The Geometry of Coalition Power: Majorization, Lattices, and Displacement in Multiwinner Elections

TL;DR

This paper analyzes the influence of coalitions in multiwinner elections, providing a mathematical framework to determine maximum displacement of winners under various scoring rules, with efficient algorithms and experimental validation.

Contribution

It introduces a novel decomposition of coalition power into prefix-majorization constraints and characterizes feasible score vectors for common scoring rules, enabling efficient computation of maximum displacement.

Findings

Exact feasibility oracle for displacing winners

Efficient algorithms for maximum displacement computation

Experimental validation on large-scale election data

Abstract

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).