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Disproportional representation is a term for a broad range of electoral systems, in which the correlation between the number of votes won by a party and the number of seats won is small or non-existent. Disproportional systems must be contrasted with non-proportional systems, such plurality voting, which, while they occasionally produce disproportional results, generally involve a correlation between vote share and seat share. Nations which have used disproportionate representation historically include the United Kingdom and Liberia, and many other African nations are modern proponents of the system.
Disproportional methods also exist for the deciding of presidential elections. The commonest are runoff, in which the two candidates who receive the most votes in the first half of the election face each other in a runoff, and the candidate with the least votes wins. Other methods include second and last past the post, in which the candidates with the second highest and lowest shares of the vote win respectively. Such methods often favor incumbent leaders.
A common criticism of disproportional representation is that it leads to post-election violence, voter apathy, and low legitimacy and transparency.  While it is widely argued that disproportional representation is a destabilizing force, the counterexamples of such nations as Egypt which have prospered under disproportional representation are often cited. It remains to be seen whether disproportional representation is a viable electoral system in any industrial or post-industrial nation, as it is almost exclusively found in the developing world.
edit Last past the post
The last past the post system is conducted with small, single-member constituencies. To win an election under the last past the post system, it is necessary for you to receive the lowest number of votes whilst still managing to pass the post. An example: if there are four candidates, and they receive respectively 11320, 9238, 5403 and 6 votes, the candidate who won six votes will have won. The post is only considered to play a significant part in the proceedings if all candidates have relatively high vote shares, as the weight of the ballot boxes normally counteracts any experience present. Last past the post is considered discriminatory in many countries due to the difficulty those with physical disabilities face in riding a horse.  Last past the post has been traditionally used in nations affiliated to the British Commonwealth, including Nigeria, Kenya and Cameroon. It should be made clear that in nations which use last past the post, the results are often converted: by giving each candidate the reciprocal of his/her vote share, summing the reciprocals and then calculating the percentage share of each candidate. The percentage share is thus applied to the vote, and gives the impression that the election was carried out by first past the post, a non-disproportional system. While this is generally done to encourage the legitimacy of the government, it tends to provoke post-election violence.  Variations on this system include second past the post etc. Anti-runoff, which has been used in the presidential elections of Zimbabwe and Iran is similar, but the post is never a part of proceedings.
edit First past the post with significant constituency size deviation
England, and its successor state Britian, used this system from 1295 to 1832, and the stability of the nation during that period is often cited as a reason to favour disproportional methods. [note 1] It was deemed appropriate to use first past the post, whereby the candidate who receives the greatest number of votes was elected as representative for the given region. However, the number of voters in each constituency varied enormously. London had a population of 1729949, and was allowed to elect one Member of Parliament. Pitton Green, a village near Swansea had a population of 32, and was allowed to elect one Member of Parliament. Old Sarum, a hill in Wiltshire had a population of 3 sheep and was allowed to elect one Member of Parliament. The Member for Old Sarum was regularly evicted from the Commons for defecating on the seating. 
edit Decided vote
This is an extremely common method of disproportional representation. The basic process is this: the government holds a private meeting to decide how many seats they want to win in the election. An election is then held, and all the ballot boxes and paper are burnt. The originally decided percentages of vote share and seats share are then published. Egypt frequently uses this system. Critics of this method are often concerned about the carbon dioxide impact of burning the ballots, as they claim that it contributes to global warming. The Egyptian government responded to this criticism by suggesting that they might use violence in future elections to reduce turnout and thus the amount of ballots which need to be printed.  The Liberian True Whig Party took decided vote to new extremes, announcing themselves to have won 253220 votes in the 1910 parliamentary election. In fact, only direct relatives of the president were allowed to vote, and the ballot papers were never read. The population of Liberia at the time was around 30000, and only 3.2% of those could read. 
- ↑ During this period, England/Britain fought 52 separate major wars, suffered two successful revolutions, including a civil war, and shut down its disproportionally elected parliament multiple times. We're just saying.
- ↑ Voter turnout and disproportional representation in the developing world, Prof. John C. Willburton, Celebration University Press, 2001, pp. 44-48
- ↑ Electoral Systems, John Whykham, Salisbury Salad Press, 1987, p. 137
- ↑ Historical elections and disability in Nigeria, John Williams (ed.), Scothbrough University Press, 2008, pp. 33-41
- ↑ Electoral Success in the Developing World, John Binan, Laman press, 1997, p. 3219
- ↑ Wiltshire elections and members before the Great Reform Act, John Wilberton, Camelford University Press, 1883, pp. 52-3
- ↑ Elections in the Middle East, 1937-2010, John Simple, Gatton University Press, 2010, pp. 139
- ↑ الانتخابات والبيئ (Elections and the Environment), Egyptian Ministry of the Environment, 2010, p. xi
- ↑ Complete directory of world elections, 1295-2003, 2005 (19th edition), Abraham Mozowsky (ed.), volume. 147, p. 1239