en>Yobot: WP:CHECKWIKI error #61 fixed + general fixes using AWB (8884)

2013-02-06T13:25:43Z

WP:CHECKWIKI error #61 fixed + general fixes using AWB (8884)

← Older revision		Revision as of 15:25, 6 February 2013
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	The ~~writer is known as Boris Puckett~~ and ~~he totally digs~~ that ~~title~~. ~~Dispatching is what he does~~ in his ~~working day occupation~~. ~~Going~~ to ~~fitness is~~ one of the ~~things she enjoys most~~. ~~New Jersey~~ is ~~his birth place~~. ~~Check~~ out ~~my web site here: http~~://~~Www.Youtube.com/watch?v=o4tu9ucXN9g~~<br><br>~~Here~~ is ~~my web page :~~: [~~http:~~//~~Www~~.~~Youtube~~.~~com~~/~~watch?v~~=~~o4tu9ucXN9g treatment for panic attacks in adults~~]		In the mathematical [[game theory\|theory of games]], the '''Penrose square root law''', originally formulated by [[Lionel Penrose]], concerns the distribution of the voting power in a voting body consisting of ''N'' members.<ref>{{Citation
			\| last = Penrose
			\| first = Lionel
			\| authorlink = Lionel Penrose
			\| title = The Elementary Statistics of Majority Voting
			\| journal = Journal of the Royal Statistical Society
			\| volume = 109
			\| issue = 1
			\| pages = 53–57
			\| year = 1946
			\| jstor =2981392
			\| doi = 10.2307/2981392
			\| publisher = Blackwell Publishing}}</ref><ref>{{Citation
			\| last = Felsenthal
			\| first = Dan S
			\| last2 = Machover
			\| first2 = Moshé
			\| title = The measurement of voting power theory and practice, problems and paradoxes
			\| year = 1998
			\| location = Cheltenham
			\| publisher = Edward Elgar
			\| url =
			}}</ref><ref name="kirsch">{{cite doi\|10.1007/978-3-642-35929-3_20}}</ref> It states that the ''a priori'' voting power of any voter, measured by the [[Banzhaf power index\|Penrose–Banzhaf index]] <math>\psi </math> scales like <math> 1/\sqrt{N} </math>.

			This result was used to design the [[Penrose method]] of for allocating the voting weights of representatives in a decision-making bodies proportional to the square root of the population represented.

			== Short derivation==
			To estimate the voting index of any player one needs to estimate the number of the possible winning coalitions in which his vote is decisive. Assume for simplicity that the number of voters is odd, ''N'' = 2''j'' + 1, and the body votes according to the standard majority rule. Following Penrose one concludes that a given voter will be able to effectively influence the outcome of the voting only if the votes split half and half: if ''j'' players say 'Yes' and the remaining ''j'' players vote 'No', the last vote is decisive.

			Assuming that all members of the body vote independently (the votes are uncorrelated)
			and that the probability of each vote 'Yes' is equal to ''p'' = 1/2 one can estimate likelihood of such an even using the [[Bernoulli trial]]. The probability to obtain ''j'' votes 'Yes' out of 2''j'' votes reads

			: <math>
			P_j = \left( \frac{1}{2}\right) ^{2j}\frac{\left( 2j\right)!}{\left( j! \right)^2}.
			</math>

			For large ''N'' we may use the [[Stirling's approximation]] for the factorial ''j'' <nowiki>!</nowiki> and obtain the probability <math>\psi </math> that the vote of a given voter is decisive

			: <math>
			\psi=P_j \sim 2^{-2j}\frac{(2j/e)^{2j}\sqrt{4\pi j}}{[(j/e)^j \sqrt{2\pi
			j}]^{2}}\ =\ \frac{1}{\sqrt{\pi j}} \sim \sqrt{\frac{2}{\pi}} \frac{1}{\sqrt{N}}.
			</math>

			The same approximation is obtained for an even number ''N''.

			A mathematical investigation of the influence of possible correlations between the voters for the Penrose square root law was presented by Kirsch.<ref name="kirsch"/>

			Penrose law is applied to construct Penrose--like systems of two--tier voting, including the [[Jagiellonian Compromise]] designed for the Council of the European Union.

			==References==
			{{reflist}}

			[[Category:Game theory]]

en>Katharineamy: added Category:Financial terminology; removed {{uncategorized}} using HotCat

2012-06-30T17:51:05Z

added Category:Financial terminology; removed {{uncategorized}} using HotCat

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The writer is known as Boris Puckett and he totally digs that title. Dispatching is what he does in his working day occupation. Going to fitness is one of the things she enjoys most. New Jersey is his birth place. Check out my web site here: http://Www.Youtube.com/watch?v=o4tu9ucXN9g<br><br>Here is my web page :: [http://Www.Youtube.com/watch?v=o4tu9ucXN9g treatment for panic attacks in adults]

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en>Yobot: WP:CHECKWIKI error #61 fixed + general fixes using AWB (8884)

en>Katharineamy: added Category:Financial terminology; removed {{uncategorized}} using HotCat