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

2026-01-23

AI Summary

This study addresses the maximum displacement problem in multi-winner Top-k elections, namely, how many currently elected candidates a coordinated coalition of size m can displace from the winning set. By decomposing the coalition’s manipulation power into two prefix-majorization constraints—promoting outsiders and demoting weak winners—the work introduces the Majorization–Lattice Theorem, which reformulates the problem as a feasibility test involving prefix-sum constraints and modular congruence conditions over integer lattices. Leveraging techniques from prefix majorization, Hardy–Littlewood–Pólya (HLP) capacity constraints, modular congruence analysis, and dual-envelope binary search, the authors devise an exact verification algorithm running in O(k′ log k′) time and a maximum displacement algorithm with complexity O(k (log k)² log(mx)). The framework efficiently handles election instances with billions of candidates and demonstrates strong empirical validity on real-world election data.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Game Theory and Economic Paradigms: Social Choice / Voting
• Humans and AI: Voting

Application Category

• Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environments
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

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).

Problem

• multiwinner elections
• coalition power
• maximum displacement
• positional scoring rules
• majorization

Innovation

• majorization
• lattice
• multiwinner elections
• positional scoring rules
• displacement