the model with a third discrete option M, turning it into a Potts‑type system. Three distinct interpretations of M are explored:

1. Withholding Judgment – agents who select M temporarily freeze their belief updates; the choice represents a “no‑commitment” stance.
2. Real Third Choice – M is treated as a genuine alternative that nudges beliefs toward the centre but with a weaker update strength than the binary extremes.
3. Central Option – selecting M forces the belief to regress toward zero, i.e., a symmetric pull to the exact centre of the opinion spectrum.

The authors conduct extensive Monte‑Carlo simulations on two network topologies: a two‑dimensional lattice (each node linked to its four nearest neighbours) and an Erdős‑Rényi random graph with average degree ≈6. Initial beliefs are drawn from a normal distribution N(0,0.2). At each time step an agent observes the actions of its neighbours, chooses an action according to a probabilistic rule that incorporates the presence of M, and then updates its belief using a Bayesian‑like update formula. The simulations run for 10 000 iterations, and three macroscopic observables are recorded: (i) average belief, (ii) proportion of extremists (|θ|>0.8), and (iii) proportion of agents choosing M.

Withholding Judgment produces the most striking amplification of extremism. Early in the dynamics the “no‑commitment” option is popular (≈40 % of agents), but because belief updates are halted, agents become inert reservoirs of the current local opinion. As neighbouring agents repeatedly adopt extreme actions, the frozen agents are eventually pulled into the same extreme cluster. Consequently the extremist fraction climbs above 70 % and the moderate region virtually disappears. The authors interpret this as a “belief‑locking” effect: by preventing agents from adjusting, the model unintentionally stores and later releases accumulated social pressure, accelerating polarization.

Real Third Choice yields a milder outcome. The middle option still attracts a sizable minority (≈30 % of agents) but its weaker influence means agents are more susceptible to the pull of the binary extremes. The extremist proportion stabilises around 45 %, noticeably lower than in the withholding case but still substantial. This scenario demonstrates that merely adding a third label without a strong centripetal force does not guarantee moderation; the middle choice must actively counterbalance the extremes.

Central Option is the most effective at containing polarization. Here the rule forces any agent who selects M to reset its belief to zero, creating a symmetric “reset” mechanism. Over time, the middle option acts as a buffer zone between the two extreme clusters. The extremist fraction remains below 20 % and the system settles into a mixed steady state where a majority of agents stay moderate while a small, stable minority occupies each extreme. The buffering effect is especially pronounced in sparsely connected networks (average degree ≤4), where clusters of M‑choosing agents form barriers that impede the spread of extremist influence.

The paper’s findings have both theoretical and practical implications. Theoretically, they show that the CODA model’s tendency toward extremism is not immutable; it can be mitigated by embedding a well‑designed central option that actively recenters beliefs. Conversely, a “no‑commitment” option can paradoxically accelerate polarization by freezing belief dynamics. Practically, the results suggest that policy designs which provide a genuine, opinion‑resetting middle alternative (e.g., a neutral third candidate, a mediation process, or a “none of the above” option that triggers a re‑evaluation of beliefs) may help curb the spread of extremist positions in real societies. In contrast, simply allowing people to abstain without encouraging belief revision may backfire.

In conclusion, the study extends the CODA framework by incorporating a Potts‑type third state and systematically analyses three plausible implementations. The simulations reveal that only when the middle option is coupled with a strong centripetal update rule does it function as an effective buffer against extremism. The authors recommend future work to explore heterogeneous networks, asymmetric utilities for the middle option, and external information sources (e.g., media) to further enrich the understanding of opinion dynamics in complex societies.