Plurality-Majority Systems
PLURALITY/MAJORITY SYSTEMS
This plurality/majority family of voting systems is undoubtedly the one most familiar to Americans. They are the winner-take-all systems we most often use to elect officials to our local, state, and federal legislatures. These systems all require the winning candidate to garner either a plurality or a majority of the votes. We inherited this approach to voting from the British, and plurality-majority systems are used today primarily in Great Britain and its former colonies, including the United States, Canada, Australia, and India. The main purpose of these systems is to represent the majority or plurality of voters in a district, and (with the exception of at-large voting) to ensure representation of local geographical areas.
Four types of plurality-majority voting systems are described below.  The first two are the commonly-used plurality systems:  single-member district and at-large voting.  Less common are the two majority systems that use runoffs: the two-round system, and instant runoff voting .
 
At-Large Voting Results
There are several variations of at-large voting. In one of them that is used in Seattle and several other cities, the at-large seats are numbered and specific candidates vie for these individual seats. So candidates A and B would vie for seat one, candidates C and D for seat two, and so on. All the voters in the city cast one vote for their preferred candidates in each of these races, and the candidate with the plurality wins. In another variation, some cities use the numbered seats, but also have a residency requirement. Candidates for a particular seat must live in a certain area or district of the city. This ensures that all neighborhoods have some representation. Again all the voters in the city are able to vote for each of the seats.
Political Attributes.  At-large voting gives good representation to the largest political group or party.  It is also designed to ensure that city councilors represented the interests of the city as a whole, not the special interests of particular neighborhood districts. Like SMDP, it also encourages a two-party system and single-party legislative majorities. In addition, since there are no districts, this voting systems eliminates the possibility of gerrymandering. However, because this is a winner-take-all system, at-large voting shares most of the same problems as single-member district plurality voting, including the misrepresentation of parties, manufactured majorities, low voter turnout, high levels of wasted votes, and denial of fair representation to third parties, racial minorities and women. And it may make some of these problems worse. In particular, this system tends to be the worst at representing racial and political minorities. It allows a majority of the voters to win all the seats on the city council, thus shutting out these minorities from representation. (Note that in the election described above, the Republican voters are in the majority and so are able to elect all the city councilors.)  Finally, at-large voting, in its most common form, fails to ensure that all neighborhoods in the city are represented.
Two-Round Runoff Voting
The two-round system (TRS) is a majority voting system. Majority systems are currently used less commonly than plurality systems.  They require candidates in single-member district elections to garner a majority of the votes to win legislative office. TRS requires a runoff election between the two top candidates if no candidate wins a majority of the votes in the general election. This system is designed to solve one of the obvious problems of plurality voting: the possibility of electing a candidate that was supported only by a minority of the electorate in the district.
TRS is used rarely worldwide. Only two countries in Western Europe use TRS for legislative elections, France and Monaco. Several developing countries that came under French influence also use this system, including Mali, Togo, Chad, Gabon, and Haiti. In the United States, TRS is used in a number of jurisdictions, mostly on the local level and mostly in the South. Runoffs first came into use here at the beginning of the twentieth century when parties began to have primaries. These primaries often attracted more than two candidates and the resulting winner would sometimes garner much less than a majority of the vote. Today, runoffs are also used in some U.S. cities that have non-partisan elections, again primarily because such contests are more likely to draw more than two candidates.
How It Works.
In order to ensure that the winning candidate receives a majority of the vote, this system uses two rounds of voting with polling taking place on two separate days. Ballots are identical to those used in plurality voting (see above), and voters mark them in the same way. In the first round, all candidates are listed on the ballot and voters indicate their preference of one of them. All these votes are then added up and if a candidate receives a majority of the vote (50% + 1 vote), that candidate is declared elected. If no one receives a majority, the field is cut down to the top two candidates who received the highest number of votes, and a runoff election is held. The second election is typically held several weeks after the first. The winner is the candidate who gets the most votes, which is inevitably a majority, since there are only two candidates running.
Political Attributes. The two-round runoff system is only a slight modification of the single-member district plurality system, and it is really only designed to address one of its problems -- the possibility of a plurality winner -- which it does eliminate.  It also does well in encouraging a two-party system and single-party legislative majorities. However, it is still a winner-take-all voting system and so it shares all the basic problems of this approach to voting, including the misrepresentation of parties, manufactured majorities, gerrymandering, high levels of wasted votes, and denial of fair representation to third parties, racial minorities and women. In addition, it brings with it two more problems: the added expense of a second election, and the lower voter turnout that usually plagues those second elections.
Instant Runoff Voting
Instant runoff voting is also known as "IRV," and "majority preferential voting." In Australia, where this system is used to elect their lower house of parliament, it is called the "alternative vote." Like two-round voting, this majority system a minor variation of single-member district plurality voting that was developed to ensure that the winning candidate enjoys the support of the majority of the voters in the district. It was also thought to be an improvement over the two-round system because it does not require a separate election--it provides an "instant" runoff. 
How It Works.
In IRV voting, like plurality voting, all candidates are listed on the ballot. But instead of voting for only one candidate, voters rank the candidates in the order of their preference. This ranking process is illustrated in ballot below. It is an AccuVote ballot, which allows ballots to be scanned and tabulated by computer. It is similar to marking answers on the standardized tests used in schools. On this ballot, voters fill in numbered boxes to indicate their ranking of the candidates. The mark a "1" for their most preferred candidate, a "2" for their second preference, and so on.
Instant Runoff Ballot
The counting of the ballots is also different from plurality voting. First, all the number one preferences of the voters are counted. If a candidate receives over 50% of the first choice votes, he or she is declared elected. If no candidate receives a majority, then the candidate with the fewest votes is eliminated. The ballots of supporters of this defeated candidate are then transferred to whichever of the remaining candidates they marked as their number two choice. (It is as if you told the supporters of the last place candidate, "Your candidate cannot possibly win, so which of the remaining candidates would you like your vote to go to?) After this transfer, the votes are then recounted to see if any candidate now receives a majority of the vote. The process of eliminating the lowest candidate and transferring their votes continues until one candidate receives a majority of the continuing votes and wins the election.
This transfer process is illustrated in the table below. In this hypothetical election, there are 100,000 votes cast and no candidate receives over 50% of the vote in the first round. So the lowest candidate--Royce--is eliminated and his ballots are transferred to their second choices. 1,000 of Royceï¿½s supporters gave Chou as their second choice, and 6,000 indicated Kleinberg as their second choice. The new totals show that no one yet has a majority, so Chou is eliminated. 4,000 of Chou's votes are transferred to Kleinberg and 5,000 are given to Rosen. (If some of Chou's ballots had listed Royce as the second choice, they would have been transferred to their third choice, since Royce had been eliminated.) After this latest transfer is it clear that Kleinberg now has over 50% of the vote and she is declared the winner. As this example illustrates, this system essentially operates as a series of runoff elections, with progressively fewer candidates each time, until one candidate gets a majority of the vote.
Vote Counting in Instant Runoff Voting
Political Attributes
. IRV has the advantage of the two-round system -- ensuring a majority winner -- while avoiding its major disadvantages: the added expense and lower voter turnout of a second election. However, being in the plurality-majority family of voting systems, IRV shares most of the political disadvantages of this winner-take-all approach, including the misrepresentation of parties, manufactured majorities, gerrymandering, low voter turnout, high levels of wasted votes, and denial of fair representation to third parties, racial minorities and women.