We study how electoral rules shape polarization dynamics when voters and candidates both adapt to repeated election outcomes. We introduce two geometric primitives for comparing rules under this feedback: the \emph{winner radius} $R_t = \max_i \|x_i - w^{(t)}\|$, the distance from the winner to the farthest voter, and the \emph{supporter centroid radius} $S_t = \max_j \|c_j - s_j^{(t)}\|$, the largest gap between any candidate and their support base. We show that $R_t$ controls a one-step contraction bound on voter disagreement and $S_t$ plays the analogous role for candidate dispersion, and that these two objectives are in tension. Rules that reduce $R_t$ tend to increase $S_t$, and vice versa. A winner close to the voter median does not resolve the tension, since proximity to the median and proximity to the Chebyshev center are different objectives. We use this framing to organize a simulation study across seven standard electoral rules and one convex-combination benchmark, comprising 1000+ runs across diverse electorate profiles, voter mechanisms, and camp-balance settings. The empirical results confirm the theoretical tradeoff: winner-take-all rules achieve small $S_t$ at the cost of large $R_t$ and weaker voter depolarization, while convex-combination rules reverse this. An oracle comparison further shows that minimizing $R_t$ per step and minimizing voter disagreement per step are distinct objectives with different long-run consequences for both voter and candidate dynamics.

翻译：我们研究了当选民和候选人双方都适应反复选举结果时，选举规则如何影响极化动态。我们引入两种几何基本量来比较这种反馈下的规则：胜选者半径 \(R_t = \max_i \|x_i - w^{(t)}\|\)（胜选者到最远选民的距离）和支持者质心半径 \(S_t = \max_j \|c_j - s_j^{(t)}\|\)（任一候选人与其支持基础的最大差距）。我们证明，\(R_t\) 控制着选民分歧的一步收缩界，而 \(S_t\) 对候选人分散度起类似作用，且这两个目标存在张力。减小 \(R_t\) 的规则往往倾向于增大 \(S_t\)，反之亦然。胜选者靠近选民中位数并不能解决这一张力，因为靠近中位数与靠近切比雪夫中心是不同的目标。我们利用这一框架，组织了涵盖七种标准选举规则和一个凸组合基准的模拟研究，涉及超过1000次运行，覆盖多样化的选民分布、选民机制和阵营平衡设置。实证结果证实了理论权衡：胜者全得规则实现了较小的 \(S_t\)，但代价是较大的 \(R_t\) 和较弱的选民去极化，而凸组合规则则相反。进一步的神谕比较表明，最小化每步 \(R_t\) 和最小化每步选民分歧是不同目标，对选民和候选人动态具有不同的长期影响。