Modeling biases in binary decision-making within the generalized nonlinear q-voter model

Collective decision-making is a process by which a group of individuals determines a shared outcome that shapes societal dynamics; from innovation diffusion to organizational choices. A common approach to model these processes is using binary dynamics, where the choices are reduced to two alternatives. One of the most popular models in this context is the $q$-voter model, which assumes that opinion changes are driven by peer pressure from a unanimous group. However, real-world decisions are also shaped by prior personal choices and external influences, such as mass media, which introduce biases that can favor certain options over others. To address this, we propose a generalized $q$-voter model that incorporates these biases. In our model, when the influence group is not unanimous, the probability that an individual changes its opinion depends on its current state, breaking the symmetry between opinions. In limiting cases, our model recovers both the original $q$-voter model and several recently introduced modifications of the $q$-voter model, while extending the framework to capture a broader range of scenarios. We analyze the model on a complete graph using analytical methods and Monte Carlo simulations. Our results highlight two key findings: (1) for larger influence groups ($q>3$), a phase emerges where both adopted and partially adopted states coexist, (2) in small systems, greater initial support for an opinion does not necessarily increase its likelihood of widespread adoption, as reflected in the unique form of the exit probability. These results point to one of the key issues in social science, the importance of group size in collective action.