Quasi-Unanimity in Voting and Aggregation

• Quasi-unanimity is a structural concept where exact consensus is relaxed through high thresholds or weakened identity requirements across various disciplines.
• It finds application in qualified-majority voting, social ranking, and distributed consensus to preserve the essence of unanimity amid practical deviations.
• Metrics such as distance-based scores and near-unanimity identities offer actionable insights for achieving almost-consensus in complex aggregation and decision processes.

Searching arXiv for papers on quasi-unanimity and related unanimity concepts. Quasi-unanimity denotes a family of weakened unanimity conditions rather than a single universally fixed concept. Across voting theory, social ranking, universal algebra, constraint satisfaction, distributed consensus, approval voting, and aggregation under ambiguity, the common idea is that exact unanimity is replaced by a high-threshold, restricted-domain, distance-based, or structurally weakened analogue. In two-alternative voting, it appears as a qualified-majority rule with quota $q>n/2$, with near-unanimity corresponding to $q=n-1$ and unanimity to $q=n$ (Hermida-Rivera, 24 Sep 2025). In social ranking, the closest formal notion is “closeness to unanimity,” defined by the minimum number of swaps needed to make an individual unanimously supported (Suzuki et al., 18 May 2026). In impartial rank aggregation, the relevant weakening is “weak unanimity,” which requires reproduction of a ranking only when all agents submit exactly the same full ranking (Cembrano et al., 2023). In universal algebra and CSP theory, the relevant language is “near-unanimity,” “quasi near-unanimity,” and “quasi weak near unanimity,” where unanimity-type identities are weakened by allowing diagonal values or by dropping idempotence (Campercholi, 2016); (Wrona, 2024); (Kazda, 2020). In distributed opinion dynamics, the closest operational analogue is stabilizing almost-consensus, where all but $O(\sqrt n)$ nodes eventually share one valid opinion (Becchetti et al., 2015). This diversity suggests that quasi-unanimity is best understood as a structural theme: unanimity relaxed just enough to remain informative in settings where literal unanimity is too strong, too rare, or formally inappropriate.

1. Qualified-majority and threshold interpretations

In voting theory, quasi-unanimity is most naturally formalized as a high-threshold version of majority rule. In the two-alternative environment $(N,A,\mathcal R,\sigma)$ with $A=\{s,r\}$, the relevant rule is the $q$-sized qualified majority $\sigma_q$, defined by

$(\forall R\in\mathcal R)\big[(\sigma_q(R)=r)\iff(n(r,R)\ge q)\big],$

where $q>n/2$ (Hermida-Rivera, 24 Sep 2025). Here $q=n-1$0 is the number of voters who strictly top-rank reform $q=n-1$1. Under this formulation, reform is selected exactly when at least $q=n-1$2 voters strictly top-rank it; otherwise the status quo $q=n-1$3 wins.

This threshold family contains the most direct social-choice analogue of quasi-unanimity. The paper explicitly identifies $q=n-1$4 as a near-unanimity or quasi-unanimity rule and $q=n-1$5 as unanimity (Hermida-Rivera, 24 Sep 2025). More generally, quotas $q=n-1$6 close to $q=n-1$7 implement “almost unanimity,” while $q=n-1$8 gives strict majority and $q=n-1$9 gives a two-thirds rule when integer-valued. The central result is that for $q=n$0, the unique anonymous, responsive, and $q=n$1-neutral voting rule is $q=n$2, expressed as

$q=n$3

This extends May’s characterization of simple majority by replacing full neutrality with $q=n$4-neutrality, a quota-indexed restricted neutrality axiom (Hermida-Rivera, 24 Sep 2025).

The interpretive significance is that quasi-unanimity is not treated as an ad hoc strengthening of majority rule. Rather, in this framework it is one member of a family of qualified-majority rules “precisely distinguished by their degree of neutrality” (Hermida-Rivera, 24 Sep 2025). As $q=n$5, the set of profiles on which neutrality is imposed becomes smaller. A plausible implication is that higher thresholds correspond to narrower domains of symmetry: near-unanimity is associated with a more selective neutrality requirement than simple majority.

A related threshold formulation appears in the theory of indecisive choice behavior by majoritarian ballots. There a quasi-choice is $q=n$6-majoritarian if

$q=n$7

for some finite family of ballots and some $q=n$8 (Alcantud et al., 2022). The paper does not define unanimity or quasi-unanimity as primitive concepts, but thresholds $q=n$9 close to $O(\sqrt n)$0 are the natural analogue of near-unanimous endorsement. At the same time, the framework formally excludes $O(\sqrt n)$1, so exact unanimity is not one of its primitive threshold cases (Alcantud et al., 2022).

2. Distance-from-unanimity and restricted-domain formulations

A second major interpretation replaces exact unanimity by measured proximity to a unanimity configuration. In social ranking, “closeness to unanimity” is the paper’s explicit near-unanimity principle. For a coalitional ranking $O(\sqrt n)$2, an individual $O(\sqrt n)$3 is unanimously supported if there exists $O(\sqrt n)$4 such that the better-ranked classes all contain $O(\sqrt n)$5 and the worse-ranked classes all exclude $O(\sqrt n)$6 (Suzuki et al., 18 May 2026). The distance to this configuration is the minimum number of swaps needed to make $O(\sqrt n)$7 unanimously supported, denoted $O(\sqrt n)$8, and on the linear domain given by

$O(\sqrt n)$9

The formal axiom, closeness to unanimity (CU), requires that on the linear symmetric domain $(N,A,\mathcal R,\sigma)$0, the social ranking compare $(N,A,\mathcal R,\sigma)$1 and $(N,A,\mathcal R,\sigma)$2 according to this inversion-number score (Suzuki et al., 18 May 2026).

This is weaker than a unanimity axiom in two senses. First, the target of measurement is an exact unanimity-like state, but the axiom ranks by nearness to that state rather than by achievement of it. Second, the requirement is restricted to the linear symmetric domain. The paper explicitly states that “quasi-unanimity” does not appear as a term, and that the faithful counterpart is CU (Suzuki et al., 18 May 2026). Theorem 1 shows that on the linear symmetric domain, several Borda-like scores coincide with the inversion-number criterion, and Theorem 2 characterizes the preferred Borda-type social ranking rule $(N,A,\mathcal R,\sigma)$3 by ECON, DCONT, NT, IPP, and CU (Suzuki et al., 18 May 2026).

A different restricted-domain weakening appears in the aggregation of incomplete preferences under ambiguity. There the dual Pareto principle,

$(N,A,\mathcal R,\sigma)$4

is a unanimity-style requirement respecting unanimous non-endorsement (Kurata et al., 18 Dec 2025). The paper argues that this is too strong because of “spurious unanimity,” and replaces it with Common-Taste Pareto$(N,A,\mathcal R,\sigma)$5, which applies only to acts with no taste disagreement (Kurata et al., 18 Dec 2025). This is a qualified unanimity principle rather than an unrestricted one. The main characterization states that Common-Taste Pareto$(N,A,\mathcal R,\sigma)$6 holds iff social utility is utilitarian and, for every combination of plausible priors, some weighted average belongs to the social prior set (Kurata et al., 18 Dec 2025). A plausible implication is that quasi-unanimity, in this environment, is unanimity filtered to exclude cases where agreement is generated by heterogeneous underlying reasons.

3. Weak unanimity in ranking and aggregation

A third family of notions weakens unanimity by shrinking the domain on which it applies. In impartial rank aggregation, the paper distinguishes weak unanimity from full pairwise unanimity. For an $(N,A,\mathcal R,\sigma)$7-ranking mechanism $(N,A,\mathcal R,\sigma)$8, weak unanimity requires

$(N,A,\mathcal R,\sigma)$9

whereas unanimity requires preservation of every unanimously agreed pairwise comparison (Cembrano et al., 2023). The paper proves that for $A=\{s,r\}$0, there exists an $A=\{s,r\}$1-ranking mechanism satisfying impartiality and weak unanimity, but for every $A=\{s,r\}$2, no mechanism satisfies both impartiality and full unanimity (Cembrano et al., 2023). It also states that the $A=\{s,r\}$3 threshold is best possible, with impossibility for $A=\{s,r\}$4 established computationally and for $A=\{s,r\}$5 by theorem (Cembrano et al., 2023).

Weak unanimity is therefore a whole-ranking consensus condition that survives under impartiality only because it does not require preservation of unanimous pairwise judgments on heterogeneous profiles. The paper explicitly identifies weak unanimity as the closest notion to quasi-unanimity in its framework (Cembrano et al., 2023). This is a particularly sharp example of how unanimity can be weakened: the condition is preserved on constant profiles but abandoned elsewhere.

In the theory of evaluation aggregation, the contrast is even more explicit. Earlier Arrow-type results classified predicates for which all unanimous aggregators are dictatorial, but the paper on aggregation without unanimity removes the unanimity assumption entirely and classifies all polymorphisms under enlarged notions of triviality (Filmus, 27 Feb 2025). Its main theorem states that a non-degenerate predicate is $A=\{s,r\}$6-trivial iff it is $A=\{s,r\}$7-trivial for $A=\{s,r\}$8, and under additional assumptions this matches the classification of impossibility domains with respect to unanimity (Filmus, 27 Feb 2025). The paper is not about quasi-unanimity in the identity-based sense, but it is directly relevant by contrast: it shows what changes when unanimity is dropped rather than merely weakened.

The paper on J. S. Mill’s liberal principle presents yet another unanimity-like weakening. It defines liberal succession by the existence of a coalition $A=\{s,r\}$9 such that coalition members weakly prefer $q$0 to $q$1, one member strictly prefers it, and outsiders are not harmed in objective interests: $q$2 This is broader in appearance than Pareto unanimity, but under additive and ordinal/topological conditions the paper proves that liberal succession and Pareto superiority coincide (Green, 2019). The structure is again quasi-unanimous in a broad sense: unanimity is replaced by coalition unanimity plus a no-harm condition, yet under sufficient conditions it collapses back to full Pareto unanimity.

4. Universal-algebraic and CSP notions

In universal algebra, the relevant terminology is not “quasi-unanimity” but near-unanimity, quasi near-unanimity, and quasi weak near unanimity. An $q$3-ary near-unanimity term $q$4 satisfies

$q$5

for $q$6; for $q$7, this is a majority term (Campercholi, 2016). The paper on epic substructures and primitive positive functions uses near-unanimity as the algebraic hypothesis that turns a pp-definability characterization of epic substructures into a finite term-based test. Its abstract states that if a quasivariety $q$8 has an $q$9-ary near-unanimity term, then $\sigma_q$0 has surjective epimorphisms iff $\sigma_q$1 has surjective epimorphisms, and that for a finite set of finite algebras with a common near-unanimity term, decidability follows (Campercholi, 2016).

In finite algebras, the paper on deciding quasi weak near unanimity terms studies a weakened version in which idempotence is removed. A $\sigma_q$2-ary qWNU operation $\sigma_q$3 satisfies

$\sigma_q$4

without requiring $\sigma_q$5 (Kazda, 2020). For each fixed $\sigma_q$6, the problem HAS-$\sigma_q$7-qWNU is solvable in polynomial time, and the proof uses a local-to-global criterion based on generated subpowers (Kazda, 2020). In finite algebras, quasi Taylor, existence of some $\sigma_q$8-qWNU term, and quasi Siggers are equivalent (Kazda, 2020). Here “quasi” means omission of idempotence, so quasi-unanimity is an identity-based weakening rather than a high-threshold voting rule.

In infinite-domain CSP theory, the most relevant term is quasi near-unanimity. The paper on quasi directed Jónsson operations states that in the infinite-domain algebraic tractability literature, “with an exception of a quasi near-unanimity operation there are no known systems of operations implying tractability in this regime” (Wrona, 2024). It recalls the quasi near-unanimity identity as

$\sigma_q$9

and proves that, for first-order expansions of finitely bounded homogeneous symmetric binary cores with free amalgamation, chains of quasi directed Jónsson operations imply relational width $(\forall R\in\mathcal R)\big[(\sigma_q(R)=r)\iff(n(r,R)\ge q)\big],$0 and hence tractability (Wrona, 2024). Since every structure preserved by a quasi near-unanimity operation is also preserved by a chain of quasi directed Jónsson operations, the paper places quasi near-unanimity as a previously exceptional tractability condition and then weakens it further (Wrona, 2024).

These algebraic uses differ from voting-theoretic ones, but the common structural feature remains clear. Exact unanimity identities are weakened either by replacing the right-hand side $(\forall R\in\mathcal R)\big[(\sigma_q(R)=r)\iff(n(r,R)\ge q)\big],$1 with the diagonal value $(\forall R\in\mathcal R)\big[(\sigma_q(R)=r)\iff(n(r,R)\ge q)\big],$2, or by dropping idempotence altogether. This suggests that quasi-unanimity in algebra is best viewed as a family of weakened Maltsev conditions rather than as a threshold or probabilistic phenomenon.

5. Consensus dynamics and distributed systems

In distributed consensus, quasi-unanimity typically refers not to a voting axiom but to a dynamical state in which almost all agents hold the same opinion. The paper on stabilizing consensus with many opinions studies the 3-majority dynamics on a complete graph with synchronous rounds and an adaptive adversary (Becchetti et al., 2015). In the adversary-free regime, if $(\forall R\in\mathcal R)\big[(\sigma_q(R)=r)\iff(n(r,R)\ge q)\big],$3, the dynamics reaches exact consensus in

$(\forall R\in\mathcal R)\big[(\sigma_q(R)=r)\iff(n(r,R)\ge q)\big],$4

rounds with high probability (Becchetti et al., 2015). Under a dynamic adversary, the formal target is stabilizing almost-consensus, and the theorem states that with $(\forall R\in\mathcal R)\big[(\sigma_q(R)=r)\iff(n(r,R)\ge q)\big],$5 and $(\forall R\in\mathcal R)\big[(\sigma_q(R)=r)\iff(n(r,R)\ge q)\big],$6 for suitable constants, the 3-majority dynamics reaches a valid stabilizing almost-consensus within

$(\forall R\in\mathcal R)\big[(\sigma_q(R)=r)\iff(n(r,R)\ge q)\big],$7

rounds, with all but $(\forall R\in\mathcal R)\big[(\sigma_q(R)=r)\iff(n(r,R)\ge q)\big],$8 nodes holding the same valid opinion (Becchetti et al., 2015). The paper explicitly identifies this as the operational analogue of quasi-unanimity.

The mechanism is not expressed as a threshold rule but as a two-stage elimination process. The minimum support among active opinions exhibits negative drift, some opinion falls below average by about $(\forall R\in\mathcal R)\big[(\sigma_q(R)=r)\iff(n(r,R)\ge q)\big],$9, and then it rapidly dies out (Becchetti et al., 2015). In the adversarial case, “small” opinions stay small and non-valid mass remains bounded, so one valid opinion holds $q>n/2$0 nodes over long polynomial time horizons (Becchetti et al., 2015). This is a direct formalization of near-unanimity as almost-consensus rather than exact consensus.

Binary majority dynamics on random graphs shows a related but graph-dependent phenomenon. On Erdős–Rényi graphs with $q>n/2$1, majority dynamics reaches exact unanimity on the initial majority opinion by time $q>n/2$2 with probability at least $q>n/2$3 (Benjamini et al., 2014). More recent work improves sparse-regime unanimity thresholds: if

$q>n/2$4

and

$q>n/2$5

then unanimity occurs on $q>n/2$6, and under uniform random initial coloring the new lower bound becomes

$q>n/2$7

(Kim et al., 10 Mar 2025). Although these papers analyze full unanimity, the proof structure in the latter explicitly passes through an intermediate regime in which by day $q>n/2$8,

$q>n/2$9

followed by geometric minority decay (Kim et al., 10 Mar 2025). A plausible implication is that quasi-unanimity appears as a precursor stage in the dynamics even when the stated theorem concerns eventual exact unanimity.

The paper on majority dynamics and polarization in the Galam model gives yet another dynamic interpretation. It does not define quasi-unanimity explicitly, but treats exact unanimity at $q=n-1$00 or $q=n-1$01 and also practical unanimity when $q=n-1$02 is “very large around $q=n-1$03” or “very low around $q=n-1$04” (Galam, 2023). In that usage, quasi-unanimity corresponds to overwhelming-majority attractors that survive below the critical contrarian threshold $q=n-1$05 or stubborn-agent thresholds $q=n-1$06 for group size $q=n-1$07 and $q=n-1$08 for group size $q=n-1$09 (Galam, 2023).

6. Agreement theorems, voting geometry, and conceptual scope

Approval voting provides a geometric version of quasi-unanimity. In an approval-voting society $q=n-1$10, the agreement number is

$q=n-1$11

and $q=n-1$12-agreeability means that among every $q=n-1$13 voters, some $q=n-1$14 approve a common platform (Mazur et al., 2017). The classical linear theorem states that in a $q=n-1$15-agreeable linear society,

$q=n-1$16

while in a circular society one has

$q=n-1$17

(Mazur et al., 2017). These are exactly local-to-global agreement theorems: strong local overlap forces a globally popular option.

The paper extends this framework to two-dimensional product societies. If $q=n-1$18 is a $q=n-1$19-agreeable product society and $q=n-1$20 are the lower bounds on agreement proportion in the factor societies, then under a size condition

$q=n-1$21

(Mazur et al., 2017). This yields explicit lower bounds for 2-box, cylindrical, and toroidal societies. For example, in a cylindrical society,

$q=n-1$22

under the large-society condition, and if $q=n-1$23, then

$q=n-1$24

(Mazur et al., 2017). Thus if a cylindrical society is $q=n-1$25-agreeable, some platform is approved by at least half the voters (Mazur et al., 2017). In this setting, quasi-unanimity is not an identity or threshold quota but a guaranteed mass of common approval extracted from local consistency conditions.

Taken together, the literature shows that quasi-unanimity has no single canonical formalization. In some papers it is a supermajority quota $q=n-1$26 close to $q=n-1$27 (Hermida-Rivera, 24 Sep 2025). In others it is distance to a unanimously supported configuration (Suzuki et al., 18 May 2026), exact reproduction of whole-ranking consensus only on constant profiles (Cembrano et al., 2023), a weakened near-unanimity identity (Campercholi, 2016); (Kazda, 2020); (Wrona, 2024), an almost-consensus state with $q=n-1$28 dissenters (Becchetti et al., 2015), or a qualified unanimity principle restricted to cases without taste disagreement (Kurata et al., 18 Dec 2025). A plausible synthesis is that quasi-unanimity is best treated as a comparative notion: it marks the point at which unanimity is relaxed, but not abandoned, in order to preserve structural force in environments where exact unanimity is either too strong or too fragile.