Approval voting
Approval voting is a single-winner electoral system in which voters may approve any number of candidates on their ballot, with the candidate receiving the most approvals declared the winner.[1] This method contrasts with plurality voting by permitting multiple approvals per voter, thereby allowing expression of support for candidates without the constraint of selecting only one.[1] Formalized in the 1970s by political scientists Steven J. Brams and Peter C. Fishburn, approval voting draws on earlier theoretical work and has roots in historical practices such as the election of doges in medieval Venice, where subsets of electors could approve multiple nominees.[2] [3] Economists like Robert J. Weber further analyzed its properties, demonstrating through game-theoretic models that it incentivizes honest voting more effectively than plurality systems in multi-candidate races, as voters face reduced pressure to strategically limit approvals to avoid aiding less-preferred options.[1] In practice, approval voting tallies each approval equally, without weights or rankings, which empirical experiments indicate promotes selection of candidates with broader acceptability.[4] For instance, a 1985 controlled trial in the election of the Institute of Management Sciences president showed the approval winner outperforming the plurality winner in subsequent head-to-head matchups, suggesting greater overall voter satisfaction.[4] Proponents argue it mitigates spoiler effects inherent in plurality voting, where similar candidates split votes, while critics contend it overlooks preference intensities, potentially favoring status-quo centrists over those with passionate minority support.[1] [5] Approval voting has been adopted by professional organizations, including the Mathematical Association of America and the American Political Science Association for internal leadership selections, and more recently by municipalities such as Fargo, North Dakota, following a 2020 ballot initiative.[1] [6] Despite these implementations, broader U.S. adoption faces resistance, with some jurisdictions rejecting it in favor of ranked-choice alternatives, amid debates over its alignment with diverse ideological preferences.[6]
Definition and Mechanics
Core Principles
Approval voting is predicated on the principle that voters should indicate support for all candidates they find acceptable, enabling the expression of broader preferences beyond a single choice. Each voter independently approves or disapproves of candidates, with no limit on the number of approvals, and the candidate accumulating the most approvals is elected. This mechanism, distinct from ordinal systems like plurality or instant-runoff voting, treats approvals as additive signals of acceptability rather than competitive rankings.[7][5] The foundational rationale emphasizes electing the candidate with the widest affirmative support, mitigating issues like vote-splitting where similar candidates divide endorsements. By allowing multiple approvals, the system aligns with dichotomous voter preferences—satisfactory or unsatisfactory—facilitating sincere ballot-casting when voters approve all candidates exceeding a personal threshold of adequacy. Brams and Fishburn formalized this approach in 1978, arguing it outperforms traditional methods in aggregating support across diverse electorates by maximizing the number of voters content with the outcome.[8][7] Computationally straightforward, approval voting requires only summing binary votes per candidate, ensuring transparency and resistance to complex manipulation in large-scale implementations. This simplicity underpins its adoption in professional societies, such as the Mathematical Association of America since 2016, where it has demonstrated practical efficacy in selecting leaders with broad consensus. Unlike systems penalizing multi-candidate support, approval voting incentivizes participation by valuing all expressed approvals equally, theoretically enhancing democratic representation through inclusive preference revelation.[6][5]Voting and Tabulation Process
In approval voting, each voter receives a ballot listing all candidates for the contested office and marks approval for any number of them, typically by placing a checkmark, "X," or "yes" beside the names of acceptable candidates.[9][10] Voters may approve zero candidates, effectively abstaining from contributing to any tally, or approve multiple, with no penalty for breadth of support.[6] Each marked approval counts as one vote for that candidate, treating approvals equally regardless of how many a voter selects.[11] Tabulation involves summing the approvals received by each candidate across all valid ballots.[10] The candidate with the greatest total approvals wins the election.[5] Ties are resolved according to predefined rules, such as lotteries or recounts, established by the election jurisdiction.[12] This summation process requires no ranking, elimination of candidates, or iterative redistribution of votes, enabling rapid manual counting or compatibility with electronic systems.[13] For instance, in a simulated election with 100 voters and three candidates, approvals tallied as 55, 60, and 45 resulted in the candidate with 60 winning.[11]Variants and Implementation Details
Approval voting ballots typically feature a list of candidates alongside checkboxes or yes/no options, allowing voters to indicate approval for any number of candidates without penalty for multiple selections.[6] The tabulation process involves summing the number of approvals for each candidate, with the winner determined by the highest total; this arithmetic simplicity facilitates implementation on both paper and electronic systems, reducing costs compared to ranked methods requiring preference sorting.[14] In practice, clear instructions on ballots prevent overvoting confusion, as approving multiple candidates is intentional rather than erroneous, unlike plurality systems. For multi-winner elections, extensions of approval voting adapt the core mechanism to select multiple candidates while aiming for proportionality. Multiwinner Approval Voting (MAV) extends single-winner approval by apportioning seats based on approval totals, treating candidates as individual vote-getters and using methods like largest remainder to allocate winners beyond simple plurality of approvals.[15] Proportional Approval Voting (PAV) further refines this by incorporating harmonic weighting in a greedy selection process, where additional winners are chosen to maximize marginal satisfaction for approve sets, satisfying axioms like justified representation under certain assumptions.[16] Other variants include Combined Approval Voting (CAV), which permits explicit disapproval alongside approval, computing scores as approvals minus disapprovals to capture opposition intensity, though this introduces potential for negative campaigning incentives not present in standard approval. Weighted Approval Voting relaxes binary approvals by assigning differential weights to candidates, enabling nuanced expression while maintaining computational tractability for small electorates.[17] The Expanding Approvals Rule generalizes to multi-winner settings by iteratively expanding approval sets based on ordinal rankings, blending approval with weak preferences to mitigate strategic truncation. These variants address limitations in standard approval, such as lack of preference strength or proportionality, but may complicate voter education and ballot design relative to the binary original.[18]Historical Origins
Pre-Modern and Early Modern Precedents
In ancient Sparta, the election of members to the Gerousia, the Council of Elders, involved a process described by Plutarch in his Life of Lycurgus as acclamation by shouting, where assembled citizens expressed approval through the volume of their cheers for each candidate in turn, with the candidate receiving the loudest acclaim declared elected.[19] This mechanism effectively aggregated collective approval signals to select elders over age 60, resembling approval voting in its reliance on positive endorsements without requiring exclusive choices among candidates.[20] The process prioritized broad support over ranked preferences, though its reliance on audible volume introduced potential biases from crowd dynamics rather than formal ballot counts.[21] During the 13th century in the Republic of Venice, approval voting was employed in electing members to the Great Council (Maggior Consiglio), where electors could vote for multiple candidates from a list of eligible nobles, and those receiving the highest number of approvals were selected to fill seats.[22] This system, implemented amid efforts to broaden participation while preventing oligarchic capture, allowed voters to endorse any number of nominees without limiting to a single choice, aiming to identify candidates with widespread acceptability.[3] Similar multi-approval elements appeared in stages of Doge elections after the 1268 reforms, such as when groups of electors nominated or voted for several candidates to advance to subsequent rounds, requiring supermajorities like 25 out of 41 approvals in filtered ballots to proceed.[23] These Venetian procedures guarded against factional dominance by favoring consensus-building over plurality wins, though they combined approval with lotteries and iterative filtering for final selection. No prominent early modern precedents (circa 1500–1800) directly replicate approval voting in governmental contexts, though ecclesiastical elections, such as those for bishops in some Catholic dioceses, occasionally permitted canons to approve multiple nominees from lists, with the most approvals determining advancement; however, these were typically hybridized with plurality or acclamation and lacked standardized tabulation akin to modern approval systems.[5]20th-Century Theoretical Development
In the mid-1970s, approval voting emerged as a theoretically grounded alternative to traditional plurality systems amid growing interest in voting theory and strategic behavior in multi-candidate elections. Robert J. Weber first formalized the method in his analysis of voting equilibria, coining the term "approval voting" and modeling it as a system where voters approve multiple candidates to maximize expected utility under uncertainty about others' ballots.[24] Weber's 1977 framework demonstrated that approval voting admits equilibria where voters approve all candidates preferred to a pivotal threshold, avoiding certain paradoxes like those in plurality voting.[25] Building on this, Steven J. Brams and Peter C. Fishburn independently developed and extensively analyzed approval voting in their seminal 1978 paper, emphasizing its normative properties such as electing the Condorcet winner whenever one exists under sincere voting.[26] They proved that, assuming dichotomous voter preferences (utility 1 for approved candidates, 0 otherwise), approval voting maximizes the expected social utility among common voting rules, as it allows expression of approval without the distortions of ranking or single-choice constraints.[26] Brams and Fishburn further showed that strategic equilibria under approval voting preclude non-monotonicity and other failures observed in plurality or runoff systems, with voters converging to approve sets that reflect genuine preferences above a compromise point.[26] Subsequent theoretical work in the late 1970s and 1980s, including multiple independent proposals, reinforced approval voting's robustness; for instance, simulations indicated it reduces the spoiler effect by enabling support for viable alternatives without wasting votes on frontrunners.[24] Brams and Fishburn's 1983 monograph synthesized these insights, advocating approval voting for professional societies and public elections based on its simplicity, incentive compatibility, and empirical applicability in reducing vote-splitting.[27] This period marked approval voting's transition from abstract game-theoretic construct to a advocated reform, distinct from earlier cumulative voting variants by its explicit approval mechanism and equilibrium analysis.[1]Initial Theoretical Advocacy and Simulations
Steven J. Brams and Peter C. Fishburn formally advocated approval voting as a superior alternative to plurality voting in their 1978 paper published in the American Political Science Review. They argued that by permitting voters to approve any number of candidates without penalty, the system enables fuller expression of preferences, reducing the strategic compulsion to select only a single "lesser evil" and thereby mitigating vote-splitting and spoiler effects inherent in plurality systems. Brams and Fishburn provided axiomatic foundations, demonstrating that approval voting satisfies key properties including anonymity (outcomes invariant to voter relabeling), neutrality (symmetric treatment of candidates), positivity (support for a candidate cannot harm their chances), and monotonicity (additional approvals for a winner cannot cause their defeat). They posited that it tends to select candidates with the widest acceptability, often aligning with the Condorcet winner—who pairwise defeats all others—more reliably than plurality, as evidenced by theoretical proofs under dichotomous preference models where voters approve candidates above a utility threshold.[26] Building on this, Robert J. Weber's 1995 analysis in the Journal of Economic Perspectives reinforced approval voting's theoretical robustness under strategic incentives. Weber showed that in equilibrium play, where voters anticipate others' strategies, approval voting precludes certain perverse outcomes possible under plurality or runoff systems, such as the election of a Pareto-dominated candidate. He highlighted its incentive compatibility: sincere approval strategies are often Nash equilibria, as deviating to approve fewer candidates risks lowering the preferred outcome's probability without gain, assuming rational utility maximization over candidate utilities. Weber also noted its computational simplicity and resistance to cloning (adding similar candidates does not manipulate outcomes), positioning it as practically viable for large electorates.[1] Early simulations complemented these arguments by quantifying performance advantages. Samuel Merrill III's 1984 study in the American Political Science Review compared approval voting to plurality, runoff, and other systems using Monte Carlo simulations under impartial culture (random preference orders) and spatial models (voters and candidates positioned in a policy space with Euclidean utilities). With 3 to 10 candidates and sincere voting assumptions, approval voting yielded higher average voter utility—measured as the expected utility of the winner relative to the social optimum—outperforming plurality by 10-20% in 5-candidate races and reducing non-Condorcet winner frequency from over 50% under plurality to under 30%. In spatial simulations with probabilistic voter error, approval elected the utilitarian optimum (highest total utility candidate) in approximately 70% of cases versus 50% for plurality, attributing gains to its aggregation of multi-candidate support without exhaustive rankings.[28] Similar results from Merrill's contemporaneous work confirmed approval's edge in equilibrium strategic voting scenarios, where it converged to sincere outcomes faster than plurality due to weaker Duverger-like coordination pressures. These simulations, grounded in probabilistic preference generation and repeated trials (e.g., 1,000+ iterations per scenario), underscored approval voting's empirical superiority in avoiding "wrong" winners—those not preferred by a majority over alternatives—though critics later noted assumptions like uniform approval thresholds may idealize real behavior.[29]Theoretical Properties
Satisfaction of Standard Voting Criteria
Approval voting satisfies the monotonicity criterion, under which increasing support for a winning candidate cannot cause it to lose. In approval voting, additional approvals for the winner strictly increase its vote total without decreasing others', preserving or strengthening its position relative to competitors.[30][31] It also satisfies the participation criterion, which requires that adding a sincere ballot supporting the winner cannot change the outcome to another candidate. A new voter approving the eventual winner adds to its tally, making reversal impossible without altering existing ballots.[30] However, approval voting fails the majority criterion, which demands that a candidate ranked first by a majority of voters must win. A counterexample involves three candidates where 51% of voters rank X first, Y second, and Z last, sincerely approving both X and Y, while 49% rank Z first, Y second, and X last, approving Z and Y; Y then receives unanimous approval and wins despite X topping most ballots.[30] Approval voting likewise fails the Condorcet criterion, requiring election of a candidate who pairwise defeats all others when such exists. Counterexamples in large electorates show equilibria where strategic or sincere approvals elect non-Condorcet winners, as voters may withhold approval from the Condorcet candidate due to threshold-based preferences or compromise incentives.[32]| Criterion | Satisfied? | Key Reason |
|---|---|---|
| Majority | No | Broad approval for compromise candidates can override top-ranked majority favorite.[30] |
| Condorcet | No | Possible to exclude pairwise-dominant candidate via approval thresholds.[32] |
| Monotonicity | Yes | Added support for winner only boosts its score.[30] |
| Participation | Yes | New supporting ballots cannot harm winner's tally.[30] |
| Independence of Irrelevant Alternatives | Partial | Unaffected if added candidate receives no approvals, but fails if it draws approvals without altering pairwise preferences between frontrunners.[33] |