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✶ Learn · Electoral systems

Same votes,
different winners.

An electoral system is the rulebook that turns votes into seats. Change the rules and the same ballots can elect a different person — or a different government.

Overview

Three families.

International IDEA groups the world's systems into three families, plus a handful of others. Within each, small rules — district size, ballot design, the counting formula — matter a lot.

I

Plurality & majority

The country is cut into districts, usually electing one member each. Whoever wins the district takes the seat; everyone else's votes elect no one.

First past the post
Most votes wins, majority or not.
Two-round
No majority? A second vote between the leaders.
Instant runoff
Voters rank; the last-placed are eliminated and their ballots transferred until someone has a majority.
Block vote
Several seats in one district; each voter has as many votes as seats, and the top vote-getters win.
II

Proportional

Larger, multi-member districts — sometimes the whole country — share their seats among parties roughly in line with their votes.

Closed list
Voters choose a party; the party's ranked list decides who takes its seats.
Open list
Voters can also back individual candidates, reordering their party's list.
Single transferable vote
Voters rank candidates in multi-member districts; surpluses and eliminations transfer until every seat reaches a quota.
III

Mixed

Two tiers in one chamber: single-member districts plus party-list seats. The difference is whether the list tier corrects the districts or runs alongside them.

Mixed-member proportional
List seats top parties up to their proportional share of the whole chamber.
Parallel
List seats are shared out separately; districts and lists simply add up.

Where they're used

Single-winner rules

Electing one person.

First past the post

Each voter marks one candidate; the candidate with the most votes wins, even with far less than half. It is simple and gives each district one clear representative. With several candidates it can elect someone most voters opposed, and it rewards parties whose support is concentrated geographically.

Two-round system

If nobody wins a majority, a second election is held — usually between the top two. France's Constitution requires an absolute majority to elect the president, with a two-candidate runoff otherwise; its legislative elections also let any candidate backed by 12.5% of registered voters into round two. Brazil and Indonesia elect presidents in two rounds as well.

Instant runoff (ranked choice)

Voters rank candidates. If nobody has a majority of first preferences, the last-placed candidate is eliminated and those ballots count for the voter's next choice, round after round. Australia has used preferential voting for its House of Representatives for a century. A ballot that ranks only eliminated candidates is "exhausted" and stops counting.

Measuring against Condorcet and Borda

Two classic yardsticks are rarely used in public elections but help judge the others. A Condorcet winner beats every rival head to head — though sometimes preferences go in a circle and none exists. A Borda count gives points by rank. No ranked system satisfies every fairness criterion at once (Arrow's theorem).

Try it · One seat, one electorate

Same ballots, different winners.

Five blocs of voters rank four candidates. Change how many voters are in each bloc, reorder their preferences, or cut a ranking short. Every rule below counts exactly the same ballots.

  1. Bloc 1
    1. Bloc 2
      1. Bloc 3
        1. Bloc 4
          1. Bloc 5

            Proportional representation

            Seats in line with votes.

            Party lists, closed or open

            In list PR, each party presents a list for a multi-member district. In a closed list (South Africa) voters choose a party and the party's ranking decides who sits. In an open list (Brazil, Indonesia) voters can also back individual candidates, so personal votes decide who fills a party's seats.

            The formula: D'Hondt, Sainte-Laguë, Hare

            Seats are rarely a whole number of vote shares, so a formula rounds them. Highest averages methods divide each party's votes by a series of divisors and hand out seats to the largest results: D'Hondt uses 1, 2, 3…, which tends to favour bigger parties; Sainte-Laguë uses 1, 3, 5…, which is closer to exact proportionality. Largest remainder methods set a quota — Hare's is votes ÷ seats — give each party one seat per full quota, then pass leftover seats to the biggest remainders.

            Thresholds

            Many systems deny list seats to parties below a legal threshold — 5% in Germany and New Zealand, for example — to limit fragmentation. Small district sizes create an effective threshold even without a legal one.

            Single transferable vote

            STV is proportional without party lists. Voters rank candidates in districts electing several members. A candidate who reaches the quota (usually the Droop quota, ⌊votes ÷ (seats + 1)⌋ + 1) is elected and their surplus passes on; otherwise the lowest candidate is eliminated and transferred. Ireland's Dáil, Malta, Australia's Senate and India's Rajya Sabha use it. Try it in the simulator above: pick STV · 3 seats.

            Mixed systems

            Districts and lists.

            Mixed-member proportional gives voters two votes: one for a district representative, one for a party. The party vote decides each party's share of the whole chamber; list seats fill the gap between that share and the districts it won. If a party wins more districts than its share, New Zealand lets it keep the extra "overhang" seats and the chamber grows. Germany's 2023 reform went the other way: a party's seats are capped at its share, and its weakest district winners can go unseated.

            Parallel systems also combine districts and lists, but the two are counted separately and added together, so list seats don't correct the districts' distortions. Japan's House of Representatives and Mexico's Chamber of Deputies are both elected this way. The results usually fall between first past the post and full proportionality.

            Try it · Many seats, one vote

            Votes in, seats out.

            Set each party's national vote and the size of the chamber. The same vote is turned into seats by three proportional formulas, then by single-member districts and two mixed systems. The Gallagher index measures how far seats stray from votes: 0 is perfectly proportional.

            Vote for each party

            List proportional representation · whole country as one district

            Districts & mixed systems · half the seats are single-member districts

            Districts use a toy geography: each party is a little stronger in its own part of the country. Ties go to the party listed first. Real systems add many details — regional lists, rounding rules, exemptions — that this model leaves out.

            Choosing a system

            No perfect rulebook.

            Systems trade one goal against another. Plurality systems tend to produce single-party majorities and a close link between a member and a district, often at the cost of large gaps between votes and seats and fewer viable parties — the pattern political scientists call Duverger's law. Proportional systems mirror opinion more closely and usually lead to coalition governments.

            One common yardstick is the Gallagher index: the square root of half the sum of the squared gaps between each party's vote share and seat share. Zero means perfectly proportional. The simulator above shows it for every rule. To see how party systems and ideologies grow out of these rules, read the political spectrum; for the executives that elections produce, see executives.