We introduce the incremental voter model (IVM), a discrete-opinion multi-agent system where agents undergo step-wise transitions biased by the opinion of a randomly selected persuader. Our incremental voter model comprises a large population of interacting agents, each holding an opinion represented by an element of the discrete set $\{-k,\ldots,0,\ldots,k\}, k \in \mathbb{N}_{+}$. At each update step as time progresses, a pair of distinct agents are selected independently and uniformly at random from the population, and the first agent (viewed as the ``listener'') updates its opinion based on that of the second (viewed as the ``persuader''), adopting a new opinion that differs from its current one by at most one unit. By deriving the mean-field system of nonlinear ordinary differential equations (ODEs) that governs the large-population limit of the agent-based model, we develop a rigorous mathematical framework to study the asymptotic behavior of the opinion distribution in the mean-field limit. These results contribute to a deeper understanding of social influence processes in complex systems, particularly in modeling opinion polarization, and may guide the formulation of more advanced models in future research.

翻译：我们引入增量投票者模型（IVM），这是一种离散意见的多智能体系统，其中智能体受随机选择的劝说者意见影响，进行逐步转变。我们的增量投票者模型包含大量交互智能体，每个智能体持有离散集合 $\{-k,\ldots,0,\ldots,k\}$（其中 $k \in \mathbb{N}_{+}$）中元素表示的意见。随着时间推移，在每个更新步骤中，独立且均匀随机地从总体中选择一对不同的智能体，第一个智能体（视为“倾听者”）基于第二个智能体（视为“劝说者”）的意见更新其意见，采用的新意见与当前意见相差最多一个单位。通过推导控制智能体模型大总体极限的非线性常微分方程（ODE）平均场系统，我们建立了一个严格的数学框架，用于研究平均场极限下意见分布的渐近行为。这些结果有助于深入理解复杂系统中的社会影响过程，特别是在意见极化建模方面，并可能为未来研究中更先进模型的制定提供指导。