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67-505: The term is used as a suffix referring to specific members (such as the U.S. congressional Representative/the Member/Rep. for Wyoming at large ). It figures as a generic prefix of its subject matter (such as Wyoming is an at-large U.S. congressional district, at present). It is commonly used when making or highlighting a direct contrast with subdivided equivalents that may be past or present, or seen in exotic comparators. It indicates that

134-585: A candidate-based PR system, has only rarely been used to elect more than 21 in a single contest. Some PR systems use at-large pooling or regional pooling in conjunction with single-member districts (such as the New Zealand MMP and the Scottish additional member system ). Other PR systems use at-large pooling in conjunction with multi-member districts ( Scandinavian countries ). Pooling is used to allocate leveling seats (top-up) to compensate for

201-707: A consequence of universal generalization : the technique shows ( c ∈ A ) ⇒ ( c ∈ B ) {\displaystyle (c\in A)\Rightarrow (c\in B)} for an arbitrarily chosen element c . Universal generalisation then implies ∀ x ( x ∈ A ⇒ x ∈ B ) , {\displaystyle \forall x\left(x\in A\Rightarrow x\in B\right),} which

268-475: A few list-PR systems). A country-wide pooling of votes to elect more than a hundred members is used in Angola, for example. Where PR is desired at the municipal level, a city-wide at-large districting is sometimes used, to allow as large a district magnitude as possible. For large districts, party-list PR is often used, but even when list PR is used, districts sometimes contain fewer than 40 or 50 members. STV,

335-480: A number of seats roughly based on its population size (see degressive proportionality ) and in each member state, the election must also be held using a PR system (with proportional results based on vote share). The most widely used families of PR electoral systems are party-list PR, used in 85 countries; mixed-member PR (MMP), used in 7 countries; and the single transferable vote (STV), used in Ireland, Malta,

402-469: A small number of members are elected by the entire European Union at large from Union-wide party lists. This is among a number of proposed reforms to deepen European integration . In Israel , elections for the Knesset (the national parliament ) are conducted at large by proportional representation from party lists. Election of municipal and town (but not regional) councils are on the same basis. In

469-514: Is less than y (an irreflexive relation ). Similarly, using the convention that ⊂ {\displaystyle \subset } is proper subset, if A ⊆ B , {\displaystyle A\subseteq B,} then A may or may not equal B , but if A ⊂ B , {\displaystyle A\subset B,} then A definitely does not equal B . Another example in an Euler diagram : The set of all subsets of S {\displaystyle S}

536-486: Is vacuously a subset of any set X . Some authors use the symbols ⊂ {\displaystyle \subset } and ⊃ {\displaystyle \supset } to indicate subset and superset respectively; that is, with the same meaning as and instead of the symbols ⊆ {\displaystyle \subseteq } and ⊇ . {\displaystyle \supseteq .} For example, for these authors, it

603-504: Is allocated seats based on its party share. Some party-list PR systems use overall country-wide vote counts; others count vote shares in separate parts of the country and allocate seats in each part according to that specific vote count. Some use both. List PR involves parties in the election process. Voters do not primarily vote for candidates (persons), but for electoral lists (or party lists ), which are lists of candidates that parties put forward. The mechanism that allocates seats to

670-420: Is also randomness – a party that receives more votes than another party might not win more seats than the other. Any such dis-proportionality produced by the district elections is addressed, where possible, by the allocation of the compensatory additional members. (Number of districts won) (party-list PR seats) under MMP MMP gives only as many compensatory seats to a party as they need to have

737-803: Is called its power set , and is denoted by P ( S ) {\displaystyle {\mathcal {P}}(S)} . The inclusion relation ⊆ {\displaystyle \subseteq } is a partial order on the set P ( S ) {\displaystyle {\mathcal {P}}(S)} defined by A ≤ B ⟺ A ⊆ B {\displaystyle A\leq B\iff A\subseteq B} . We may also partially order P ( S ) {\displaystyle {\mathcal {P}}(S)} by reverse set inclusion by defining A ≤ B  if and only if  B ⊆ A . {\displaystyle A\leq B{\text{ if and only if }}B\subseteq A.} For

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804-419: Is described here. The mixed-member proportional system combines single member plurality voting (SMP), also known as first-past-the-post (FPTP), with party-list PR in a way that the overall result of the election is supposed to be proportional. The voter may vote for a district candidate as well as a party. The main idea behind MMP is compensation , meaning that the list-PR seat allocation is not independent of

871-533: Is equivalent to A ⊆ B , {\displaystyle A\subseteq B,} as stated above. If A and B are sets and every element of A is also an element of B , then: If A is a subset of B , but A is not equal to B (i.e. there exists at least one element of B which is not an element of A ), then: The empty set , written { } {\displaystyle \{\}} or ∅ , {\displaystyle \varnothing ,} has no elements, and therefore

938-456: Is more complicated than first-past-the-post voting , but the following example shows how the vote count is performed and how proportionality is achieved in a district with 3 seats. In reality, districts usually elect more members than that in order to achieve more proportional results. A risk is that if the number of seats is larger than, for example, 10 seats, the ballot will be so large as to be inconvenient and voters may find it difficult to rank

1005-551: Is no need for a single office (e.g. a president, or mayor) to be elected proportionately if no votes are for parties (subgroups). In the European Parliament , for instance, each member state has a number of seats that is (roughly) proportional to its population, enabling geographical proportional representation. For these elections, all European Union (EU) countries also must use a proportional electoral system (enabling political proportional representation): When n % of

1072-489: Is not considered to make an electoral system "proportional" the way the term is usually used. For example, the US House of Representatives has 435 members, who each represent a roughly equal number of people; each state is allocated a number of members in accordance with its population size (aside from a minimum single seat that even the smallest state receives), thus producing equal representation by population. But members of

1139-476: Is over-representation. They have high apportionment but are ethnically diverse and of exceptional geographical size. Provinces are divided to make up the other 335 electoral districts (ridings or comtés). Canada's 102 senators are appointed to serve a large region within a province or a city.) The Conference on the Future of Europe has proposed adding "transnational lists" to the European Parliament , in which

1206-458: Is possible for A and B to be equal; if they are unequal, then A is a proper subset of B . The relationship of one set being a subset of another is called inclusion (or sometimes containment ). A is a subset of B may also be expressed as B includes (or contains) A or A is included (or contained) in B . A k -subset is a subset with k elements. When quantified, A ⊆ B {\displaystyle A\subseteq B}

1273-548: Is represented as ∀ x ( x ∈ A ⇒ x ∈ B ) . {\displaystyle \forall x\left(x\in A\Rightarrow x\in B\right).} One can prove the statement A ⊆ B {\displaystyle A\subseteq B} by applying a proof technique known as the element argument : Let sets A and B be given. To prove that A ⊆ B , {\displaystyle A\subseteq B,} The validity of this technique can be seen as

1340-405: Is true of every set A that A ⊂ A . {\displaystyle A\subset A.} (a reflexive relation ). Other authors prefer to use the symbols ⊂ {\displaystyle \subset } and ⊃ {\displaystyle \supset } to indicate proper (also called strict) subset and proper superset respectively; that is, with

1407-580: Is used with 5-member districts, it is common for successful candidates to receive 16.6 percent of the vote in the district. This produces a high effective threshold in the districts, and the country maintains a very strong two-party system. However, about 4000 voters in a district would be enough to elect a third-party candidate if voters desired but this seldom happens. Conversely, New South Wales, which uses STV to elect its legislative council in 21-seat contests, sees election of representatives of seven or eight parties each time. In this election about 1/22nd of

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1474-653: The Australian Senate , and Indian Rajya Sabha . Proportional representation systems are used at all levels of government and are also used for elections to non-governmental bodies, such as corporate boards . All PR systems require multi-member election contests, meaning votes are pooled to elect multiple representatives at once. Pooling may be done in various multi-member voting districts (in STV and most list-PR systems) or in single countrywide – a so called at-large  – district (in only

1541-687: The Netherlands , elections for the House of Representatives (the lower house of the States-General , the national parliament ) are conducted at large by proportional representation from party lists. This manner of election applies to the Senate . All voters can cast twelve votes to refresh half of the Senate, namely twelve senators, from a longer list of candidates. The simple tally determines

1608-500: The k -tuple from { 0 , 1 } k , {\displaystyle \{0,1\}^{k},} of which the i th coordinate is 1 if and only if s i {\displaystyle s_{i}} is a member of T . The set of all k {\displaystyle k} -subsets of A {\displaystyle A} is denoted by ( A k ) {\displaystyle {\tbinom {A}{k}}} , in analogue with

1675-456: The state has a single representative, that will be a representative at large. This is a table of every such instance. It shows the situation applied to a small, varying group of states in three periods. The 33rd Congress began in 1853; it ended two years later. The 38th began in 1863; the 50th ended in 1889. The 53rd began in 1893; the 89th ended in January 1967, the final such period. This

1742-619: The 1990s. After the 2010 United States redistricting cycle , Nevada eliminated their two remaining multi-member senate districts and implemented single-winner districts in both houses. In 2018, West Virginia passed a law switching all remaining multi-winner House of Delegates seats to single-winner districts following the 2020 United States Census . Since passage of the Voting Rights Act of 1965 and lessening of some historic barriers to voter registration and voting, legal challenges have been made based on at-large election schemes at

1809-543: The 200-seat legislature as large as in the examples that follow, about 67 three-seat districts would be used. Districts with more seats would provide more proportional results – one form of STV in Australia uses a district with 21 members being elected at once. With a larger district magnitude, it is more likely that more than two parties will have some of their candidates elected. For example, in Malta , where STV

1876-562: The House are elected in single-member districts generally through first-past-the-post elections : a single-winner contest does not produce proportional representation as it has only one winner. Conversely, the representation achieved under PR electoral systems is typically proportional to a district's population size (seats per set amount of population), votes cast (votes per winner), and party vote share (in party-based systems such as party-list PR ). The European Parliament gives each member state

1943-670: The US House of Representatives). Votes and seats often cannot be mathematically perfectly allocated, so some amount of rounding has to be done. The various methods deal with this in different ways, although the difference is reduced if there are many seats – for example, if the whole country is one district. Party-list PR is also more complicated in reality than in the example, as countries often use more than one district, multiple tiers (e.g. local, regional and national), open lists or an electoral threshold . This can mean that final seat allocations are frequently not proportional to

2010-433: The absence or insufficient number of leveling seats (in list PR, MMP or AMS) may produce disproportionality. Other sources are electoral tactics that may be used in certain systems, such as party splitting in some MMP systems. Nonetheless, PR systems approximate proportionality much better than other systems and are more resistant to gerrymandering and other forms of manipulation. Proportional representation refers to

2077-632: The black minority makes up more than one-third of the population. In another instance, in 2013 Fayette County, Georgia , which had an estimated 70% white majority and 20% black minority, was ordered by a federal district court to develop single-member districts for election of members to its county council and its school board. Due to at-large voting, African Americans had been unable to elect any candidate of their choice to either of these boards for decades. Such local election systems have become subject to litigation since enabling more representative elections can create entry points for minorities and women into

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2144-412: The candidates determine the winner. This is done using a preferential ballot . The ranking is used to instruct election officials of how the vote should be transferred in case the first preference is marked for an un-electable candidate or for an already elected candidate. Each voter casts one vote and the district used elects multiple members (more than one, usually 3 to 7). Because parties play no role in

2211-570: The city council from single-member districts. The town of Islip, New York was sued by four residents in 2018 for violating the Voting Rights Act by maintaining a discriminatory at-large council system. One-third of Islip's population is Hispanic, but only one person of color has ever been elected to a town seat. As part of the settlement reached in 2020, the at-large system will be abolished and replaced by four council districts by 2023. Some states have passed laws that further discourage

2278-571: The city or province may be split into as many as seven districts, and each then elects at least two members. Article One of the United States Constitution provides for direct election of members of the House of Representatives . The Uniform Congressional District Act , enacted in 1967 and codified as 2 U.S.C.   § 2c , dictates that representatives must be elected from geographical districts and that these must be single-member districts . Indeed it confirms when

2345-510: The context of voting systems, PR means that each representative in an assembly is elected by a roughly equal number of voters. In the common case of electoral systems that only allow a choice of parties, the seats are allocated in proportion to the vote tally or vote share each party receives. The term proportional representation may be used to mean fair representation by population as applied to states, regions, etc. However, representation being proportional with respect solely to population size

2412-464: The county or city level, including in school board elections, in numerous jurisdictions where minorities had been effectively excluded from representation on local councils or boards. An example is Charleston County, South Carolina , which was sued in 2001 and reached a settlement in 2004. Its county commission changed to nine members elected from single-member districts; in 2015 they included six white Republicans and three African-American Democrats, where

2479-416: The described zone has no further subsets used for the same representative purpose. An exception is a nil-exceptions arrangement of overlapping tiers (resembling or being district and regional representatives, one set of which is at large ) for return to the very same chamber, and consequent issue of multiple ballots for plural voting to every voter. This avoids plural voting competing with single voting in

2546-529: The disproportional results produced in single-member districts using FPTP or to increase the fairness produced in multi-member districts using list PR. PR systems that achieve the highest levels of proportionality tend to use as general pooling as possible (typically country-wide) or districts with large numbers of seats. Due to various factors, perfect proportionality is rarely achieved under PR systems. The use of electoral thresholds (in list PR or MMP), small districts with few seats in each (in STV or list PR), or

2613-504: The elected body. The concept applies mainly to political divisions ( political parties ) among voters. The essence of such systems is that all votes cast – or almost all votes cast – contribute to the result and are effectively used to help elect someone. Under other election systems, a bare plurality or a scant majority are all that are used to elect candidates. PR systems provide balanced representation to different factions, reflecting how votes are cast. In

2680-469: The election are as follows (popular vote). Under party-list PR, every party gets a number of seats proportional to their share of the popular vote. This is done by a proportional formula or method; for example, the Sainte-Laguë method  – these are the same methods that may be used to allocate seats for geographic proportional representation (for example, how many seats each states gets in

2747-432: The electorate support a particular political party or set of candidates as their favourite, then roughly n % of seats are allotted to that party or those candidates. All PR systems aim to provide some form of equal representation for votes but may differ in their approaches on how they achieve this. Party-list PR is the most commonly used version of proportional representation. Voters cast votes for parties and each party

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2814-450: The example below, the Droop quota is used and so any candidate who earns more than 25 percent of the vote is declared elected. Note that it is only possible for 3 candidates to each achieve that quota. In the first count, the first preference (favourite candidate) marked on each of the ballots is counted. Candidates whose vote tally equals or passes the quota are declared elected as shown in

2881-497: The example below. (first preferences) Next, surplus votes belonging to those already elected, votes the candidates received above the quota (votes that they did not need to be elected), are transferred to the next preference marked by the voters who voted for them. Continuing the example, suppose that all voters who marked first preference for Jane Doe marked John Citizen as their second choice. Based on this, Jane Doe's surplus votes are transferred to John Citizen, John Citizen passes

2948-530: The general principle found in any electoral system in which the popularly chosen subgroups (parties) of an electorate are reflected proportionately in the elected body. To achieve that intended effect, proportional electoral systems need to either have more than one seat in each district (e.g. single transferable vote ), or have some form of compensatory seats (e.g. mixed-member proportional representation apportionment methods ). A legislative body (e.g. assembly, parliament) may be elected proportionally, whereas there

3015-543: The jurisdiction, an inherent different level of democratic power . Examples of a democratic power disparity were found in a small number of states at certain U.S. Congresses, between 1853 and 1967, and in the old lower houses of the United Kingdom and Ireland , whereby certain voters could vote for (and lobby) at-large (whole-state/County) and district(-based) representatives to them, giving zones of plural voting and thus representation contrasting with zones, for

3082-401: The many candidates, although 21 are elected through STV in some elections with no great difficulty. (In many STV systems, voters are not required to mark more choices than desired. Even if all voters marked only one preference, the resulting representation would be more balanced than under single-winner FPTP.) Under STV, an amount that guarantees election is set, which is called the quota . In

3149-481: The mayor is elected at large as well. Municipal election at large is widespread in small towns to avoid " them and us " cultural dissociation produced by partition of voters into wards and their representatives thus being seen to represent only a specific part of the city. It is also used in many large cities in Canada. The voting method in all such elections and multi-member wards today is plurality block voting . (In

3216-920: The notation for binomial coefficients , which count the number of k {\displaystyle k} -subsets of an n {\displaystyle n} -element set. In set theory , the notation [ A ] k {\displaystyle [A]^{k}} is also common, especially when k {\displaystyle k} is a transfinite cardinal number . Proportional representation Condorcet methods Positional voting Cardinal voting Quota-remainder methods Approval-based committees Fractional social choice Semi-proportional representation By ballot type Pathological response Strategic voting Paradoxes of majority rule Positive results Proportional representation ( PR ) refers to any type of electoral system under which subgroups of an electorate are reflected proportionately in

3283-491: The number of seats of each party be proportional. Another way to say this is that MMP focuses on making the outcome proportional. Compare the MMP example to a mixed-member majoritarian system, where the party-list PR seat allocation is independent of the district results (this is also called parallel voting ). There is no compensation (no regard to how the district seats were filled) when allocating party-list seats so as to produce

3350-597: The other members. It follows that such true or quasi- local government units do not in the purest sense elect members at large when their geography is analysed, as each member co-exists with the others who have territorial overlap, as representing greater or lower-rank districts. The members are by law chosen by the public directly or indirectly. Members chosen by the City Council or the Sangguniang Panlalawigan (Provincial Board) are elected such that

3417-419: The parties' vote share. The single transferable vote is an older method than party-list PR, and it does not need to formally involve parties in the election process. Instead of parties putting forward ordered lists of candidates from which winners are drawn in some order, candidates run by name, each voter marks preferences for candidates, with only one marked preference used to place the vote, and votes cast for

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3484-408: The parties/lists is how these systems achieve proportionality. Once this is done, the candidates who take the seats are based on the order in which they appear on the list. This is the basic, closed list version of list PR. An example election where the assembly has 200 seats to be filled is presented below. Every voter casts their vote for the list created by their favourite party and the results of

3551-511: The party's seats. 81 percent of the voters saw their first choice elected. At least 15 percent of them (the Doe first, Citizen second voters) saw both their first and second choices elected – there were likely more than 15 percent if some "Citizen first" votes gave their second preference to Doe. Every voter had satisfaction of seeing someone of the party they support elected in the district. for candidates of party Under STV, to make up

3618-568: The past, single transferable voting (STV) was used in 20 Canadian municipalities, either in at-large contests or multi-member wards.) Notable larger instances are, from west to east: (As well, the three territories - Yukon , Nunavut , and the Northwest Territories - are federally served in the Parliament of Canada by one Member of Parliament and Senator each. These are so sparsely populated that even one member each

3685-515: The political system and provide more representative government. In the late 1980s, several major cities in Tennessee reached settlement in court cases to adopt single-member districts to enable minorities to elect candidates of their choice to city councils, who had previously been excluded by at-large voting favoring the majority population. By 2015, voters in two of these cities had elected women mayors who had gotten their start in being elected to

3752-1000: The power set P ⁡ ( S ) {\displaystyle \operatorname {\mathcal {P}} (S)} of a set S , the inclusion partial order is—up to an order isomorphism —the Cartesian product of k = | S | {\displaystyle k=|S|} (the cardinality of S ) copies of the partial order on { 0 , 1 } {\displaystyle \{0,1\}} for which 0 < 1. {\displaystyle 0<1.} This can be illustrated by enumerating S = { s 1 , s 2 , … , s k } , {\displaystyle S=\left\{s_{1},s_{2},\ldots ,s_{k}\right\},} , and associating with each subset T ⊆ S {\displaystyle T\subseteq S} (i.e., each element of 2 S {\displaystyle 2^{S}} )

3819-409: The quota and so is declared elected to the third and last seat that had to be filled. Even if all of Fred Rubble's surplus had gone to Mary Hill, the vote transfer plus Hill's original votes would not add up to quota. Party B did not have two quotas of votes so was not due two seats, while Party A was. It is possible, in realistic STV elections, for a candidate to win without quota if they are still in

3886-528: The results of the district level voting. First-past-the-post is a single winner system and cannot be proportional (winner-takes-all), so these disproportionalities are compensated by the party-list component. A simple, yet common version of MMP has as many list-PR seats as there are single-member districts. In the example it can be seen, as is often the case in reality, that the results of the district elections are highly disproportional: large parties typically win more seats than they should proportionally, but there

3953-421: The running when the field of candidates has thinned to the number of remaining open seats. In this example, the district result is balanced party-wise. No one party took all the seats, as frequently happens under FPTP or other non-proportional voting systems. The result is fair – the most popular party took two seats; the less popular party took just one. The most popular candidates in each party won

4020-736: The same meaning as and instead of the symbols ⊊ {\displaystyle \subsetneq } and ⊋ . {\displaystyle \supsetneq .} This usage makes ⊆ {\displaystyle \subseteq } and ⊂ {\displaystyle \subset } analogous to the inequality symbols ≤ {\displaystyle \leq } and < . {\displaystyle <.} For example, if x ≤ y , {\displaystyle x\leq y,} then x may or may not equal y , but if x < y , {\displaystyle x<y,} then x definitely does not equal y , and

4087-507: The same national assembly, of single voting and representation. In 1964 the U.S. Supreme Court banned such plural voting for the U.S. Congress (thus banning at-large, whole-state congressional districts which overlap state subdivision congressional districts). Universal principles apply regardless whether election(s) are for a member at large, or not. Many municipalities in Canada elect part or all of their city councils at large. In most,

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4154-509: The use of at-large districts. For example, the California Voting Rights Act removes one of the criteria required for a successful federal Voting Rights Act challenge, which has thus resulted in hundreds of cities, school districts, and special districts moving to single member area-based elections. Some jurisdictions have kept at-large city councils and boards. The solution adopted by Cambridge, Massachusetts in 1941

4221-402: The vote count, STV may be used for nonpartisan elections, such as the city council of Cambridge, Massachusetts . A large proportion of the votes cast are used to actually elect someone so the result is mixed and balanced with no one voting block taking much more than its due share of the seats. Where party labels are indicated, proportionality party-wise is noticeable. Counting votes under STV

4288-481: The vote in the state is enough to take a seat, and seven or eight parties take at least that many votes, demonstrating a different voting pattern than Malta exhibits. Mixed-member proportional representation combines election of district members with election of additional members as compensatory top-up. Often MMP systems use single-member districts (SMDs) to elect district members. (Denmark, Iceland and Sweden use multi-member districts in their MMP systems.) MMP with SMDs

4355-619: The winners ( plurality-at-large voting ). Provinces with smaller populations elect their House representative at large. Some cities that has its own congressional district also elect its representatives this way. Most other provinces and a few cities are divided into two or more districts. Party-list elections are elected nationwide at-large. Likewise, the Sangguniang Kabataan (Youth Councils), Sangguniang Barangay (Village Councils), Sangguniang Bayan (Municipal Councils) and some Sangguniang Panlungsod (City Councils) elect

4422-542: Was due to the 1964 case of Reynolds v. Sims : the United States Supreme Court determined that the general basis of apportionment must be " one person, one vote ." As of 2021, ten U.S. states have at least one legislative chamber which uses multi-winner at-large districts: In the 1980s, Florida, Hawaii, Illinois, South Carolina, and Virginia all moved entirely from multi-winner districts in either chamber, followed by Alaska, Georgia, and Indiana in

4489-466: Was to elect all council officials via Ranked-choice proportional representation , Single transferable voting . Five territorial governments in the United States elect some or all of their members at large or in multi-member districts: Subset In mathematics, a set A is a subset of a set B if all elements of A are also elements of B ; B is then a superset of A . It

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