Stable Voting Mechanisms

• Stable voting is a framework that defines stability through core and λ-approximate concepts, ensuring committees resist blocking coalitions in approval-based elections.
• It incorporates noise-stability measures and refined ranked-choice methods like SV and SSV to mitigate perturbations and resolve cycles in voting results.
• The framework extends to meta-stability and power indices, offering insights into self-equivalence and minimal-stable rules that inform robust institutional designs.

Stable voting encompasses a set of rigorous concepts, results, and voting rule designs focused on the resilience and robustness of election outcomes to individual, group, or systematic perturbations. It arises in social choice theory in several distinct but interrelated forms: as core or approximate-core stability in group decision/committee selection, as resilience to noise or adversarial manipulation in aggregation functions, and as a self-referential meta-criterion for the stability of voting rules under their own potential replacement. Stable voting thus unifies advances in core stability, noise-stability analysis, meta-stable constitutions, and robust randomized committee rules, situating itself as a cornerstone for both theoretical and applied mechanism design.

1. Core and Approximate Stability in Committee Selection

Core stability in approval-based committee (ABC) elections demands that no alternative committee $T$ of any size $|T|$ can be strictly preferred by a proportion $(|T|/K)\cdot n$ or more of the voters, given a selected committee $S$ of fixed size $K$—formalizing resistance to blocking coalitions. However, the core may be empty, even with simple approval ballots. To address this, recent work introduces $\lambda$-approximate stability: for $\lambda\geq 1$, a committee $S$ is $\lambda$-approximately-stable if, for every $T\subseteq C$, strictly fewer than $(\lambda|T|/K)\cdot n$ voters strictly prefer $T$ to $S$. When $\lambda=1$, this recovers exact core-stability.

Gao, Sun, and Vondrák establish that for any ABC instance and any $K$, there exists a $3.651$-approximately-stable committee, constructible in randomized polynomial time via a Lindahl equilibrium and strongly Rayleigh rounding to bound voter losses in the rounding process at each recursion level (Gao et al., 31 Jul 2025). This algorithm achieves stability through a sequence of: (1) finding a fractional Lindahl equilibrium (yielding a fractional, core-stable certificate), (2) rounding via a strongly Rayleigh distribution, and (3) a greedy cleanup step that recursively completes committee selection. The method extends and improves prior work that only guaranteed existence at $\lambda=16$ for general monotonic preferences and establishes efficiently computable, tight approximate guarantees.

In the more general, ordinal, and weighted setting with monotonic preferences over all committees, Jain, Munagala, and Wang show that a lottery over committees always exists with $2$-approximate stability, and that iterative rounding yields a deterministic committee with $32$-approximate stability. The approach harnesses linear programming duality and dependent rounding, albeit with computational tractability only in preference classes amenable to polynomial support (Jiang et al., 2019).

Table: Core vs. Approximate Stability in Committee Voting

Stability Type	Definition	Algorithmic Guarantee
Core (λ = 1)	No $T$: $|\{v : T \succ_v S\}| \geq \frac{|T|}{K} n$	Non-emptiness open for large $K$
λ-approximate	No $T$ with $|\{v : T \succ_v S\}| \geq \frac{\lambda|T|}{K} n$	Efficient for $\lambda=3.651$

2. Noise Stability and Robustness to Perturbations

Noise-stability, initially developed in the Boolean function and social choice literature, quantifies the probability that the winner remains unchanged under random, independent perturbations of ballots. For two-candidate elections, the majority function is stablest among all balanced, low-influence aggregation rules in the presence of random noise—a result formalized by Mossel–O’Donnell–Oleszkiewicz. In the multi-candidate setting ($k\geq3$), the natural analogue is the “Plurality is Stablest” Conjecture: among all rules with small individual influences (i.e., no voter with decisive power), plurality uniquely maximizes noise stability.

This conjecture is resolved for three-candidate elections and correlation $\rho \in (-1/43, 1/10)$: plurality uniquely achieves maximal noise stability in this regime (Heilman, 2023). The proof leverages a reduction to Gaussian space partitions—specifically the Standard Simplex Conjecture, demonstrating that the simplex cone partition is optimal for stability—and transfers the extremal property back via an invariance principle.

Furthermore, in ranked-choice voting, the Borda count is conjectured (and, for $k=3$ and small correlations, proven) to be stablest among low-influence voting methods satisfying the Condorcet Loser Criterion (Heilman, 2022). The theoretical reductions again rely on viewing pairwise majority comparisons as a lift to the k=3 case, and on dimension-reduction arguments in Gaussian space.

3. Self-Equivalence, Meta-Stability, and Rule Replacement

Stability at the meta-level arises when considering a society's resistance to replacing its own voting rule under its own decision procedure. The self-equivalence (or “stable” voting rule) axiom, introduced in (Hermida-Rivera, 18 Jun 2025), requires that if the society contemplates switching to a convex combination (i.e., lottery) over deterministic, neutral voting rules, then, when the current rule itself is applied to that set of rules—treated as alternatives—it reproduces the mixture. Under anonymity, neutrality, Pareto-optimality, and monotonicity, only the uniform random dictatorship satisfies self-equivalence. Thus, within this axiomatic frame, random dictatorship is uniquely meta-stable: all classical rules (plurality, Borda, Condorcet, IRV) fail this strong property.

Binary self-selectivity, a weaker variant considering only pairwise replacement, is shown equivalent to universal self-selectivity for neutral rules (Hermida-Rivera et al., 18 Jun 2025). Under unrestricted preference domains, dictatorial rules are the only stable choice. On the domain of profiles with a unique strong Condorcet winner, the only neutral, anonymous, unanimous, and self-selective (binary or universal) rule is majority.

Table: Meta-Stable Voting Rules under Key Axioms

Domain / Axioms	Only Stable Rule
Anonymity, neutrality, Pareto, mono	Random Dictatorship
Unanimity, neutrality, unrestricted	Dictatorship
Strong Condorcet domain	Majority

4. Stable Voting in Cycle Resolution and Single-Winner Elections

Stable Voting (SV), introduced by Holliday and Pacuit, is a resolute single-winner method on ranked ballots that recursively selects the candidate whose “claims” to victory persist when opponents are eliminated, subject to the Stability for Winners principle: if $A$ would win without $B$ and beats $B$ head-to-head, then $A$ should still win when $B$ is included, modulo tie-breaking among equally “claims-strong” candidates (Holliday et al., 2021). SV is a refinement of Split Cycle (SC), which itself sits among advanced path- and cycle-resolving Condorcet methods such as Ranked Pairs and Schulze Beat Path.

Simple Stable Voting (SSV) further simplifies the recursive procedure and, for up to six alternatives with all pairwise margins distinct, is proven to always select an SC-undefeated candidate; for seven or more, explicit SAT-found counterexamples exist (Holliday et al., 29 Nov 2025). SV/SSV provide practical, tie-minimizing, Condorcet-consistent single-winner rules with deep theoretical and computational ties to cycle-busting, and permit fine-grained SAT-based analysis of voting rule properties up to small candidate numbers.

5. Algorithmic and Probabilistic Stability in Committee Voting

Recent advances leverage randomized algorithms to ensure various robust forms of stability, continuity, and privacy in committee elections. The Softmax-GJCR rule provides a randomized extension of the Greedy Justified Candidate Rule, achieving ex-ante neutrality, monotonicity, $O(k^3/n)$-TV-stability, and ex-post EJR+ proportionality in polynomial time, with tighter bounds incurred by relaxing to approximate EJR+ or JR (Kehne et al., 23 Jun 2025). TV-stability guarantees that the distribution over committees changes only $O(k^3/n)$ in total variation distance when a single voter modifies their approvals within a bounded set. This stability property is equivalent to strong continuity in selection probabilities and, via standard differential privacy reductions, ensures robust privacy guarantees as well.

Furthermore, TV-stable rules yield low-recourse dynamic voting algorithms—i.e., in dynamic settings where voters update their ballots, the expected number of committee changes (recourse) is bounded by $O(k^4/n)$ per time step. Exponential-time sampling from PAV under a softmax distribution further refines the trade-off, reaching $O(k^2/n)$-stability in favorable regimes.

6. Minimal Stable Voting Rules and Power Indices

When the analysis centers on power—rather than preference representation—the notion of stability pertains to the invariance of voting rules to potential Pareto improvements by coalitions formed on the basis of their status as winning, veto, or swing players. A minimal-stable voting rule is such that no minimal winning coalition can prefer any alternative rule via Pareto domination. Oligarchic rules (unique minimal winning coalitions) are always minimal-stable under the standard power-ordering axioms: non-dominance, anonymity, null player, and swing player. Constitutions (pairs of ordinary and extraordinary voting rules) are minimal-self-stable if and only if the swing set of the ordinary rule is contained in the oligarchic set of the amendment rule (Hermida-Rivera, 18 Jun 2025). This framework makes precise the link between real-world institutions with high group discipline (e.g., political parties as decisive actors) and minimal-stability at the constitutional level.

7. Open Problems and Future Directions

The domain of stable voting faces several outstanding challenges:

• Existence and efficient computation of exact core-stable committees in ABC elections for general $K$ remain unresolved (Gao et al., 31 Jul 2025).
• Achieving $\lambda$-approximate stability with $\lambda < 3$ faces fundamental tail-bound and fractional-core constraints.
• The computational complexity of deterministically stable (meta-stable) cycle-resolving rules for large alternative sets remains open.
• No deterministic, neutral rule meets the full self-equivalence criterion, and relaxations involving probabilistic or restricted-menu self-equivalence could yield richer rule catalogs.
• Dimension-reduction in noise-stability of ranked choice voting methods signals the possibility of low-dimensional, robust procedures, but the optimal stablest method under the Condorcet Winner Criterion remains unknown (Heilman, 2022).
• In probabilistic committee settings, closing the gap between JR-triggered continuity lower bounds and best-known upper bounds for stable randomized algorithms is an active area.

Stable voting thus bridges social choice, algorithmic game theory, and computational complexity, providing both rigorous theoretical foundations and practical robust mechanisms for group decision-making.