Voting Systems

A voting system, also called an electoral method, is a procedure that takes a set of ballots as input and produces a winner, a ranking, or a distribution of seats as output. The procedure must specify three things: how voters express preferences, how those expressions are aggregated, and how the result is declared.²

The study of voting systems sits at the intersection of political science, mathematics, and computer science. Political scientists examine how rules shape party formation and incumbency. Mathematicians prove which combinations of fairness criteria can simultaneously be satisfied. Computer scientists model the complexity of tabulation and the resilience of methods to strategic manipulation.

Three layers of any voting system

• Ballot design — single-mark, ranked, scored, or approval-based.
• Aggregation rule — plurality, runoff, Condorcet, Schulze, Borda, or quota-based.
• Reporting standard — winner, full social ranking, or proportional allocation.

Plurality, often called "first past the post," declares as winner the candidate with the most votes, regardless of whether that candidate received a majority. It is the simplest method to administer and remains the most widely used in single-winner contests across the Anglosphere.³

Majority methods require more than half of valid votes for a candidate to win. When no candidate clears the threshold on the first count, a runoff is held — either as a separate election (two-round system) or simulated through a ranked ballot (instant-runoff voting).

Use in single-winner offices

Country	Office	Method	Threshold
United Kingdom	House of Commons	Plurality	—
France	President	Two-round runoff	50%
United States	House of Representatives	Plurality	—
Australia	House of Representatives	Instant-runoff	50%
Brazil	President	Two-round runoff	50%

Ranked methods ask voters to order candidates by preference. Different aggregation rules then transform the same set of rankings into very different winners. Instant-runoff voting eliminates the lowest-ranked candidate iteratively. The Borda count assigns points by rank position. Condorcet methods examine every pairwise contest implied by the ballots.⁴

Approval and score variants

Approval voting permits each voter to mark any number of candidates as acceptable; the candidate with the most marks wins. Score voting (also called range or cardinal voting) lets voters rate candidates on a numeric scale, and the highest mean score wins. Both methods avoid the rank-vs-rate question by treating ballots as additive evaluations rather than ordinal lists.⁵

Proportional representation (PR) systems aim to seat parties or groups in proportion to the votes they receive. They are used for legislatures rather than single offices. The two principal families are party-list PR, where voters choose a party and seats are allocated by formula, and single transferable vote (STV), which uses ranked ballots and quota-based elimination.⁶

Allocation formulas

Formula	Family	Bias	Notable users
D'Hondt	Highest averages	Slight large-party	Spain, Portugal, Israel
Sainte-Laguë	Highest averages	Neutral	Norway, New Zealand, Germany
Hare quota	Largest remainder	Slight small-party	Hong Kong (until 2021)
Droop quota	Largest remainder	Mild large-party	South Africa, Slovakia
Imperiali	Largest remainder	Strong large-party	Italy (1946–1992)

Mixed-member systems

Mixed-member proportional (MMP) systems combine single-member districts with party-list seats, using the list seats to compensate for disproportionality in the district results. New Zealand adopted MMP in 1996 after a binding referendum; Germany has used a variant since 1949.

• MMP preserves geographic representation while maintaining proportionality.
• Voters cast two ballots — one for a local candidate, one for a party.
• List seats top up under-represented parties to match the vote share.

• The two-ballot structure adds cognitive load.
• Overhang seats can inflate the legislature beyond its nominal size.

Social choice theory evaluates voting systems against formal criteria. A method either satisfies a criterion universally or admits at least one ballot profile that violates it. The body of results known collectively as impossibility theorems shows that no single method can satisfy all desirable criteria simultaneously.⁷

Common criteria

Criterion	Plurality	IRV	Borda	Condorcet	Approval
Majority	yes	yes	no	yes	no
Condorcet winner	no	no	no	yes	no
Monotonicity	yes	no	yes	yes	yes
Independence of clones	no	yes	no	yes	yes
Later-no-harm	yes	yes	no	no	no

The Gibbard–Satterthwaite theorem extends the impossibility result to strategic manipulation: every non-dictatorial deterministic rule with three or more outcomes is susceptible to tactical voting. The practical question is therefore not whether a method can be manipulated, but how often and how easily.⁸

Reform movements have shifted several jurisdictions toward ranked or proportional methods over the last quarter-century. Adoption is uneven — driven by court orders, citizen-led ballot initiatives, or constitutional review. Reversals are also common.

Selected adoption events

Year	Jurisdiction	Reform	Mechanism
1996	New Zealand	FPTP → MMP	Binding referendum
2002	San Francisco, US	Plurality → IRV	City charter amendment
2018	Maine, US	Plurality → IRV (federal)	Citizen initiative
2020	Alaska, US	Top-four primary + IRV	Ballot measure
2022	Slovenia	Open list adjustment	Constitutional review

Empirical research on the effects of these reforms is ongoing. Early studies suggest that ranked methods reduce negative campaigning in some contexts, while proportional systems tend to broaden the spectrum of represented parties.⁹ The evidence on turnout effects is mixed and highly context-dependent.

Open questions for further research

• How do ranked ballots affect candidate entry decisions in low-information races?
• Does proportional representation change the distribution of legislative seniority?

• Claims that reform alone resolves polarisation are unsupported.
• Single-jurisdiction case studies do not generalise without controls.

1. 1 Tideman, T. N. Collective Decisions and Voting. Ashgate, 2006, ch. 1, pp. 3–18.
2. 2 Brams, S. J. & Fishburn, P. C. "Voting Procedures." Handbook of Social Choice and Welfare, vol. 1, Elsevier, 2002.
3. 3 Norris, P. Electoral Engineering: Voting Rules and Political Behavior. Cambridge, 2004.
4. 4 Saari, D. G. Decisions and Elections: Explaining the Unexpected. Cambridge, 2001.
5. 5 Brams, S. J. & Fishburn, P. C. Approval Voting, 2nd ed. Springer, 2007.
6. 6 Gallagher, M. & Mitchell, P., eds. The Politics of Electoral Systems. Oxford, 2005.
7. 7 Arrow, K. J. Social Choice and Individual Values, 2nd ed. Yale, 1963.
8. 8 Gibbard, A. "Manipulation of Voting Schemes: A General Result." Econometrica, vol. 41, no. 4, 1973, pp. 587–601.
9. 9 Donovan, T., Tolbert, C. & Gracey, K. "Campaign Civility under Preferential and Plurality Voting." Electoral Studies, 2016.