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

TL;DR

We address the maximum displacement problem in Top-$k$ multiwinner elections under positional scoring rules: given a coalition of size $m$, how many current top-$k$ winners can be displaced? The core approach decomposes manipulation into two independent feasibility problems—boosting the strongest outsiders via the top-$k'$ positions and suppressing the weakest incumbents via the bottom-$k'$ positions—and encapsulates the coalition’s continuous score-feasibility by the Block--HLP envelope, a Minkowski-sum of ballot permutahedra described by prefix-sum inequalities. For the broad AP ladder family with a common step $g$, we obtain an exact discrete characterization: an integer aggregate is realizable if and only if it satisfies the Block--HLP prefix constraints plus a single congruence condition modulo $g$. This enables a polynomial-time oracle to test feasibility and a dual-binary-search algorithm to compute the maximum displacement $k^*$. Experiments on Mallows and PrefLib data confirm exact cutoffs and demonstrate the congruence effects for $g>1$, while large-scale experiments show near-linear scalability in the number of candidates and independence from coalition size. The framework provides a canonical, geometry-driven understanding of coalition power, with practical implications for manipulation auditing and the design of robust multiagent systems in voting and beyond.

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).

Figures (7)

• Figure 1: Majorization vs. Lattice Realizability. Each panel shows a two-dimensional projection of the Block--HLP polytope $\mathcal{P}_{\mathrm{conv}}$ (blue) for $m=3$ and $k'=3$ onto the prefix-sum coordinates $(y_1,\; y_1+y_2)$, together with integer points in the projected region. Both figures are plotted using the same coordinate range to facilitate direct visual comparison. (a) Unit-step AP ladder ($g=1$). For the unit-step ladder $r=(6,5,4)$, the congruence constraint of the AP--Ladder Lattice Theorem is vacuous. Consequently, every integer point satisfying the Block--HLP prefix inequalities is realizable, and the discrete feasible set (red) fills the convex envelope. (b) Sparse-step AP ladder ($g=3$). For the common-step ladder $r=(8,5,2)$ with step size $g=3$, realizable aggregates are restricted to a single residue class modulo 3. Accordingly, only a strict sublattice of integer points (red) is realizable, while other interior points (grey) satisfy all Block--HLP prefix inequalities yet violate the congruence condition. Together, the panels illustrate that Block--HLP majorization determines the continuous feasibility envelope, while arithmetic step size imposes an additional modular constraint on integer realizability.
• Figure 2: Maximal displacement $k^\ast$ as a function of coalition size. (a) Synthetic Mallows profiles (Borda; $n=1000$; $x=500$; $k=250$), averaged over 50 trials for each dispersion parameter $\phi$ (see legend). (b) Real PrefLib elections (Glasgow, Irish, SUSHI), plotting $k^\ast$ against coalition fraction $m/n$ up to $20\%$ (with $k=\min\{10,\lfloor x/2\rfloor\}$ for cross-dataset comparability).
• Figure 3: Boost (blue) and suppress (orange) cutoff envelopes for a synthetic election with $x=100$ candidates and $m=20$ colluding voters. For each displacement level $k'$, the blue curve shows the largest boost-feasible cutoff $B_{\max}(k')$, while the orange curve shows the smallest suppress-feasible cutoff $B_{\min}(k')$. Displacement at level $k'$ is feasible exactly when $B_{\min}(k') \le B_{\max}(k')$, i.e., over the range of $k'$ where the two envelopes overlap. The total number of feasible displacement levels coincides with the value $k^\ast$ returned by MaximizeDisplacement.
• Figure 4: Scalability to large-scale elections. Runtime of the maximum-achievable-displacement oracle (computing $k^\ast$) as a function of the number of candidates $x$ (log--log scale), ranging from $10^{3}$ up to $10^{9}$. Results are shown for five representative coalition sizes ($m=10^{3}, 10^{4}, 10^{5}, 10^{6}, 10^{7}$), whose curves nearly overlap, demonstrating that runtime is effectively independent of the coalition size. The observed approximately linear scaling in $x$ reflects the cost of scanning the score array and extracting the boundary-relevant top and bottom segments via linear-time selection, while the Block--HLP feasibility checks and dual binary search contribute only negligible overhead.
• Figure 5: Comparison with heuristic baselines. Synthetic Mallows elections with $x=400$ candidates and Borda scoring, averaged over 50 i.i.d. instances per coalition size. Shaded regions indicate 95% confidence intervals.

Theorems & Definitions (34)

• Lemma 1: Canonical Manipulation
• proof : Proof sketch
• Lemma 2: Independent Feasibility
• proof : Proof sketch
• Theorem 1: Guaranteed displacement at level $k'$
• proof : Proof sketch
• Theorem 2: Impossibility when independent feasibility fails
• proof : Proof sketch
• Corollary 1: Feasibility characterization
• Lemma 3: Monotonicity of boost and suppress feasibility