Proportional representation (PR) refers to a type of electoral system under which subgroups of an electorate are reflected proportionately in the elected body. The concept applies mainly to geographical and political divisions of the electorate. The essence of such systems is that all votes cast - or almost all votes cast - contribute to the result and are actually used to help elect someone—not just a plurality, or a bare majority—and that the system produces mixed, balanced representation reflecting how votes are cast.
Party-list proportional representation (list-PR) is a subset of proportional representation electoral systems in which multiple candidates are elected through their position on an electoral list. They can also be used as part of mixed-member electoral systems.
Mixed-member proportional representation is a mixed electoral system in which votes cast are considered in local elections and also to determine overall party vote tallies, which are used to allocate additional members to produce or deepen overall Proportional representation.
The D'Hondt method, also called the Jefferson method or the greatest divisors method, is a method for allocating seats in parliaments among federal states, or in party-list proportional representation systems. It belongs to the class of highest-averages methods.
The Webster method, also called the Sainte-Laguë method or the major fractions method, is a method for allocating seats in a parliament among federal states, or among parties in a party-list proportional representation system.
A highest-averages method, also called a divisor method, is a class of methods for allocating seats in a parliament among agents such as political parties or federal states. A divisor method is an iterative method: at each iteration, the number of votes of each party is divided by its divisor, which is a function of the number of seats currently allocated to that party. The next seat is allocated to the party whose resulting ratio is largest.
The largest remainder method is one way of allocating seats proportionally for representative assemblies with party list voting systems. It contrasts with various highest averages methods.
An apportionment paradox exists when the rules for apportionment in a political system produce results which are unexpected or seem to violate common sense.
The Huntington–Hill method is a way of allocating seats proportionally for representative assemblies such as the United States House of Representatives. The method assigns seats by finding a modified divisor D such that each constituency's priority quotient, using the geometric mean of the lower and upper quota for the divisor, yields the correct number of seats that minimizes the percentage differences in the size of subconstituencies. When envisioned as a proportional electoral system, it is effectively a highest averages method of party-list proportional representation in which the divisors are given by ,n being the number of seats a state or party is currently allocated in the apportionment process and n + 1 is the number of seats the state or party would have if it is assigned to the party list. Although no legislature uses this method of apportionment to assign seats to parties after an election, it was considered for House of Lords elections under the ill-fated House of Lords Reform Bill.
Leveling seats, commonly known also as adjustment seats, are an election mechanism employed for many years by all Nordic countries in elections for their national legislatures. In 2013, Germany also introduced national leveling seats for their national parliament, the Bundestag. Leveling seats are seats of additional members elected to supplement the members directly elected by each constituency. The purpose of these additional seats is to ensure that each party's share of the total seats is roughly proportional to the party's overall shares of votes at the national level.
An electoral system or voting system is a set of rules that determine how elections and referendums are conducted and how their results are determined. Electoral systems are used in politics to elect governments, while non-political elections may take place in business, non-profit organisations and informal organisations. These rules govern all aspects of the voting process: when elections occur, who is allowed to vote, who can stand as a candidate, how ballots are marked and cast, how the ballots are counted, how votes translate into the election outcome, limits on campaign spending, and other factors that can affect the result. Political electoral systems are defined by constitutions and electoral laws, are typically conducted by election commissions, and can use multiple types of elections for different offices.
A mixed electoral system or mixed-member electoral system combines methods of majoritarian and proportional representation (PR). The majoritarian component is usually first-past-the-post voting (FPTP/SMP), whereas the proportional component is most often based on party-list PR. The results of the combination may be mixed-member proportional (MMP), where the overall results of the elections are proportional, or mixed-member majoritarian, in which case the overall results are semi-proportional, retaining disproportionalities from the majoritarian component.
In mathematics and political science, the quota rule describes a desired property of a proportional apportionment or election method. It states that the number of seats that should be allocated to a given party should be between the upper or lower roundings of its fractional proportional share. As an example, if a party deserves 10.56 seats out of 15, the quota rule states that when the seats are allotted, the party may get 10 or 11 seats, but not lower or higher. Many common election methods, such as all highest averages methods, violate the quota rule.
Mathematics of apportionment describes mathematical principles and algorithms for fair allocation of identical items among parties with different entitlements. Such principles are used to apportion seats in parliaments among federal states or political parties. See apportionment (politics) for the more concrete principles and issues related to apportionment, and apportionment by country for practical methods used around the world.
House monotonicity is a property of apportionment methods and multiwinner voting systems. These are methods for allocating seats in a parliament among federal states. The property says that, if the number of seats in the "house" increases, and the method is re-activated, then no state should have less seats than it previously had. A method that fails to satisfy house-monotonicity is said to have the Alabama paradox.
State-population monotonicity is a property of apportionment methods, which are methods of allocating seats in a parliament among federal states. The property says that, if the population of a state increases faster than that of other states, then it should not lose a seat. An apportionment method that fails to satisfy this property is said to have a population paradox.
Seat bias is a property describing methods of apportionment. These are methods used to allocate seats in a parliament among federal states or among political parties. A method is biased if it systematically favors small parties over large parties, or vice versa. There are various ways to compute the bias of apportionment methods.
Coherence, also called uniformity or consistency, is a criterion for evaluating rules for fair division. Coherence requires that the outcome of a fairness rule is fair not only for the overall problem, but also for each sub-problem. Every part of a fair division should be fair.
Vote-ratio monotonicity (VRM) is a property of apportionment methods, which are methods of allocating seats in a parliament among political parties. The property says that, if the ratio between the number of votes won by party A to the number of votes won by party B increases, then it should NOT happen that party A loses a seat while party B gains a seat.
Balance or balancedness is a property of apportionment methods, which are methods of allocating identical items between among agens, such as dividing seats in a parliament among political parties or federal states. The property says that, if two agents have exactly the same entitlements, then the number of items they receive should differ by at most one. So if two parties win the same number of votes, or two states have the same populations, then the number of seats they receive should differ by at most one.
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