Talk:Penrose method

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Talk:Penrose method/CSV[edit]

Here is a CSV file that you can use to do your own Penrose analysis. It uses data from List of countries by population as of December 15, 2005. At the bottom are world totals. See Talk:Penrose method/CSV Captain Zyrain 17:57, 15 December 2005 (UTC)


Is it just me, or is everyone else seeing that this list shows fractional seats? The article doesn't propose any method for rounding these numbers. Wouter Lievens 13:34, 3 July 2006 (UTC)

For me the relevant question is how strong is the population increase before another seat is added. I've checked and come to the conclusion that consistent rounding up of seats results in the lowest average span of population change covered by an increase of seats if that makes it clear. 09:52, 1 November 2007 (UTC)

I have written a computer utility to calculate the seat distribution for number of districts (or Countries in this case) according to population, while ensuring that each district gets at least one representative. The utility uses the Huntington-Hill method used to distrubute seats in the United States House of Representatives. I've modified this also include the Penrose Method calculation. The result is an in integer list of representatives each Country would be entitled to under the Penrose method. I would like feedback from the creators/maintainers of this page before I update the table with this informaton.

--Romping Cloud (talk) 09:57, 10 July 2010 (UTC)

What do you want to update? Tomeasy T C 10:30, 11 July 2010 (UTC)

I was thinking of adding another column to the table, showing the seat distribution when using the Huntington-Hill method combined with the Penrose method. The numbers would be integers and therefore make more sense in the real world, at least I think so. This would deal with the fractional issue. --Romping Cloud (talk) 13:37, 11 July 2010 (UTC)

I do not see why we should add the voting powers allocated by a proportional method to this article. For comparison? This comparison can well be made by just looking at the percentage of population with the percentage of weighting power. Tomeasy T C 15:26, 11 July 2010 (UTC)

Information vs. filler[edit]

  1. Does this article really need the 200 row table detailing countries' populations and their Penrose numbers? I'm not an expert in Wikipedia mark-up, but is there some way to minimize (and to expand) the table on demand? Give it another, related page?
  2. An account of Penrose's reasoning would be much more illuminating than just the results. Does anyone have the mathematical background for this?

-- 08:51, 11 June 2007 (UTC)

The reasoning behind the square root method is:
1) Penrose defines the "voting power" of an individual as the probability that his vote can tip the balance, all other votes are random events with equal probability of 1/2 for yes/no.
2) When there are N votes you need to calculate the probability p for half of them voting yes, which can be done using the binomial distribution (probability mass function)
3) You need to get rid of the the nasty factorials using Stirling's approximation and get the surprisingly simple equation p=(2/(pi N))^1/2 which is the voting power (as defined by Penrose) for an idivisual and it is proportional to 1/sqrt(N).
4) To compensate for this you need to weight the representative's votes with the square root of N. see e.g.


I suggest to add a section on critics, i.e. claims that the Penrose method is not the fairest one. See e.g. . --Roentgenium111 (talk) 22:54, 5 June 2009 (UTC)

Done (or at least started). Edratzer (talk) 20:26, 5 January 2010 (UTC)
Thank you very much. However, the second criticism ("odd number of voters") seems dubious to me. With an even number of voters, a single vote can decide between (a) a tie (say, between two alternatives) and (b) a "decisive" result (in one or the other direction). In the case of a tie the usual solution would be drawing of lots, which means that the single vote is decisive with a 50% probability. I know the claim is referenced, but the reference does not seem reliable to me if they make such mistakes. Is its author really a studied mathematician/politician?
A criticism that might be added is that the Penrose method makes sense only for distributing voting weights between contries, not for distributing seats in a parliament. In the latter case, proportional representation is the fairest solution, since the individual MPs of a given countries will not generally vote en bloc. --Roentgenium111 (talk) 19:21, 14 April 2010 (UTC)
I agree on your first point, i.e., the possibility of creating a tie is also powerful.
The last thing you mentioned is of course true. However, it is not a criticism to the Penrose method, but merely a limitation. The Penrose method is developed and only appropriate for indirect elections, where the representatives cast their votes en bloc. Typical cases where this may make sense are the Council of the European Union, the election of the President of the USA, or the German Bundesrat. So not just when talking about countries.
Obviously (well more or less perhaps), the Penrose method does not apply to the distribution of seats in a parliament where each seat makes their own free decision. However, nobody claims this would be the case. So, I see no point in adding this to the criticism. Perhaps we should state more clearly as to the situation where the Penrose method applies. Tomeasy T C 19:40, 14 April 2010 (UTC)
The problem is that the article does mention the proposed use of the Penrose method for a UN parliamentary assembly, and the corresponding chapter speaks of "seats". So maybe we should remove this "non-example" from the article (even the reference given for it at INFUSA does not mention "Penrose method", only "seat distribution by square root"). --Roentgenium111 (talk) 20:20, 14 April 2010 (UTC)
I would guess that the UN parliamentary assembly was given as an example, because it was assumed that the countries must vote en bloc. If this condition was not met by the proposal, I could not imagine anyone would seriously propose the Penrose method. No, i am sure the proposal assumes en bloc voting and is therefore a good example.
I agree that we have to clarify that the Penrose method is limited to en bloc indirect elections. Tomeasy T C 20:36, 14 April 2010 (UTC)
The proposal ( does not mention en bloc voting anywhere as far as I can see. On the contrary, they say that representatives should be "elected" and not be national government members. And who except the national government could force the national representatives to vote en bloc? --Roentgenium111 (talk) 14:37, 19 April 2010 (UTC)
You are right. Last week, I also went through this proposal trying to find statement on how the representatives would vote in the assembly. There is nothing written about it, either way. I see two options: (a) It is assumed that countries' votes are cast en bloc, which I thought since otherwise application of the Penrose method does not make sense and the explanations given in the document as to the fair share of voting powers among the earth's population would be wrong; or (b) all representatives are individually free to make their choices in decision, which you assume for good reasons. In any case, we can say that this whole proposal lacks professionalism by either (a) excluding an important fact about the decision making; or (b) proposing the Penrose method where it is clearly not appropriate. Tomeasy T C 20:21, 20 April 2010 (UTC)
I agree completely. So I think we should remove the UN proposal from this article for being unprofessional (thus not encyclopedic) and possibly (case (b)) inappropriate. --Roentgenium111 (talk) 17:06, 23 April 2010 (UTC)
Yes, you are right the even/odd number of voters is a minor technical issue. The first criticism is far more important. I've adjusted the article to reflect this. Edratzer (talk) 09:48, 17 April 2010 (UTC)
Thanks, but I would prefer to remove it completely, by my above reasoning, since the argument is plain wrong IMO. Note that Penrose himself included even numbers of voters in his original article. --Roentgenium111 (talk) 17:23, 23 April 2010 (UTC)
We could keep taking about the UN assembly, but reference Penrose's original paper. Therein, he outlines the appropriateness of the square root function for an indirect en bloc vote (Section 3b, p. 55). On the following page, he suggest an application of this method to a "world assembly". Tomeasy T C 17:41, 23 April 2010 (UTC)
OK, I corrected the UN stuff, so that the table is in line with Penrose's idea. I also added the demanded reference. Let me know if that helped. Tomeasy T C 21:24, 27 April 2010 (UTC)
I think that it's important to include this technicality. Without this technicality you need to add extra rules for how heats are handled. Penrose specifically talks about the analysis of a odd size population in his derivation of square root power: "The power, thus defined, is the same as half the likelihood a situation in which an individual vote can be decisive - that is to say, a situation in which the remaining votes are equally divided upon the issue at stake. The general formula for the probability of equal division of n random votes, where n is an even number, approaches when n is large." To go beyond this and include rules for ties may be verging upon Wikipedia:No original research. Edratzer (talk) 21:33, 1 May 2010 (UTC)
Wikipedia is concerned about the common usage of terms, documented by reliable sources. When people refer to the Penrose method (e.g., when they suggest using it as a means of allocating voting powers in the Council of the European Union), they do not worry whether all member states' population figures are odd number.
Penrose used the example of an odd size to analytically proof his point. When an even n approaches millions, the numerical error of Penrose's method, which is exact only for odd n, becomes really negligible. So please abstain from labeling as OR, when the article applies the Penrose method to even populations sizes. BTW, no state in the world (well perhaps the Vatican) knows whether its population size is odd or even. So what are we practically talking about ...
However, I would really appreciate if you wanted to add something along the lines of your quotes: A formula and an explanation that it has to do with game theory and the probability of breaking a tie. In this context, it would be very appropriate to mention that the analytic tools are valid for odd sizes of the electorate and that Penrose published the theory this way. Tomeasy T C 22:28, 1 May 2010 (UTC)

I have absouletely NO idea why this gives fair representation.[edit]

"Penrose showed that in terms of statistical theory the square-root method gives to each voter in the world an equal influence on decision-making in a world assembly"

This sentence baffles me. It should be explained. It seems logical to me that if the system gave proportional representation, that then each voter has equal influence. —Preceding unsigned comment added by CoincidentalBystander (talkcontribs) 10:57, 8 August 2010 (UTC)

It is counter-intuitive, I agree. The most important thing to know first is that the fairness follows only if the delegates representing a group of citizens have to vote en bloc, i.e., no splitting of the votes. This is the case in only a few parliament-like bodies (e.g., German Bundesrat, Council of the EU, US presidential elections). Contrarily, in all parliaments that I know of, the delegates are free to vote according to their will. In this case, your intuition is correct and proportional representation is the fair approach.
Once you have understood this difference (i.e., square-root only for indirect voting where delegates vote en bloc), you should think about how a citizen influences a decision made by the body of delegations. This happens when the citizen's vote changes the delegation of their polity, and this change leads to a change of the decision made in the body of delegations. Now, every citizen should have the same likelihood of achieving this by their vote when the delegation are elected. To this end, the delegations' voting powers are proportional to the square root of the people they represent.
Let me know if you need more explanation. I will try to make the preconditions of the square-root method more transparent in the article. Tomeasy T C 21:07, 26 April 2011 (UTC)


Is there a (relatively) simple algorithm for the Penrose Method? If so, where is it and why isn't it here?Bayowolf (talk) 16:58, 14 June 2012 (UTC)

I think, "method" is somehow misleading. The Penrose law/rule states that allocating seats with respect to the squareroot achieves certain conditions.
So, if are the population counts of the countries, the "algorithm" is:
  1. Allocate the seats using a proportional allocation method in respect of the . --Arno Nymus (talk) 21:02, 14 June 2012 (UTC)

I agree with Arno about the algorithm. There is not much more to say than what he wrote down. Just as a side note: It should be understood that are the voting powers (or weights) of the countries - not the number of seats. The number of seats can be any number, as long as voting is en block. This single block vote has varying power depending on the country's population. If were the number of seats with independent votes, than the square root method would not be fair. Tomeasy T C 22:18, 14 June 2012 (UTC)