How much influence can a coordinated coalition exert in a multiwinner Top-$k$ election under a positional scoring rule? We study the maximum displacement problem: with coalition size $m$, how many of the current top-$k$ winners can be forced out? We show coalition power decomposes into two independent prefix-majorization constraints, capturing how much the coalition can (i) boost outsiders and (ii) suppress weak winners. For arbitrary scoring rules these prefix inequalities are tight, efficiently checkable necessary conditions (exact in the continuous relaxation). For common-step arithmetic-progression (AP) score ladders, including Borda, truncated Borda, $k$-approval/$k$-veto, plurality, and multi-level rules such as $3$--$2$--$1$, we prove a Majorization--Lattice Theorem: feasible aggregate score vectors are exactly the integer points satisfying the Block--HLP prefix-sum capacity constraints plus a single global congruence condition modulo the step size $g$. For Borda ($g=1$) the congruence vanishes, yielding a pure prefix-majorization test. This characterization yields an $O(k'\log k')$ exact feasibility oracle for displacing $k'$ winners, and an $O(k(\log k)^2\log(mx))$ algorithm (via dual-envelope binary search) for computing the maximum achievable displacement $k^\ast$. Experiments on Mallows profiles and PrefLib elections confirm exact cutoffs, diminishing returns, and substantial gains over baseline heuristics; for $g>1$ they also demonstrate the predicted congruence effect, where prefix-only tests produce false positives. The oracle scales to extreme instances, processing $10^9$ candidates in under 28 seconds (memory permitting).

翻译：在一个基于位置计分规则的多席位Top-$k$选举中，一个协调的联盟能施加多大的影响力？我们研究最大席位替换问题：给定联盟规模为$m$，当前前$k$名获胜者中最多能被挤出多少位？我们证明联盟权力可分解为两个独立的前缀优超约束，分别刻画了联盟能够（i）提升外部候选人和（ii）压制弱势获胜者的程度。对于任意计分规则，这些前缀不等式是紧的、可高效验证的必要条件（在连续松弛下是精确的）。对于常见的等差（AP）计分阶梯，包括Borda、截断Borda、$k$-认可/$k$-否决、多数制，以及多层级规则如$3$--$2$--$1$，我们证明了一个优超-格定理：可行的总得分向量恰好是满足Block-HLP前缀和容量约束加上一个关于步长$g$的全局同余条件的整数点。对于Borda（$g=1$），同余条件消失，从而得到一个纯粹的前缀优超检验。这一特征刻画产生了一个$O(k'\log k')$的精确可行性预言机，用于判断能否替换$k'$个获胜者，以及一个$O(k(\log k)^2\log(mx))$的算法（通过双包络二分搜索）用于计算最大可实现的替换席位数量$k^\ast$。在Mallows偏好分布和PrefLib选举数据上的实验证实了精确的临界点、收益递减现象，以及相对于基线启发式方法的显著提升；对于$g>1$的情况，实验还展示了预测的同余效应，即仅基于前缀的检验会产生误报。该预言机可扩展至极端实例，在内存允许的情况下，能在28秒内处理$10^9$名候选人。