Vote by Veto: Minority Protection & Efficiency

• Vote by Veto is a decision procedure where voters block their least-favored candidates rather than selecting a favorite.
• It encompasses models like the proportional veto core, sequential elimination, metric voting, and quantum veto protocols.
• The framework balances minority protection with utilitarian efficiency, offering polynomial-time computation and insights into strategic manipulation.

Vote By Veto denotes a family of decision procedures in which blocking disfavored outcomes is the primary operation. In social choice, the term is used for rules that select a candidate from the proportional veto core, for sequential and continuous elimination rules that combine initial support with vetoes, and, in more recent work, for approval-based and committee-selection principles that quantify how large and how flexible a group must be to block an outcome. Outside classical voting, the same expression also appears in quantum anonymous-veto protocols and in veto bargaining models, where the decisive action is acceptance or rejection of a proposal rather than the aggregation of full rankings (Kizilkaya et al., 23 Jul 2025, Ianovski et al., 2020, Sangwan et al., 19 Sep 2025).

1. Veto power as a formal voting concept

A standard formalization of “voting by veto” treats a group’s power as the ability to block disliked candidates rather than to force a favorite. In the ranked-ballot framework of majority and veto power, a voting rule satisfies a $(q,l)$-veto criterion if, whenever a share of voters strictly greater than $q$ ranks some set $L$ of $l$ candidates in the bottom $l$ positions, none of those $l$ candidates can win. The minimal such quota $q$ is the rule’s measure of veto power: the smaller the quota, the stronger the guaranteed blocking power of like-minded voters (Kondratev et al., 2018).

This formalization makes clear that veto power is dual to majority power. For a given number of candidates $m$, vetoing $l$ least-preferred candidates guarantees that one of the remaining $k=m-l$ candidates must win. The majority loser criterion is the special case $q$0 and $q$1. Within this framework, instant-runoff voting has veto quota $q$2 for all $q$3, whereas plurality has $q$4 for all $q$5; the proportional veto core, Black’s rule, and Borda appear as exceptions that combine relatively low majority power with high veto power and thus provide minority protection (Kondratev et al., 2018).

This literature establishes a useful distinction. “Vote by veto” can mean a concrete ballot operation in which voters cast negative judgments, but it can also mean a normative guarantee: how large a coalition must be to ensure that a set of disliked candidates is excluded. Much of the later work on veto cores, metric distortion, and approval ballots can be read as refinements of this second idea (Kondratev et al., 2018).

2. Proportional veto core and the classical Vote By Veto rule

In the cooperative-game formulation due to Moulin, a coalition $q$6 has veto power

$q$7

where $q$8 is the number of voters and $q$9 the number of candidates. A coalition blocks candidate $L$0 if there exists a set $L$1 such that every voter in $L$2 strictly prefers every $L$3 to $L$4, and the coalition can veto all candidates outside $L$5, that is,

$L$6

The proportional veto core is the set of candidates not blocked by any coalition; Moulin’s existence theorem implies that this core is always nonempty (Ianovski et al., 2020).

In a precise sense used in recent metric-social-choice work, Vote By Veto is exactly the rule that picks a candidate from the proportional veto core. The rule’s defining intuition is that a candidate should survive only if no coalition is large enough, relative to the size of the alternative set it unanimously prefers, to exclude that candidate by proportional veto rights (Kizilkaya et al., 23 Jul 2025).

The proportional veto core resolved two structural defects of earlier sequential veto procedures. Earlier “voting by veto” rules could fail anonymity because they processed voters in a fixed order, and some versions required specific relations between the numbers of voters and candidates. The proportional veto core is anonymous, is defined for arbitrary $L$7 and $L$8, and gives each coalition veto power proportional to its size (Ianovski et al., 2020).

A central algorithmic result is that the veto core can be computed in polynomial time. The naïve approach is exponential because it ranges over coalitions and blocking sets, but the core-membership problem can be reduced to a biclique condition in a bipartite graph and then to maximum flow. The resulting running time is $L$9 (Ianovski et al., 2020).

The same paper introduces veto by consumption, an anonymous and neutral selection rule from the core. Each candidate starts with capacity 1; every voter continuously “eats” her least-preferred remaining candidate at speed 1; candidates disappear when their capacity is exhausted; and the last candidate or candidates to be eaten are the winners. Veto by consumption is core-consistent, Pareto-efficient, and monotonic, and it can be implemented in $l$0 arithmetic operations (Ianovski et al., 2020).

3. Metric voting: Plurality Veto and simultaneous veto dynamics

In metric social choice, voters and candidates are embedded in a metric space, each voter ranks candidates by increasing distance, and the objective is to choose a candidate minimizing total distance to voters. Deterministic rules cannot achieve distortion below 3, but that lower bound is tight (Kizilkaya et al., 2022).

Plurality Veto is a simple deterministic rule that reaches this optimal distortion. Each candidate begins with score equal to the number of first-place votes, $l$1. Then, in an $l$2-round veto process, voters are processed in some order, and each voter decrements the score of her bottom choice among the currently standing candidates. A candidate is eliminated when the score reaches zero, and the last-eliminated candidate wins. The rule uses only two ordinal queries per voter—a top query and one bottom-among-standing query—and each query communicates $l$3 bits, so the communication overhead is low (Kizilkaya et al., 2022).

Plurality Veto is therefore a literal “vote by veto” rule in which positive support and negative judgment are combined. Each voter first gives one point to a favorite candidate, then later receives one veto against the worst remaining candidate. The winner is the candidate whose initial support survives this sequence of vetoes the longest. The rule differs structurally from plain plurality, plain veto, and instant-runoff voting (Kizilkaya et al., 2022).

Although Plurality Veto is simple, its winner depends on the order in which voters are processed, so it is not anonymous. This led to a generalized-veto-core reinterpretation. In that framework, a candidate is in a $l$4-veto core if no coalition with voter weights $l$5 can block the candidate given candidate-support weights $l$6, and this is equivalent to the existence of an appropriate $l$7-matching in a domination graph. For plurality weights, the resulting plurality veto core coincides with the set of plurality-dominant candidates, each of which has distortion at most 3 (Kizilkaya et al., 2023).

The same line of work introduces SimultaneousPluralityVeto, a continuous-time rule that removes the dependence on voter order. Each candidate starts with support equal to the plurality score. From time 0 to 1, every voter continuously brings down, at rate 1, the support of her bottom choice among not-yet-eliminated candidates, and a candidate is eliminated if opposed by a voter after support reaches 0. This rule is anonymous and neutral, returns a nonempty winner set, and every winner has distortion at most 3 (Kizilkaya et al., 2023).

4. $l$8-Approval Veto and Vote By Veto as the $l$9 endpoint

The rule family $l$0-Approval Veto places Vote By Veto on a continuous spectrum between optimal metric efficiency and maximal minority protection. In the sequential version, each voter contributes $l$1-approval support to the top $l$2 candidates, so each candidate $l$3 starts with score $l$4, the number of voters who rank $l$5 in their top $l$6. Then $l$7 veto tokens are processed in sequence; each token walks upward from the voter’s worst remaining candidate, eliminating zero-score candidates until it reaches a positive-score candidate, and then decreases that score by 1 (Kizilkaya et al., 23 Jul 2025).

The set of possible winners is the $l$8-approval veto core, denoted $l$9. This core has an equivalent characterization through $l$0-domination graphs and perfect matchings. The same paper proves that every candidate in $l$1 satisfies the $l$2-mutual minority criterion with $l$3, formalized as

$l$4

Thus $l$5 directly controls the guaranteed level of mutual minority protection (Kizilkaya et al., 23 Jul 2025).

Vote By Veto is the special case $l$6. When $l$7, every voter approves all $l$8 candidates, every candidate has approval score $l$9, and the $q$0-approval veto core is exactly Moulin’s proportional veto core. In the paper’s formulation, Vote By Veto is therefore precisely the $q$1 endpoint of $q$2-Approval Veto and “picks a candidate from the proportional veto core” (Kizilkaya et al., 23 Jul 2025).

This endpoint has the strongest minority-protection guarantee in the family. Every Vote By Veto winner satisfies the $q$3-mutual minority criterion: if a coalition $q$4 solidly vetoes a set $q$5 and

$q$6

then no candidate in $q$7 can win. In particular, if more than $q$8 voters rank some candidate last, that candidate cannot be chosen by Vote By Veto (Kizilkaya et al., 23 Jul 2025).

The cost of this protection is reduced utilitarian efficiency. For candidates in $q$9, utilitarian distortion is

$m$0

so for Vote By Veto, $m$1, the bound is $m$2, and it is tight. For the $m$3-percentile objective, distortion is at most 5 only when $m$4; for Vote By Veto this becomes $m$5, and distortion is unbounded below that threshold. By contrast, egalitarian distortion remains 3 for all $m$6, including Vote By Veto, and that bound is optimal for deterministic rules (Kizilkaya et al., 23 Jul 2025).

5. Approval ballots, thumbs up/down voting, and committee generalizations

The proportional veto idea has also been reformulated for approval ballots. In that setting, each voter $m$7 has approval set $m$8 over $m$9 candidates, and voter flexibility is

$l$0

For a threshold $l$1, a voter is $l$2-flexible if $l$3. The key single-winner guarantee is flexible-voter representation $l$4, the smallest $l$5 such that any group of more than $l$6 voters, all $l$7-flexible, must contain at least one approver of the selected candidate. This is an approval-ballot version of the proportional veto principle (Halpern et al., 2 May 2025).

A universal lower bound holds: $l$8 for every rule $l$9. Approval voting achieves

$k=m-l$0

so it is near-optimal for some thresholds but not optimal uniformly. The paper then identifies a scoring rule with weight function

$k=m-l$1

that is FVR-optimal for all thresholds simultaneously: $k=m-l$2 Moreover, if a scoring rule is FVR-optimal for all $k=m-l$3, then its weight function must be $k=m-l$4 for some constant $k=m-l$5 (Halpern et al., 2 May 2025).

This scoring rule has an explicit veto interpretation. Each voter distributes $k=m-l$6 points to each candidate they disapprove, so every voter distributes exactly $k=m-l$7 disapproval points in total. The winner is the candidate with the fewest total disapproval points. In that sense, approval-based vote-by-veto becomes a weighted disapproval minimization rule whose weights are determined by voter flexibility (Halpern et al., 2 May 2025).

The same paper extends the principle to committee elections. For a committee $k=m-l$8 of size $k=m-l$9, a voter $q$00-approves $q$01 if $q$02. The multiwinner FVR benchmark is a hypergeometric bound,

$q$03

and there are both an expanded-candidate construction and a polynomial-time sequential algorithm that are FVR-optimal for all $q$04, for fixed $q$05 and $q$06 (Halpern et al., 2 May 2025).

A related 2025 paper on thumbs-up/down committee voting makes the interpretive split explicit. One approach treats electing approved candidates and vetoing disapproved candidates as comparable and studies combined proportionality guarantees; the other treats veto power separately from traditional proportionality. The paper formalizes axioms for both perspectives and studies suitable adaptations of Phragmén’s rule, Proportional Approval Voting, and the Method of Equal Shares (Kraiczy et al., 3 Mar 2025). This suggests that, once positive and negative ballots are both available, vote-by-veto can be embedded either as a symmetric satisfaction model or as a distinct veto-rights model.

6. Computation, manipulation, and uncertainty

Algorithmic questions around veto rules show a sharp contrast between simple veto and richer negative-voting systems. For the proportional veto core, exact winner determination is polynomial-time, and the anonymous veto-by-consumption rule is also polynomial-time computable (Ianovski et al., 2020). In the same framework, the veto core is strategy-proof for optimists, whereas a pessimist can manipulate it in polynomial time (Ianovski et al., 2020).

For the simple veto rule with weighted votes and three candidates, the manipulation problem has a smooth phase transition. The probability that a coalition can elect a desired candidate rises from about $q$07 at coalition size $q$08 to near 1 as the coalition grows, and after rescaling by $q$09 the empirical curve is well fit by

$q$10

The same paper argues that manipulation is asymptotically easy for many independent and identically distributed votes, becomes computationally hard in highly correlated “hung” elections, and becomes easy again when even a single uncorrelated voter is introduced (0905.3720).

The computational profile of veto also stands out in partial-information settings. For pure scoring rules with unweighted voters and an unbounded number of candidates, the Possible Winner problem is solvable in polynomial time for plurality and veto, whereas it is NP-complete for all other pure scoring rules, including the rule with scoring vector $q$11 (Baumeister et al., 2011).

A related tractability result holds under uncertain turnout. When voter attendance is independent and each voter attends with probability $q$12, the probability that a candidate wins can be computed in polynomial time for plurality and veto. For $q$13-approval and $q$14-veto with $q$15, Borda, Condorcet, and Maximin, exact winning probability is #P-hard. At the same time, there is a fully polynomial-time randomized approximation scheme for the probability of losing for every positional scoring rule with polynomial scores, as well as for the Condorcet rule (Imber et al., 2021).

7. Quantum and bargaining formulations

Outside classical social choice, vote-by-veto also denotes a binary collective-decision task: detect whether at least one veto has occurred while revealing as little else as possible. In a recent quantum anonymous veto protocol, there are $q$16 voters $q$17, each with binary input “support” or “veto,” and the goal is to reveal only whether $q$18 or $q$19, where $q$20 is the number of vetoes. The protocol uses

$q$21

Bell pairs, is single-round and deterministic, and preserves anonymity, correctness, and verifiability; the voting authority learns only whether a veto is present, not who vetoed or even the exact value of $q$22 (Sangwan et al., 19 Sep 2025).

A different quantum formulation, Quantum Logical Veto, treats vote by veto as a quantum-computed voting rule. Classical “disagree” and “agree” are encoded as $q$23 and $q$24, and ballots are aggregated by quantum AND. The proposal is rejected with probability 1 if at least one voter’s ballot is the pure disagree state $q$25. The protocol uses only Toffoli-based AND, NOT, and measurement primitives, and the paper emphasizes constant coherent width of 3 qubits (Sun et al., 2022).

In political economy, vote by veto appears as a bargaining institution. In persuasion-based veto bargaining, a proposer chooses a policy $q$26, a veto player either accepts it or preserves the status quo at 0, and the proposer may also design an information experiment about the veto player’s ideal point. With quadratic loss for the vetoer, the acceptance rule is

$q$27

where $q$28 is the posterior belief about the ideal point. The proposer-optimal outcome can be achieved either by providing no information or by a simple binary experiment, and partial revelation is optimal when expected misalignment is sufficiently large (Kim et al., 2023).

A related delegation model studies menus rather than single proposals. There, the proposer chooses a delegation set $q$29, the vetoer chooses from $q$30, and “full delegation” $q$31 is optimal under identifiable conditions. More generally, interval delegation $q$32 is optimal under explicit monotonicity conditions, and can be a Pareto improvement over cheap talk (Kartik et al., 2020). In this broader institutional sense, vote by veto is not a tally rule but a governance structure in which the decisive act is approval or blocking of a proposed outcome.