The research question addresses how groups reach consensus without centralized authority, focusing on the resilience of decision-making processes to strongly opinionated minorities and individualistic behavior. The study's context is the existing literature on opinion dynamics, particularly the voter model and its limitations in handling inflexible individuals or asocial behavior. The purpose is to introduce and analyze a cross-inhibition model, comparing it with existing models to determine its effectiveness in achieving stable consensus under challenging conditions. The importance of this study lies in its potential to explain the prevalence of inhibitory signals in natural collective decision-making systems and to guide the design of robust decentralized artificial systems. The voter model, despite its simplicity and wide use across various fields, has been shown to be vulnerable to even a small number of inflexible individuals (zealots) or to frequent asocial opinion changes (noise), resulting in persistent indecision within the population. This limitation motivates the exploration of alternative models that can overcome these challenges. This research provides insights into the development of resilient and adaptable collective decision making in various contexts.

The authors review existing literature on opinion dynamics models, focusing on the voter model and its variants. The voter model, a simple and mathematically tractable model, describes consensus formation through pairwise interactions. However, studies have demonstrated the voter model's fragility in the presence of inflexible individuals (zealots) or asocial behavior (noise), leading to persistent disagreements. The authors highlight the mathematical equivalence between noise and zealotry at the macroscopic level, while emphasizing their distinct microscopic mechanisms. The review also touches upon the prevalence of inhibitory mechanisms in various biological systems that achieve coordinated actions despite noise and conflicting information, such as in honeybee swarms and neuronal populations. The paper points to the need for models that incorporate more realistic aspects of group interactions while maintaining mathematical tractability.

The research employed a multi-faceted approach combining theoretical modeling, mathematical analysis, and robotic experiments. The core of the study involves comparing three models: the voter model, the weighted voter model (for situations involving alternatives with differing qualities), and the cross-inhibition model. The cross-inhibition model distinguishes itself by incorporating an "undecided" state, where individuals encountering conflicting opinions become temporarily uninvolved before adopting a new opinion. A mean-field analysis using ordinary differential equations (ODEs) provided a macroscopic description of the dynamics of each model, predicting the likelihood of achieving consensus. The analysis focused on the impact of different levels of asocial behavior, implemented either as noise (random opinion switches) or zealotry (inflexible individuals). Furthermore, a detailed analysis was conducted using master equations to study the stochastic dynamics of finite-sized systems and to quantify the stability of the fixed points identified in the ODE analysis. Analytical solutions for the stationary probability distributions were derived for several scenarios. These theoretical analyses were validated through experiments using a swarm of 100 Kilobot robots, implementing both the voter model and the cross-inhibition model under different levels of zealot robots (simulating inflexible minorities). The experiments measured the swarm's ability to reach and maintain consensus given varying levels of asocial behavior. The accuracy and reward in the best-of-*n* problem were investigated by comparing the performance of the weighted voter model and the weighted cross-inhibition model in scenarios with options of varying qualities. The analysis was based on both numerical simulations and analytical calculations to evaluate the models' ability to select the best option, considering metrics like accuracy and reward.

The mean-field analysis of ODEs revealed that the cross-inhibition model predicts a stable group consensus (two stable fixed points) even with a significant proportion of zealots (less than 40%) or a high level of noise, unlike the voter model which falls into a state of permanent indecision with even minimal asocial behavior. Finite-sized system analysis using master equations and Gillespie's stochastic simulation algorithm confirmed the resilience of the cross-inhibition model against high levels of asocial behavior, maintaining a stable consensus for a wide range of parameters. The voter model, in contrast, transitioned from a regime of decision to indecision as noise or system size increased, demonstrating its fragility. Robot swarm experiments validated the theoretical findings. Swarms employing the voter model failed to reach agreement in the presence of even a small number of zealot robots. Swarms using the cross-inhibition model, however, rapidly and stably converged to a consensus, even with a substantial number of zealots. The study examined the trade-off between accuracy and reward in the best-of-*n* problem. The weighted voter model demonstrated higher accuracy (selecting the best option more frequently) but lower stability. Conversely, the cross-inhibition model showed greater stability in its consensus but occasionally locked into a suboptimal decision. The authors argue that in many real-world scenarios, a higher reward rate (overall benefit from the chosen option) outweighs the need for maximal accuracy. The cross-inhibition model, by prioritizing stability, often leads to a higher reward rate.

The findings address the research question by demonstrating that cross-inhibition is a robust mechanism for achieving consensus in the face of strongly opinionated minorities and asocial behavior. The significance lies in explaining the prevalence of inhibitory signals in biological systems and offering a design principle for resilient artificial systems. The results challenge the conventional understanding of consensus formation, highlighting the limitations of models that do not account for inhibitory mechanisms. The study's relevance extends to various fields, including biology, social sciences, and robotics, providing insights into the design of effective, decentralized systems that can function reliably even under noise and adverse conditions. The trade-off between accuracy and reward rate, as observed with the cross-inhibition model, provides an important consideration for decision-making in value-based contexts.

This study demonstrates that the cross-inhibition model offers a superior approach to consensus-building compared to voter models, particularly in noisy or adversarial environments. The key contribution is the successful application of cross-inhibition as a robust mechanism for achieving stable group decisions despite the presence of inflexible minorities and asocial behavior. Future research could explore the cross-inhibition model's performance under more complex interaction topologies, investigate its ability to handle a broader range of asocial behavior, and adapt the model to deal with situations involving more than two options.

The study's limitations primarily involve the simplifying assumptions inherent in the models used. The assumption of homogeneous populations, identical individual behavior, and a well-mixed interaction topology may not accurately capture the complexities of real-world systems. While the robot experiments provide valuable validation, the controlled nature of the experimental setup might not fully replicate the dynamism and variability of natural or complex social systems. Further research is needed to investigate the model's robustness under more realistic conditions.