The same ballots, counted by different rules, can produce different winners. Here are the main systems, in plain terms.

Choosing one winner

• Plurality (“first past the post”): each voter picks one candidate, and whoever gets the most votes wins, even without a majority. It is used in the United Kingdom, Canada and most United States elections. Its main weakness is the spoiler effect: two similar candidates can split their supporters' votes and let a less popular third candidate win. Plurality systems tend to produce two large parties, a pattern often called Duverger's law.
• Two-round runoff: if no one wins a majority, the top two face each other in a second election. France uses this for its president.
• Ranked choice voting (also called instant runoff): voters rank candidates in order. If no one has a majority of first choices, the last-place candidate is dropped and those ballots move to their next choice, repeating until someone has a majority. Australia uses it for its House of Representatives.
• Approval voting: voters mark every candidate they find acceptable, and the candidate approved by the most voters wins. It avoids most spoilers, but voters cannot say which approved candidate they prefer most.

Choosing a legislature

In proportional representation (PR), parties win seats in proportion to their share of the vote: a party with 20 percent of the vote gets about 20 percent of the seats. In list PR, used in the Netherlands, voters choose a party and seats are filled from its list. Germany and New Zealand use mixed systems that combine local districts with party lists to keep the overall result proportional. Many PR systems set a threshold, such as 5 percent, below which a party wins no seats.

PR gives smaller parties and minority views a voice and makes most votes count. Critics say it often requires coalition governments, which can be slow to form and can give small parties outsized bargaining power. Plurality districts give each area a clear local representative, but can award a majority of seats to a party with well under half the votes.

Two puzzles

In the 1780s, the French thinker the Marquis de Condorcet noticed something odd. Suppose three voters rank options A, B and C. Voter 1 prefers A to B to C; voter 2 prefers B to C to A; voter 3 prefers C to A to B. Then a majority prefers A to B, a majority prefers B to C, and a majority prefers C to A. The group's majority preferences go in a circle. This is the Condorcet paradox: majorities can be inconsistent even when every individual is consistent.

In 1951, the economist Kenneth Arrow proved a deeper result. Arrow's theorem shows that when there are three or more options, no ranked voting method can satisfy a short list of reasonable fairness conditions all at once (for example, that adding or removing an irrelevant option should not flip the result between two others, and that no single voter should decide everything). Simply put: every ranked voting system involves trade-offs. There is no perfect method, only better and worse fits for particular goals.

Weighted voting

In weighted voting, some votes count more than others. Shareholders vote in proportion to their shares. The Council of the European Union passes many decisions only with a majority of member states representing a large share of the EU's population. Some cooperatives and online communities weight votes by participation, membership time or track record.

Supporters argue that weighting can reflect real differences in stake, effort or knowledge and reward participation. Critics answer that it breaks the principle of one person, one vote, can entrench those already powerful, and depends on whoever designs the weights. A further subtlety: a vote's weight is not the same as its power, meaning how often it actually changes the outcome.

Worked example: one election, four results

One hundred voters choose among three candidates: Avery, Blake and Casey. Their rankings fall into three groups:

• 40 voters: Avery first, then Blake, then Casey.
• 35 voters: Casey first, then Blake, then Avery.
• 25 voters: Blake first, then Casey, then Avery.

Plurality: Avery has the most first choices (40) and wins.

Ranked choice: no one has a majority, so Blake (fewest first choices) is dropped. Blake's 25 ballots go to their next choice, Casey. Casey now has 60 and wins.

Head to head: Blake beats Avery 60 to 40 (the Casey and Blake groups both prefer Blake) and beats Casey 65 to 35. Blake is the Condorcet winner, preferred to each rival by a majority, yet Blake lost under both systems above.

Approval: if each voter approves their top two, Avery gets 40, Casey gets 60, and Blake gets 100. Blake wins.

The same voters, with the same preferences, produced different winners under different rules. This is the practical side of Arrow's lesson: every method builds in trade-offs. Plurality rewards intense support from the largest single group. Ranked choice rewards being a common second choice once weaker candidates drop out. Head-to-head comparison and approval reward broad acceptability. Choosing a voting system is never neutral: it is a choice about which kinds of support matter most, and that choice is best made openly, before anyone knows who it will help.