Method to make collective decisions
Voting refers to the process of choosing officials or policies by casting a ballot, a document used by people to formally express their preferences. Republics and representative democracies are governments where the population chooses representatives by voting. 
The procedure for identifying the winners based on votes varies depending on both the country and the political office. Political scientists call these procedures electoral systems, while mathematicians and economists call them social choice rules. The study of these rules and what makes them good or bad is the subject of a branch of welfare economics known as social choice theory.
In smaller organizations, voting can occur in many different ways: formally via ballot to elect others for example within a workplace, to elect members of political associations, or to choose roles for others; or informally with a spoken agreement or a gesture like a raised hand. In larger organizations, like countries, voting is generally confined to periodic elections.
In politics.
In a democracy, the government is elected by the people who vote in an election: a way for an electorate to "elect", that is "choose", from several different candidates. It is more than likely that elections will be between two opposing parties. These two will be the most established and most popular parties in the country. For example, in the US, the competition is between the Republicans and the Democrats. In an indirect democracy, "voting" is the method by which the person elected (in charge) represents the people, whilst making decisions. Direct democracy is the complete opposite, the people make the policy decisions directly without selecting a representative to do it for them.
A majority vote is when more than half of voters vote for the same person or party. However, whilst it is usually said each individual's vote does count, many countries use a combination of factors to decide who has power, not the at-large "popular vote". Most influential of these factors are districts that divide the electorate. For example, in the UK a party winning plurality in a majority of constituencies wins majority government, but they may not always have the most individual votes (i.e. they may have lose the popular vote but still win the seat count). (It is also possible for a party to win plurality in a minority of constituencies (but more than any other one party) and thus win minority government.)
All modern liberal democracies use voting by secret ballot to prevent individuals from becoming influenced by other people and to protect their political privacy. The objective of secret ballots is to try to achieve the most authentic outcome, without any risk of pressure, threat, or services linked to one's vote; this way, a person is able to express their actual preferences.
Voting often takes place at a polling station but voting can also be done remotely by mail or using internet voting (such as in Estonia). Voting is voluntary in some countries, like the UK, but it may be required by law in others, such as Australia.
Electoral systems.
There are many electoral methods. The purpose of an election may be to choose one person, such as a president, or a group, such as a committee or a parliament. In electing a parliament, either each of many small constituencies can elect a single representative, as in Britain; or each of a lesser number of multi-member constituencies may elect two or more representatives, as in Ireland; or multi-member districts and some single-member districts can be used; or the entire country can be treated as one "at-large" district, as in The Netherlands.
Different voting systems use different ballot designs. Some ballots allow only one choice to be selected (single X voting); others allow ranking or selecting multiple options (Ranked ballots).
Different voting systems allow each voter to cast a different number of votes - only one (single voting as in First-past-the-post voting, Single non-transferable voting and Single transferable voting); as many as are being elected in a multiple-member district (multiple voting as used in Plurality block voting; more than one but fewer than are being elected in a multiple-member district (Limited voting). Most allow a voter to put just one vote on each candidate, but others allow a voter's votes to be piled on to one candidate.
Different voting systems require different levels of support to be elected. Plurality voting (First-past-the-post voting) elects the candidate with more votes than any other single candidate. It does not require the winner to achieve a voting majority, to have more than half of the total votes cast. In First-past-the-post voting, when more than two candidates run, the winner commonly has less than half of the vote, as few as 18 percent was recorded in 2014 in Toronto. In Instant-runoff voting, a candidate must have a majority of votes to be elected, although presence of exhausted votes may mean that the majority at time of final count is not majority of votes cast.
In STV, any candidate who takes quota (usually set at much less than half of the votes) is elected; others without quota (but with more votes than any other single candidate) may be declared elected as well.
Side effects of First-past-the-post voting include a waste of votes due to vote splitting, a two-party system and political polarization due to electing candidates that do not support centrism. To understand why a race using First-past-the-post voting tends to favor less-centric candidates, consider a simple lab experiment where students in a class vote for their favorite marble. If five marbles are assigned names and are placed "up for election", and if three of them are green, one is red, and one is blue, then a green marble will rarely win the election. The reason for the green's lack of success is vote splitting. The three green marbles will split the votes of those who prefer green. In fact, in this analogy, the only way that a green marble is likely to win is if more than three-fifths of the voters prefer green. If the same number of people prefer green as those who prefer red and blue, that is to say, if one-third of the voters prefer green, one-third prefer blue, and one-third prefer red, then each green marble will only get one-ninth of the vote, if the green marbles each take same number of votes, while the red and blue marbles will each get one-third, putting the green marbles at a serious disadvantage. If the experiment is repeated with other colors, the color that is in the majority (if the majority is split among multiple choices) will still rarely win. In other words, from a purely mathematical perspective, a single-winner system tends to favor a winner that is different from the majority, if the majority runs multiple candidates, and if the minority group runs just one candidate. This minority rule success can also result if multiple winners are elected and voters cast multiple votes (Plurality block voting). But even if the majority is split among multiple candidates, proportionate results can still be produced if votes can be transferred, as under STV, or if multiple winners are elected and each voter has just one vote.
Alternatives to First-past-the-post voting include approval voting, two-round, proportional representation, and instant-runoff voting. With approval voting, voters are encouraged to vote for as many candidates as they approve of, so the winner is much more likely to be any one of the five marbles because people who prefer green will be able to vote for every one of the green marbles. With two-round elections, the field of candidates is thinned prior to the second round of voting. In most cases, the winner must receive a majority of the votes, which is more than half. If no candidate obtains a majority in the first round, then the two candidates with the most significant plurality run again for the second round of voting. Variants exist regarding these two points: the requirement for being elected at the first round is sometimes less than 50%, and the rules for participation in the runoff may vary.
With single-round ranked voting, such as instant-runoff voting system as used in some elections in Australia and the United States, voters rank each candidate in order of preference (1,2,3,4 etc.). Votes are distributed to each candidate according to the first preferences. If no single candidate has 50% of the vote, then the candidate with the fewest votes is excluded and their votes are redistributed according to the voter's nominated order of preference. The process repeats itself until a candidate has 50% or more votes. The system is designed to produce the same result as an exhaustive ballot but using only a single round of voting.
Ranked voting is also used in a PR format. PR-STV is used in Australia, Ireland and Malta. Quota is calculated. In say a four-seat constituency, quota (if Droop quota is used) is 20 percent of the valid vote plus 1. Every candidate with quota (of 1st preferences alone or combination of first preferences and later preferences) will be elected. If a candidate has more than a quota and seats are yet to be filled, his/her surplus will be distributed to the other candidates in proportion to all of that candidate's 2nd preferences, in line with secondary preferences marked on the vote if any. If there are still candidates to be elected and no surplus votes to be transferred, the least-popular is eliminated, as above in AV or IRV, and the process continues until four candidates have reached a quota or are declared elected when the field of candidates is thinned to the number of remaining open seats.
In the Quota Borda System (QBS), the voters also cast their preferences, 1,2,3,4... as they wish. In the analysis, all 1st preferences are counted; all 2nd preferences are counted; after these preferences have been translated into points per the rules of a Modified Borda Count (MBC), the candidates' points are also counted. Seats are awarded to any candidates with a quota of 1st preferences; to any pair of candidates with two quotas of 1st/2nd preferences; and if seats are still to be filled, to those candidates with the highest MBC scores.
In a voting system that uses "multiple votes" (Plurality block voting), the voter can vote for any subset of the running candidates. So, a voter might vote for Alice, Bob, and Charlie, rejecting Daniel and Emily. Approval voting uses such multiple votes.
In a voting system that uses a "ranked vote", the voter ranks the candidates in order of preference. For example, they might mark a preference for Bob in the first place, then Emily, then Alice, then Daniel, and finally Charlie. Ranked voting systems, such as those used in Australia and Ireland, use a ranked vote.
In a voting system that uses a "scored vote" (or "range vote"), the voter gives each alternative a number between one and ten (the upper and lower bounds may vary). See cardinal voting systems.
Some "multiple-winner" systems such as the Single Non-Transferable Vote, SNTV, used in Afghanistan and Vanuatu give a single vote or one vote per elector per available position.
STV uses single ranked votes; block voting (Plurality-at-large voting) are often used for at-large positions such as on some city councils.
Finally, the Condorcet rule is used (sometimes) in decision-making. The voters or elected representatives cast their preferences on one, some, or all options, 1,2,3,4... as in PR-STV or QBS. In the analysis, option A is compared to option B, and if A is more popular than B, then A wins this pairing. Next, A is compared with option C, then D, and so on. Likewise, B is compared with C, D, etc. The option which wins the most pairings, (if there is one), is the Condorcet winner.-- ;
Referendums.
When the citizens of a country are invited to vote, they are participating in an election. However, people can also vote in referendums and initiatives. Since the end of the eighteenth century, more than five hundred national referendums (including initiatives) were organized in the world; among them, more than three hundred were held in Switzerland. Australia ranked second with dozens of referendums.
Most referendums are binary. The first multi-option referendum was held in New Zealand in 1894, and most of them are conducted under a two-round system. New Zealand had a five-option referendum in 1992, while Guam had a six-option plebiscite in 1982, which