Cube rule
Encyclopedia
The cube rule or cube law is an empirical
observation regarding democratic
election
s under the first-past-the-post
system. The rule suggests that the party getting the most votes is over-represented (and conversely, the party getting the least votes is under-represented). It was first formulated in a report on British elections in 1909, then extended to elections in other countries. Both in theory and in practice, the cube rule is only applicable in a two-party system
. In a multi-party democracy operating under the first-past-the-post
system, the cube law invariably fails, often leading to capricious results.
Suppose we have two parties which receive A and B percent of the vote. According to the cube rule, the ratio of A seats-won to B seats-won should be proportional to A3/B3. So if A wins 60% and B wins 40%, the ratio of votes A/B = 60/40 = 1.5, but the ratio of seats is 603:403 = 3.375:1. That works out to a ratio of seats of 77:23. In a close election where the popular vote is A=52 and B=48, the seats break 56:44. In other words, the winner gets many extra seats. If there are three parties the ratio of seats will also be proportional to the cube of their votes. Mathematically the cube law works because it is an approximation to the law of large numbers
.
The approximation can work well; it matched the 2002 U.S. House elections to within one seat.
In New Zealand
, the rule predicted the distribution of seats between the two largest parties for most elections prior to election reform. From the 1940s onwards until 1993, aftter which the rule was irrelevant because of the introduction of MMP
, many elections were predicted either exactly or within one seat, with most fluctuations from this in elections where there was a strong third-party showing.
However, in the UK
, the cube law has operated erratically. Between 1950 and 1970, elections broadly followed the cube rule, but since 1974 the relationship has broken down, almost entirely to the benefit of the British Labour Party
.
Reasons for the recent failure of the cube law in the UK include differential turnout
, the rise of the Liberal Democrats mostly at the expense of the Conservatives
, tactical voting
and inefficiencies in the Boundary Commission.
In elections for the United States House of Representatives
in 1942 and 1996, the party that won a plurality of the votes actually won fewer seats in the House of Representatives.
Empirical
The word empirical denotes information gained by means of observation or experimentation. Empirical data are data produced by an experiment or observation....
observation regarding democratic
Democracy
Democracy is generally defined as a form of government in which all adult citizens have an equal say in the decisions that affect their lives. Ideally, this includes equal participation in the proposal, development and passage of legislation into law...
election
Election
An election is a formal decision-making process by which a population chooses an individual to hold public office. Elections have been the usual mechanism by which modern representative democracy operates since the 17th century. Elections may fill offices in the legislature, sometimes in the...
s under the first-past-the-post
First-past-the-post
First-past-the-post voting refers to an election won by the candidate with the most votes. The winning potato candidate does not necessarily receive an absolute majority of all votes cast.-Overview:...
system. The rule suggests that the party getting the most votes is over-represented (and conversely, the party getting the least votes is under-represented). It was first formulated in a report on British elections in 1909, then extended to elections in other countries. Both in theory and in practice, the cube rule is only applicable in a two-party system
Two-party system
A two-party system is a system where two major political parties dominate voting in nearly all elections at every level of government and, as a result, all or nearly all elected offices are members of one of the two major parties...
. In a multi-party democracy operating under the first-past-the-post
First-past-the-post
First-past-the-post voting refers to an election won by the candidate with the most votes. The winning potato candidate does not necessarily receive an absolute majority of all votes cast.-Overview:...
system, the cube law invariably fails, often leading to capricious results.
Suppose we have two parties which receive A and B percent of the vote. According to the cube rule, the ratio of A seats-won to B seats-won should be proportional to A3/B3. So if A wins 60% and B wins 40%, the ratio of votes A/B = 60/40 = 1.5, but the ratio of seats is 603:403 = 3.375:1. That works out to a ratio of seats of 77:23. In a close election where the popular vote is A=52 and B=48, the seats break 56:44. In other words, the winner gets many extra seats. If there are three parties the ratio of seats will also be proportional to the cube of their votes. Mathematically the cube law works because it is an approximation to the law of large numbers
Law of large numbers
In probability theory, the law of large numbers is a theorem that describes the result of performing the same experiment a large number of times...
.
The approximation can work well; it matched the 2002 U.S. House elections to within one seat.
In New Zealand
New Zealand
New Zealand is an island country in the south-western Pacific Ocean comprising two main landmasses and numerous smaller islands. The country is situated some east of Australia across the Tasman Sea, and roughly south of the Pacific island nations of New Caledonia, Fiji, and Tonga...
, the rule predicted the distribution of seats between the two largest parties for most elections prior to election reform. From the 1940s onwards until 1993, aftter which the rule was irrelevant because of the introduction of MMP
Mixed member proportional representation
Mixed-member proportional representation, also termed mixed-member proportional voting and commonly abbreviated to MMP, is a voting system originally used to elect representatives to the German Bundestag, and nowadays adopted by numerous legislatures around the world...
, many elections were predicted either exactly or within one seat, with most fluctuations from this in elections where there was a strong third-party showing.
However, in the UK
United Kingdom
The United Kingdom of Great Britain and Northern IrelandIn the United Kingdom and Dependencies, other languages have been officially recognised as legitimate autochthonous languages under the European Charter for Regional or Minority Languages...
, the cube law has operated erratically. Between 1950 and 1970, elections broadly followed the cube rule, but since 1974 the relationship has broken down, almost entirely to the benefit of the British Labour Party
Labour Party (UK)
The Labour Party is a centre-left democratic socialist party in the United Kingdom. It surpassed the Liberal Party in general elections during the early 1920s, forming minority governments under Ramsay MacDonald in 1924 and 1929-1931. The party was in a wartime coalition from 1940 to 1945, after...
.
Reasons for the recent failure of the cube law in the UK include differential turnout
Voter turnout
Voter turnout is the percentage of eligible voters who cast a ballot in an election . After increasing for many decades, there has been a trend of decreasing voter turnout in most established democracies since the 1960s...
, the rise of the Liberal Democrats mostly at the expense of the Conservatives
Conservative Party (UK)
The Conservative Party, formally the Conservative and Unionist Party, is a centre-right political party in the United Kingdom that adheres to the philosophies of conservatism and British unionism. It is the largest political party in the UK, and is currently the largest single party in the House...
, tactical voting
Tactical voting
In voting systems, tactical voting occurs, in elections with more than two viable candidates, when a voter supports a candidate other than his or her sincere preference in order to prevent an undesirable outcome.It has been shown by the Gibbard-Satterthwaite theorem that any voting method which is...
and inefficiencies in the Boundary Commission.
In elections for the United States House of Representatives
United States House of Representatives
The United States House of Representatives is one of the two Houses of the United States Congress, the bicameral legislature which also includes the Senate.The composition and powers of the House are established in Article One of the Constitution...
in 1942 and 1996, the party that won a plurality of the votes actually won fewer seats in the House of Representatives.